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</ul> </li> <li id="toc-Partial_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_functions"> <div class="vector-toc-text"> 1.2 Partial functions </div> </a> <ul id="toc-Partial_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multivariate_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multivariate_functions"> <div class="vector-toc-text"> 1.3 Multivariate functions </div> </a> <ul id="toc-Multivariate_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> 2 Notation </div> </a> <button aria-controls="toc-Notation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Notation subsection </button> <ul id="toc-Notation-sublist" class="vector-toc-list"> <li 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id="toc-Dot_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specialized_notations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specialized_notations"> <div class="vector-toc-text"> 2.5 Specialized notations </div> </a> <ul id="toc-Specialized_notations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functions_of_more_than_one_variable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functions_of_more_than_one_variable"> <div class="vector-toc-text"> 2.6 Functions of more than one variable </div> </a> <ul id="toc-Functions_of_more_than_one_variable-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_terms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_terms"> <div class="vector-toc-text"> 3 Other terms </div> </a> <ul id="toc-Other_terms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specifying_a_function" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Specifying_a_function"> <div class="vector-toc-text"> 4 Specifying a function </div> </a> <button aria-controls="toc-Specifying_a_function-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Specifying a function subsection </button> <ul id="toc-Specifying_a_function-sublist" class="vector-toc-list"> <li id="toc-By_listing_function_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#By_listing_function_values"> <div class="vector-toc-text"> 4.1 By listing function values </div> </a> <ul id="toc-By_listing_function_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-By_a_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#By_a_formula"> <div class="vector-toc-text"> 4.2 By a formula </div> </a> <ul id="toc-By_a_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse_and_implicit_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverse_and_implicit_functions"> <div class="vector-toc-text"> 4.3 Inverse and implicit functions </div> </a> <ul id="toc-Inverse_and_implicit_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_differential_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_differential_calculus"> <div class="vector-toc-text"> 4.4 Using differential calculus </div> </a> <ul id="toc-Using_differential_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-By_recurrence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#By_recurrence"> <div class="vector-toc-text"> 4.5 By recurrence </div> </a> <ul id="toc-By_recurrence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representing_a_function" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Representing_a_function"> <div class="vector-toc-text"> 5 Representing a function </div> </a> <button aria-controls="toc-Representing_a_function-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Representing a function subsection </button> <ul id="toc-Representing_a_function-sublist" class="vector-toc-list"> <li id="toc-Graphs_and_plots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graphs_and_plots"> <div class="vector-toc-text"> 5.1 Graphs and plots </div> </a> <ul id="toc-Graphs_and_plots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tables"> <div class="vector-toc-text"> 5.2 Tables </div> </a> <ul id="toc-Tables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bar_chart" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bar_chart"> <div class="vector-toc-text"> 5.3 Bar chart </div> </a> <ul id="toc-Bar_chart-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_properties"> <div class="vector-toc-text"> 6 General properties </div> </a> <button aria-controls="toc-General_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle General properties subsection </button> <ul id="toc-General_properties-sublist" class="vector-toc-list"> <li id="toc-Standard_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standard_functions"> <div class="vector-toc-text"> 6.1 Standard functions </div> </a> <ul id="toc-Standard_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Function_composition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Function_composition"> <div class="vector-toc-text"> 6.2 Function composition </div> </a> <ul id="toc-Function_composition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Image_and_preimage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Image_and_preimage"> <div class="vector-toc-text"> 6.3 Image and preimage </div> </a> <ul id="toc-Image_and_preimage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Injective,_surjective_and_bijective_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Injective,_surjective_and_bijective_functions"> <div class="vector-toc-text"> 6.4 Injective, surjective and bijective functions </div> </a> <ul id="toc-Injective,_surjective_and_bijective_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Restriction_and_extension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Restriction_and_extension"> <div class="vector-toc-text"> 6.5 Restriction and extension </div> </a> <ul id="toc-Restriction_and_extension-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_calculus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_calculus"> <div class="vector-toc-text"> 7 In calculus </div> </a> <button aria-controls="toc-In_calculus-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle In calculus subsection </button> <ul id="toc-In_calculus-sublist" class="vector-toc-list"> <li id="toc-Real_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_function"> <div class="vector-toc-text"> 7.1 Real function </div> </a> <ul id="toc-Real_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector-valued_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector-valued_function"> <div class="vector-toc-text"> 7.2 Vector-valued function </div> </a> <ul id="toc-Vector-valued_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Function_space" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Function_space"> <div class="vector-toc-text"> 8 Function space </div> </a> <ul id="toc-Function_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multi-valued_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Multi-valued_functions"> <div class="vector-toc-text"> 9 Multi-valued functions </div> </a> <ul id="toc-Multi-valued_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_the_foundations_of_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_the_foundations_of_mathematics"> <div class="vector-toc-text"> 10 In the foundations of mathematics </div> </a> <ul id="toc-In_the_foundations_of_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_computer_science" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_computer_science"> <div class="vector-toc-text"> 11 In computer science </div> </a> <ul id="toc-In_computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle See also subsection </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Subpages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subpages"> <div class="vector-toc-text"> 12.1 Subpages </div> </a> <ul id="toc-Subpages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 12.2 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_topics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_topics"> <div class="vector-toc-text"> 12.3 Related topics </div> </a> <ul id="toc-Related_topics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 14 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 15 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 16 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 17 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Function (mathematics)</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 119 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-119" aria-hidden="true" > 119 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Funksie" title="Funksie – Afrikaans" lang="af" hreflang="af" data-title="Funksie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik) – Alemannic" lang="gsw" hreflang="gsw" data-title="Funktion (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%88%B5%E1%88%A8%E1%8A%AB%E1%89%A2" title="አስረካቢ – Amharic" lang="am" hreflang="am" data-title="አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Funktio" title="Funktio – Inari Sami" lang="smn" hreflang="smn" data-title="Funktio" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target">Anarâškielâ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9" title="دالة – Arabic" lang="ar" hreflang="ar" data-title="دالة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Funci%C3%B3n_matematica" title="Función matematica – Aragonese" lang="an" hreflang="an" data-title="Función matematica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática – Asturian" lang="ast" hreflang="ast" data-title="Función matemática" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Funksiya_(riyaziyyat)" title="Funksiya (riyaziyyat) – Azerbaijani" lang="az" hreflang="az" data-title="Funksiya (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="অপেক্ষক (গণিত) – Bangla" lang="bn" hreflang="bn" data-title="অপেক্ষক (গণিত)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/H%C3%A2m-s%C3%B2%CD%98" title="Hâm-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – Bashkir" lang="ba" hreflang="ba" data-title="Функция (математика)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Функцыя (матэматыка) – Belarusian" lang="be" hreflang="be" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Функцыя (матэматыка) – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%AB%E0%A4%82%E0%A4%95%E0%A5%8D%E0%A4%B6%E0%A4%A8_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="फंक्शन (गणित) – Bhojpuri" lang="bh" hreflang="bh" data-title="फंक्शन (गणित)" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Функция – Bulgarian" lang="bg" hreflang="bg" data-title="Функция" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – Bosnian" lang="bs" hreflang="bs" data-title="Funkcija (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3" title="Funció – Catalan" lang="ca" hreflang="ca" data-title="Funció" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функци (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Функци (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Funkce_(matematika)" title="Funkce (matematika) – Czech" lang="cs" hreflang="cs" data-title="Funkce (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Murimo_(Masvomhu)" title="Murimo (Masvomhu) – Shona" lang="sn" hreflang="sn" data-title="Murimo (Masvomhu)" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffwythiant" title="Ffwythiant – Welsh" lang="cy" hreflang="cy" data-title="Ffwythiant" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Funktion_(matematik)" title="Funktion (matematik) – Danish" lang="da" hreflang="da" data-title="Funktion (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9" title="دالة – Moroccan Arabic" lang="ary" hreflang="ary" data-title="دالة" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik) – German" lang="de" hreflang="de" data-title="Funktion (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Funktsioon_(matemaatika)" title="Funktsioon (matemaatika) – Estonian" lang="et" hreflang="et" data-title="Funktsioon (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Συνάρτηση – Greek" lang="el" hreflang="el" data-title="Συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_(matem%C3%A1tica)" title="Función (matemática) – Spanish" lang="es" hreflang="es" data-title="Función (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Funkcio_(matematiko)" title="Funkcio (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Funkcio (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_(matematika)" title="Funtzio (matematika) – Basque" lang="eu" hreflang="eu" data-title="Funtzio (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9" title="تابع – Persian" lang="fa" hreflang="fa" data-title="تابع" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Function" title="Function – Fiji Hindi" lang="hif" hreflang="hif" data-title="Function" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Funksj%C3%B3n" title="Funksjón – Faroese" lang="fo" hreflang="fo" data-title="Funksjón" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_(math%C3%A9matiques)" title="Fonction (mathématiques) – French" lang="fr" hreflang="fr" data-title="Fonction (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Feidhm_(matamaitic)" title="Feidhm (matamaitic) – Irish" lang="ga" hreflang="ga" data-title="Feidhm (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n" title="Función – Galician" lang="gl" hreflang="gl" data-title="Función" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%87%BD%E6%95%B8" title="函數 – Gan" lang="gan" hreflang="gan" data-title="函數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Функция – Kalmyk" lang="xal" hreflang="xal" data-title="Функция" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A8%EC%88%98" title="함수 – Korean" lang="ko" hreflang="ko" data-title="함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Ֆունկցիա (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Ֆունկցիա (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="फलन – Hindi" lang="hi" hreflang="hi" data-title="फलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – Croatian" lang="hr" hreflang="hr" data-title="Funkcija (matematika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Funciono" title="Funciono – Ido" lang="io" hreflang="io" data-title="Funciono" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_(matematika)" title="Fungsi (matematika) – Indonesian" lang="id" hreflang="id" data-title="Fungsi (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Function_(mathematica)" title="Function (mathematica) – Interlingua" lang="ia" hreflang="ia" data-title="Function (mathematica)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fall_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Fall (stærðfræði) – Icelandic" lang="is" hreflang="is" data-title="Fall (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_(matematica)" title="Funzione (matematica) – Italian" lang="it" hreflang="it" data-title="Funzione (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה – Hebrew" lang="he" hreflang="he" data-title="פונקציה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/K%C9%A9lab%C9%A9m" title="Kɩlabɩm – Kabiye" lang="kbp" hreflang="kbp" data-title="Kɩlabɩm" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target">Kabɩyɛ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%90_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ფუნქცია (მათემატიკა) – Georgian" lang="ka" hreflang="ka" data-title="ფუნქცია (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – Kazakh" lang="kk" hreflang="kk" data-title="Функция (математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Fonksyon_(mat%C3%A9matik)" title="Fonksyon (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Fonksyon (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio" title="Functio – Latin" lang="la" hreflang="la" data-title="Functio" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Funkcija" title="Funkcija – Latvian" lang="lv" hreflang="lv" data-title="Funkcija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Funktioun_(Mathematik)" title="Funktioun (Mathematik) – Luxembourgish" lang="lb" hreflang="lb" data-title="Funktioun (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – Lithuanian" lang="lt" hreflang="lt" data-title="Funkcija (matematika)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/fancu" title="fancu – Lojban" lang="jbo" hreflang="jbo" data-title="fancu" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Fonzion_(matematega)" title="Fonzion (matematega) – Lombard" lang="lmo" hreflang="lmo" data-title="Fonzion (matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/F%C3%BCggv%C3%A9ny_(matematika)" title="Függvény (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Függvény (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функција (математика) – Macedonian" lang="mk" hreflang="mk" data-title="Функција (математика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AB%E0%B4%99%E0%B5%8D%E0%B4%B7%E0%B5%BB" title="ഫങ്ഷൻ – Malayalam" lang="ml" hreflang="ml" data-title="ഫങ്ഷൻ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Funzjonijiet_(matematika)" title="Funzjonijiet (matematika) – Maltese" lang="mt" hreflang="mt" data-title="Funzjonijiet (matematika)" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AB%E0%A4%B2_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="फल (गणित) – Marathi" lang="mr" hreflang="mr" data-title="फल (गणित)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi" title="Fungsi – Malay" lang="ms" hreflang="ms" data-title="Fungsi" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA)" title="Функц (математик) – Mongolian" lang="mn" hreflang="mn" data-title="Функц (математик)" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%96%E1%80%94%E1%80%BA%E1%80%9B%E1%80%BE%E1%80%84%E1%80%BA" title="ဖန်ရှင် – Burmese" lang="my" hreflang="my" data-title="ဖန်ရှင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Cakacaka_(fika)" title="Cakacaka (fika) – Fijian" lang="fj" hreflang="fj" data-title="Cakacaka (fika)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Functie_(wiskunde)" title="Functie (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Functie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%96%A2%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="関数 (数学) – Japanese" lang="ja" hreflang="ja" data-title="関数 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Funksion" title="Funksion – Northern Frisian" lang="frr" hreflang="frr" data-title="Funksion" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Funksjon_(matematikk)" title="Funksjon (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Funksjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_funksjon" title="Matematisk funksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Aplicacion_(matematicas)" title="Aplicacion (matematicas) – Occitan" lang="oc" hreflang="oc" data-title="Aplicacion (matematicas)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Warroomii_(faankishinii)" title="Warroomii (faankishinii) – Oromo" lang="om" hreflang="om" data-title="Warroomii (faankishinii)" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Funksiya_(matematika)" title="Funksiya (matematika) – Uzbek" lang="uz" hreflang="uz" data-title="Funksiya (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AB%E0%A9%B0%E0%A8%95%E0%A8%B8%E0%A8%BC%E0%A8%A8_(%E0%A8%B9%E0%A8%BF%E0%A8%B8%E0%A8%BE%E0%A8%AC)" title="ਫੰਕਸ਼ਨ (ਹਿਸਾਬ) – Punjabi" lang="pa" hreflang="pa" data-title="ਫੰਕਸ਼ਨ (ਹਿਸਾਬ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%81%D9%86%DA%A9%D8%B4%D9%86" title="فنکشن – Western Punjabi" lang="pnb" hreflang="pnb" data-title="فنکشن" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Fongshan_(matimatix)" title="Fongshan (matimatix) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Fongshan (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Fonsion" title="Fonsion – Piedmontese" lang="pms" hreflang="pms" data-title="Fonsion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Afbillen_(Mathematik)" title="Afbillen (Mathematik) – Low German" lang="nds" hreflang="nds" data-title="Afbillen (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja" title="Funkcja – Polish" lang="pl" hreflang="pl" data-title="Funkcja" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Função (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Função (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie" title="Funcție – Romanian" lang="ro" hreflang="ro" data-title="Funcție" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Kinraysuyu" title="Kinraysuyu – Quechua" lang="qu" hreflang="qu" data-title="Kinraysuyu" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – Russian" lang="ru" hreflang="ru" data-title="Функция (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F._%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%87%D1%8D%D1%80%D1%87%D0%B8%D1%82%D1%8D,_%D1%81%D1%83%D0%BE%D0%BB%D1%82%D0%B0%D0%BB%D0%B0%D1%80%D1%8B%D0%BD_%D1%82%D2%AF%D0%BC%D1%81%D1%8D%D1%8D%D0%BD%D1%8D" title="Функция. Функция чэрчитэ, суолталарын түмсээнэ – Yakut" lang="sah" hreflang="sah" data-title="Функция. Функция чэрчитэ, суолталарын түмсээнэ" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics) – Scots" lang="sco" hreflang="sco" data-title="Function (mathematics)" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni" title="Funksioni – Albanian" lang="sq" hreflang="sq" data-title="Funksioni" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Funzioni_(matim%C3%A0tica)" title="Funzioni (matimàtica) – Sicilian" lang="scn" hreflang="scn" data-title="Funzioni (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Function (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zobrazenie_(matematika)" title="Zobrazenie (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Zobrazenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – Slovenian" lang="sl" hreflang="sl" data-title="Funkcija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Funkcyjo" title="Funkcyjo – Silesian" lang="szl" hreflang="szl" data-title="Funkcyjo" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Shaqada_(xisaabta)" title="Shaqada (xisaabta) – Somali" lang="so" hreflang="so" data-title="Shaqada (xisaabta)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="فانکشن (ماتماتیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فانکشن (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функција (математика) – Serbian" lang="sr" hreflang="sr" data-title="Функција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Funkcija (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Fungsi_(matematika)" title="Fungsi (matematika) – Sundanese" lang="su" hreflang="su" data-title="Fungsi (matematika)" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Funktio" title="Funktio – Finnish" lang="fi" hreflang="fi" data-title="Funktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Funktion" title="Funktion – Swedish" lang="sv" hreflang="sv" data-title="Funktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Punsiyon_(matematika)" title="Punsiyon (matematika) – Tagalog" lang="tl" hreflang="tl" data-title="Punsiyon (matematika)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="சார்பு – Tamil" lang="ta" hreflang="ta" data-title="சார்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tas%C9%A3ent_(tusnakt)" title="Tasɣent (tusnakt) – Kabyle" lang="kab" hreflang="kab" data-title="Tasɣent (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – Tatar" lang="tt" hreflang="tt" data-title="Функция (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ฟังก์ชัน (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="ฟังก์ชัน (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fonksiyon" title="Fonksiyon – Turkish" lang="tr" hreflang="tr" data-title="Fonksiyon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-udm mw-list-item"><a href="https://udm.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – Udmurt" lang="udm" hreflang="udm" data-title="Функция (математика)" data-language-autonym="Удмурт" data-language-local-name="Udmurt" class="interlanguage-link-target">Удмурт</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функція (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Функція (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D9%81%D8%A7%D8%B9%D9%84_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="تفاعل (ریاضیات) – Urdu" lang="ur" hreflang="ur" data-title="تفاعل (ریاضیات)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D9%81%DB%87%D9%86%D9%83%D8%B3%D9%89%D9%8A%DB%95" title="فۇنكسىيە – Uyghur" lang="ug" hreflang="ug" data-title="فۇنكسىيە" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" class="interlanguage-link-target">ئۇيغۇرچە / Uyghurche</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Funkcii_(matematik)" title="Funkcii (matematik) – Veps" lang="vep" hreflang="vep" data-title="Funkcii (matematik)" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_s%E1%BB%91" title="Hàm số – Vietnamese" lang="vi" hreflang="vi" data-title="Hàm số" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%98%A0%E5%B0%84" title="映射 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="映射" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Funsiyon_(matematika)" title="Funsiyon (matematika) – Waray" lang="war" hreflang="war" data-title="Funsiyon (matematika)" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%87%BD%E6%95%B0" title="函数 – Wu" lang="wuu" hreflang="wuu" data-title="函数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%A2" title="פונקציע – Yiddish" lang="yi" hreflang="yi" data-title="פונקציע" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%87%BD%E6%95%B8" title="函數 – Cantonese" lang="yue" hreflang="yue" data-title="函數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Funkc%C4%97j%C4%97" title="Funkcėjė – Samogitian" lang="sgs" hreflang="sgs" data-title="Funkcėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%87%BD%E6%95%B0" title="函数 – Chinese" lang="zh" hreflang="zh" data-title="函数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B5%9C%E2%B4%B0%E2%B5%99%E2%B5%96%E2%B5%8F%E2%B5%9C_(%E2%B5%9C%E2%B5%93%E2%B5%99%E2%B5%8F%E2%B4%B0%E2%B4%BD%E2%B5%9C)" title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ) – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ)" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target">ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11348#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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function">domain</a> and <a href="/wiki/Codomain" title="Codomain">codomain</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Boolean-valued_function" title="Boolean-valued function"><var>X</var> → 𝔹</a></li> <li><a href="/wiki/Ordered_pair" title="Ordered pair"> 𝔹 → <var>X</var></a></li> <li><a href="/wiki/Boolean_function" title="Boolean function"> 𝔹<var>n</var> → <var>X</var></a></li> <li><a href="/wiki/Integer-valued_function" title="Integer-valued function"> <var>X</var> → ℤ</a></li> <li><a href="/wiki/Sequence" title="Sequence"> ℤ → <var>X</var></a></li> <li><a href="/wiki/Real-valued_function" title="Real-valued function"> <var>X</var> → ℝ</a></li> <li><a href="/wiki/Function_of_a_real_variable" title="Function of a real variable"> ℝ → <var>X</var></a></li> <li><a href="/wiki/Function_of_several_real_variables" title="Function of several real variables"> ℝ<var>n</var> → <var>X</var></a></li> <li><a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function"> <var>X</var> → ℂ</a></li> <li><a href="/wiki/Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable"> ℂ → <var>X</var></a></li> <li><a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables"> ℂ<var>n</var> → <var>X</var></a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> <a href="/wiki/List_of_types_of_functions" title="List of types of functions">Classes/properties</a> </th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Constant_function" title="Constant function">Constant</a></li> <li><a href="/wiki/Identity_function" title="Identity function">Identity</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear</a></li> <li><a href="/wiki/Polynomial" title="Polynomial">Polynomial</a></li> <li><a href="/wiki/Rational_function" title="Rational function">Rational</a></li> <li><a href="/wiki/Algebraic_function" title="Algebraic function">Algebraic</a></li> <li><a href="/wiki/Analytic_function" title="Analytic function">Analytic</a></li> <li><a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable</a></li> <li><a href="/wiki/Injective_function" title="Injective function">Injective</a></li> <li><a href="/wiki/Surjective_function" title="Surjective function">Surjective</a></li> <li><a href="/wiki/Bijection" title="Bijection">Bijective</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> Constructions</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li> <li><a href="/wiki/Function_composition" title="Function composition">Composition</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">λ</a></li> <li><a href="/wiki/Inverse_function" title="Inverse function">Inverse</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> Generalizations </th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a>)</li> <li><a href="/wiki/Set-valued_function" title="Set-valued function">Set-valued</a></li> <li><a href="/wiki/Multivalued_function" title="Multivalued function">Multivalued</a></li> <li><a href="/wiki/Partial_function" title="Partial function">Partial</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit</a></li> <li><a href="/wiki/Function_space" title="Function space">Space</a></li> <li><a href="/wiki/Higher-order_function" title="Higher-order function">Higher-order</a></li> <li><a href="/wiki/Morphism" title="Morphism">Morphism</a></li> <li><a href="/wiki/Functor" title="Functor">Functor</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-size: 117%; letter-spacing: 0.0125em; font-weight: 500; border-top: 1px solid black; padding: 5px 0 3px"> <a href="/wiki/List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></th></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functions" title="Template:Functions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functions" title="Template talk:Functions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functions" title="Special:EditPage/Template:Functions"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a function from a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> X to a set Y assigns to each element of X exactly one element of Y.<a href="#cite_note-halmos-1">[1]</a> The set X is called the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of the function<a href="#cite_note-2">[2]</a> and the set Y is called the <a href="/wiki/Codomain" title="Codomain">codomain</a> of the function.<a href="#cite_note-codomain-3">[3]</a> Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a <a href="/wiki/Planet" title="Planet">planet</a> is a function of time. <a href="/wiki/History_of_the_function_concept" title="History of the function concept">Historically</a>, the concept was elaborated with the <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a> at the end of the 17th century, and, until the 19th century, the functions that were considered were <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and this greatly increased the possible applications of the concept. A function is often denoted by a letter such as f, g or h. The value of a function f at an element x of its domain (that is, the element of the codomain that is associated with x) is denoted by f(x); for example, the value of f at x = 4 is denoted by f(4). Commonly, a specific function is defined by means of an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> depending on x, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{2}+1;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}+1;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cfd777bae6ce11f1c33372d52fa61a5bb2d0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.55ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}+1;}"> in this case, some computation, called <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>function evaluation, may be needed for deducing the value of the function at a particular value; for example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{2}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}+1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31eec96067f891efa5a072d1781924daf9ed0003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.55ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}+1,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(4)=4^{2}+1=17.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>17.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(4)=4^{2}+1=17.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38062c42739e5d2fdd7b53aa0c9a94617cdc881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.639ex; height:3.176ex;" alt="{\displaystyle f(4)=4^{2}+1=17.}"> Given its domain and its codomain, a function is uniquely represented by the set of all <a href="/wiki/Pair_(mathematics)" class="mw-redirect" title="Pair (mathematics)">pairs</a> (x, f (x)), called the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of the function</a>, a popular means of illustrating the function.<a href="#cite_note-4">[note 1]</a><a href="#cite_note-5">[4]</a> When the domain and the codomain are sets of real numbers, each such pair may be thought of as the <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of a point in the plane. Functions are widely used in <a href="/wiki/Science" title="Science">science</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a>, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.<a href="#cite_note-FOOTNOTESpivak200839-6">[5]</a> The concept of a function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See <a href="/wiki/History_of_the_function_concept" title="History of the function concept">History of the function concept</a> for details. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Function_machine2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Function_machine2.svg/220px-Function_machine2.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Function_machine2.svg/330px-Function_machine2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Function_machine2.svg/440px-Function_machine2.svg.png 2x" data-file-width="191" data-file-height="189" /></a><figcaption>Schematic depiction of a function described metaphorically as a "machine" or "<a href="/wiki/Black_box" title="Black box">black box</a>" that for each input yields a corresponding output</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Example_Function.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Example_Function.png/220px-Example_Function.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Example_Function.png/330px-Example_Function.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Example_Function.png/440px-Example_Function.png 2x" data-file-width="640" data-file-height="480" /></a><figcaption>The red curve is the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of a function</a>, because any <a href="/wiki/Vertical_line_test" title="Vertical line test">vertical line</a> has exactly one crossing point with the curve.</figcaption></figure> A function f from a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> X to a set Y is an assignment of one element of Y to each element of X. The set X is called the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of the function and the set Y is called the <a href="/wiki/Codomain" title="Codomain">codomain</a> of the function. If the element y in Y is assigned to x in X by the function f, one says that f maps x to y, and this is commonly written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea153ca7e4a53e290c347ab731724265f98c0b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle y=f(x).}"> In this notation, x is the <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> or <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a> of the function. A specific element x of X is a value of the variable, and the corresponding element of Y is the value of the function at x, or the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of x under the function. The image of a function, sometimes called its <a href="/wiki/Range_of_a_function" title="Range of a function">range</a>, is the set of the images of all elements in the domain.<a href="#cite_note-EOM_Function-7">[6]</a><a href="#cite_note-T&K_Calc_p.3-8">[7]</a><a href="#cite_note-Trench_RA_pp.30-32-9">[8]</a><a href="#cite_note-TBB_RA_pp.A4-A5-10">[9]</a> A function f, its domain X, and its codomain Y are often specified by the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b918aeefba8721a6732102a5848bd4238615ec55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y.}"> One may write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2452f2d32e5424f3db361de033fd49a73f9dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:2.176ex;" alt="{\displaystyle x\mapsto y}"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}">, where the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mapsto }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↦</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mapsto }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc09de045e7d82eef9fe078e7e7606576640c11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \mapsto }"> (read '<a href="/wiki/Maps_to" title="Maps to">maps to</a>') is used to specify where a particular element x in the domain is mapped to by f. This allows the definition of a function without naming. For example, the <a href="/wiki/Square_function" class="mw-redirect" title="Square function">square function</a> is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724da729a8df718591408d7bdfb38ce2b47035db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.975ex; height:2.676ex;" alt="{\displaystyle x\mapsto x^{2}.}"> The domain and codomain are not always explicitly given when a function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a10a3ad05781f5cf9c2d875a02227e21a8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> is a <a href="/wiki/Real_function" class="mw-redirect" title="Real function">real function</a>, the determination of the domain of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto 1/f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto 1/f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cde09bf037011b15f092d47999d05b417088dde1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.686ex; height:2.843ex;" alt="{\displaystyle x\mapsto 1/f(x)}"> requires knowing the <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of f. This is one of the reasons for which, in <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, "a function from X to Y " may refer to a function having a proper subset of X as a domain.<a href="#cite_note-11">[note 2]</a> For example, a "function from the reals to the reals" may refer to a <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued</a> function of a <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">real variable</a> whose domain is a proper subset of the <a href="/wiki/Real_number" title="Real number">real numbers</a>, typically a subset that contains a non-empty <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open interval</a>. Such a function is then called a <a href="/wiki/Partial_function" title="Partial function">partial function</a>. A function f on a set S means a function from the domain S, without specifying a codomain. However, some authors use it as shorthand for saying that the function is f : S → S. <div class="mw-heading mw-heading3"><h3 id="Formal_definition">Formal definition</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=2" title="Edit section: Formal definition">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Injection_keine_Injektion_2a.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Injection_keine_Injektion_2a.svg/220px-Injection_keine_Injektion_2a.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Injection_keine_Injektion_2a.svg/330px-Injection_keine_Injektion_2a.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Injection_keine_Injektion_2a.svg/440px-Injection_keine_Injektion_2a.svg.png 2x" data-file-width="200" data-file-height="200" /></a><figcaption>Diagram of a function</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Injection_keine_Injektion_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Injection_keine_Injektion_1.svg/220px-Injection_keine_Injektion_1.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Injection_keine_Injektion_1.svg/330px-Injection_keine_Injektion_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Injection_keine_Injektion_1.svg/440px-Injection_keine_Injektion_1.svg.png 2x" data-file-width="200" data-file-height="200" /></a><figcaption>Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair. Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair therein</figcaption></figure> The above definition of a function is essentially that of the founders of <a href="/wiki/Calculus" title="Calculus">calculus</a>, <a href="/wiki/Leibniz" class="mw-redirect" title="Leibniz">Leibniz</a>, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>. However, it cannot be <a href="/wiki/Formal_proof" title="Formal proof">formalized</a>, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of <a href="/wiki/Set_theory" title="Set theory">set theory</a>. This set-theoretic definition is based on the fact that a function establishes a relation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> between two sets X and Y is a <a href="/wiki/Subset" title="Subset">subset</a> of the set of all <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b14bc7ecf2f86320b4a930f4961919ae45a34d9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle y\in Y.}"> The set of all these pairs is called the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of X and Y and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times Y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f39ef78be28dc2b9015ff7f82e9a1ef719a9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.241ex; height:2.176ex;" alt="{\displaystyle X\times Y.}"> Thus, the above definition may be formalized as follows. A function with domain X and codomain Y is a binary relation R between X and Y that satisfies the two following conditions:<a href="#cite_note-12">[10]</a> <ul><li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in R.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in R.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421ae0833b9c96e25c691320faca968cfe9a3d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.58ex; height:2.843ex;" alt="{\displaystyle (x,y)\in R.}"></li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/564266a1c3efe90b1974df60a445161fdf58f14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.933ex; height:2.843ex;" alt="{\displaystyle (x,y)\in R}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,z)\in R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,z)\in R,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b26066404c28748e3db2e9e6c00255332fb3ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.513ex; height:2.843ex;" alt="{\displaystyle (x,z)\in R,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=z.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d03c8b5a2276379635012caa46307f02a63e08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.989ex; height:2.009ex;" alt="{\displaystyle y=z.}"></li></ul> This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (including <a href="/wiki/Set-builder_notation" title="Set-builder notation">set-builder notation</a>): A function is formed by three sets, the domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> the codomain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}"> and the graph <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> that satisfy the three following conditions. <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subseteq \{(x,y)\mid x\in X,y\in Y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\subseteq \{(x,y)\mid x\in X,y\in Y\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c364ed56d7bba17484ddaf407c2741cde3f775b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.407ex; height:2.843ex;" alt="{\displaystyle R\subseteq \{(x,y)\mid x\in X,y\in Y\}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in X,\exists y\in Y,\left(x,y\right)\in R\qquad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mi>R</mi> <mspace width="2em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in X,\exists y\in Y,\left(x,y\right)\in R\qquad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc8b441d7668696926e1ff3b9acbc4576bc6b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.151ex; height:2.843ex;" alt="{\displaystyle \forall x\in X,\exists y\in Y,\left(x,y\right)\in R\qquad }"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in R\land (x,z)\in R\implies y=z\qquad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>R</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>R</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mspace width="2em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in R\land (x,z)\in R\implies y=z\qquad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2294cdf37e849e059f75d1826cda8b5fef7231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.755ex; height:2.843ex;" alt="{\displaystyle (x,y)\in R\land (x,z)\in R\implies y=z\qquad }"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Partial_functions">Partial functions</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=3" title="Edit section: Partial functions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Partial_function" title="Partial function">Partial function</a></div> Partial functions are defined similarly to ordinary functions, with the "total" condition removed. That is, a partial function from X to Y is a binary relation R between X and Y such that, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}"> there is at most one y in Y such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in R.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in R.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421ae0833b9c96e25c691320faca968cfe9a3d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.58ex; height:2.843ex;" alt="{\displaystyle (x,y)\in R.}"> Using functional notation, this means that, given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}"> either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is in Y, or it is undefined. The set of the elements of X such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> is defined and belongs to Y is called the domain of definition of the function. A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X, one often says that the partial function is a total function. In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain. In <a href="/wiki/Calculus" title="Calculus">calculus</a>, a real-valued function of a real variable or <a href="/wiki/Real_function" class="mw-redirect" title="Real function">real function</a> is a partial function from the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> of the <a href="/wiki/Real_number" title="Real number">real numbers</a> to itself. Given a real function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:x\mapsto f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:x\mapsto f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6abc5d2f5d3b6f1ed1d94c74c2b9f255fc118e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.577ex; height:2.843ex;" alt="{\displaystyle f:x\mapsto f(x)}"> its <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto 1/f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto 1/f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cde09bf037011b15f092d47999d05b417088dde1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.686ex; height:2.843ex;" alt="{\displaystyle x\mapsto 1/f(x)}"> is also a real function. The determination of the domain of definition of a multiplicative inverse of a (partial) function amounts to compute the <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of the function, the values where the function is defined but not its multiplicative inverse. Similarly, a <a href="/wiki/Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable">function of a complex variable</a> is generally a partial function with a domain of definition included in the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> of the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. The difficulty of determining the domain of definition of a <a href="/wiki/Complex_function" class="mw-redirect" title="Complex function">complex function</a> is illustrated by the multiplicative inverse of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>: the determination of the domain of definition of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto 1/\zeta (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\mapsto 1/\zeta (z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328b3ae5767865f93350e2f503cce630ec5e968b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.02ex; height:2.843ex;" alt="{\displaystyle z\mapsto 1/\zeta (z)}"> is more or less equivalent to the proof or disproof of one of the major open problems in mathematics, the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>. In <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>, a <a href="/wiki/General_recursive_function" title="General recursive function">general recursive function</a> is a partial function from the integers to the integers whose values can be computed by an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see <a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a>). <div class="mw-heading mw-heading3"><h3 id="Multivariate_functions">Multivariate functions </h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=4" title="Edit section: Multivariate functions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Multivalued_function" title="Multivalued function">Multivalued function</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_operations_as_black_box.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/220px-Binary_operations_as_black_box.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/330px-Binary_operations_as_black_box.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Binary_operations_as_black_box.svg/440px-Binary_operations_as_black_box.svg.png 2x" data-file-width="142" data-file-height="142" /></a><figcaption>A binary operation is a typical example of a bivariate function which assigns to each pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> the result <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\circ y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∘</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\circ y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ba86902ee98c41deb1275ddb8693977f27e1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.68ex; height:2.009ex;" alt="{\displaystyle x\circ y}">.</figcaption></figure> A multivariate function, multivariable function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. Formally, a function of n variables is a function whose domain is a set of n-tuples.<a href="#cite_note-13">[note 3]</a> For example, multiplication of <a href="/wiki/Integer" title="Integer">integers</a> is a function of two variables, or bivariate function, whose domain is the set of all <a href="/wiki/Ordered_pairs" class="mw-redirect" title="Ordered pairs">ordered pairs</a> (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a>. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a <a href="/wiki/Surface_(mathematics)#Graph_of_a_bivariate_function" title="Surface (mathematics)">parametric surface</a>, as used in, e.g., <a href="/wiki/Bivariate_interpolation" class="mw-redirect" title="Bivariate interpolation">bivariate interpolation</a>. Commonly, an n-tuple is denoted enclosed between parentheses, such as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,2,\ldots ,n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,2,\ldots ,n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a956b235b4d9e7046aeabf286c020fa1020e0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.388ex; height:2.843ex;" alt="{\displaystyle (1,2,\ldots ,n).}"> When using <a href="/wiki/Functional_notation" class="mw-redirect" title="Functional notation">functional notation</a>, one usually omits the parentheses surrounding tuples, writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1},\ldots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d6311d8c66acc1a5755c4c7cb688d3b1fa0fcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.198ex; height:2.843ex;" alt="{\displaystyle f(x_{1},\ldots ,x_{n})}"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f((x_{1},\ldots ,x_{n})).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f((x_{1},\ldots ,x_{n})).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ea486934809be3b7f5d97c159b9a116ef18837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.654ex; height:2.843ex;" alt="{\displaystyle f((x_{1},\ldots ,x_{n})).}"> Given n sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0ca82f7c1f167358d7122823318450410ae5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.946ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n},}"> the set of all n-tuples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7935f7983d8a5ae59fea84efe65415235fa7c47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n})}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6041cb619899bcfa55a26ff377e7eb55ffbfdae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.913ex; height:2.509ex;" alt="{\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}}"> is called the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0ca82f7c1f167358d7122823318450410ae5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.946ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n},}"> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}\times \cdots \times X_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}\times \cdots \times X_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9332eafc39443cb3785081f0d3e889b64b0961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.172ex; height:2.509ex;" alt="{\displaystyle X_{1}\times \cdots \times X_{n}.}"> Therefore, a multivariate function is a function that has a Cartesian product or a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of a Cartesian product as a domain. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\to Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2bd504d53f52b40bbc45ebb2bae1ab5336723f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.033ex; height:2.509ex;" alt="{\displaystyle f:U\to Y,}"></dd></dl> where the domain U has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X_{1}\times \cdots \times X_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X_{1}\times \cdots \times X_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ced9a7c3754c3f7bdb7949396b20124a6ff96d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.053ex; height:2.509ex;" alt="{\displaystyle U\subseteq X_{1}\times \cdots \times X_{n}.}"></dd></dl> If all the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> are equal to the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> of the <a href="/wiki/Real_number" title="Real number">real numbers</a> or to the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> of the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, one talks respectively of a <a href="/wiki/Function_of_several_real_variables" title="Function of several real variables">function of several real variables</a> or of a <a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables">function of several complex variables</a>. <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=5" title="Edit section: Notation">edit</a>]</div> There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below. <div class="mw-heading mw-heading3"><h3 id="Functional_notation">Functional notation</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=6" title="Edit section: Functional notation">edit</a>]</div> The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter f. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),\quad \sin(3),\quad {\text{or}}\quad f(x^{2}+1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x),\quad \sin(3),\quad {\text{or}}\quad f(x^{2}+1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c30f49866bcc96fc4b8daf973844abe48f0eee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.476ex; height:3.176ex;" alt="{\displaystyle f(x),\quad \sin(3),\quad {\text{or}}\quad f(x^{2}+1).}"></dd></dl> The argument between the parentheses may be a <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a>, often x, that represents an arbitrary element of the domain of the function, a specific element of the domain (3 in the above example), or an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> that can be evaluated to an element of the domain (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3a8d23f9f8123651e496dcf8490990c65cf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}+1}"> in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sin(x^{2}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\sin(x^{2}+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3b1c9d9e56b3327baceebfd1e530cc5b132a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.568ex; height:3.176ex;" alt="{\displaystyle f(x)=\sin(x^{2}+1)}">". When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write sin x instead of sin(x). Functional notation was first used by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in 1734.<a href="#cite_note-14">[11]</a> Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a <a href="/wiki/Roman_type" title="Roman type">roman type</a> is customarily used instead, such as "sin" for the <a href="/wiki/Sine_function" class="mw-redirect" title="Sine function">sine function</a>, in contrast to italic font for single-letter symbols. The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> be a function". This is an <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a> that is useful for a simpler formulation. <div class="mw-heading mw-heading3"><h3 id="Arrow_notation">Arrow notation</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=7" title="Edit section: Arrow notation">edit</a>]</div> Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced "<a href="/wiki/Maps_to" title="Maps to">maps to</a>". For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/092c91a7a2ccb9481d3da9c5e59cd410c66dd87f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.276ex; height:2.343ex;" alt="{\displaystyle x\mapsto x+1}"> is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> is implied. The domain and codomain can also be explicitly stated, for example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {sqr} \colon \mathbb {Z} &\to \mathbb {Z} \\x&\mapsto x^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sqr</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {sqr} \colon \mathbb {Z} &\to \mathbb {Z} \\x&\mapsto x^{2}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c83da8e3fd28ad40fade38a4a75b3de3be02e792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.036ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {sqr} \colon \mathbb {Z} &\to \mathbb {Z} \\x&\mapsto x^{2}.\end{aligned}}}"></dd></dl> This defines a function sqr from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>;</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4649a27b503cf52487d3e9252faf20b362db2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32ex; height:2.843ex;" alt="{\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)}"> is a function in two variables, and we want to refer to a <a href="/wiki/Partial_application" title="Partial application">partially applied function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290b16963d52e4a7995aae01ee854b97a6ea10c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle X\to Y}"> produced by fixing the second argument to the value t0 without introducing a new function name. The map in question could be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto f(x,t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto f(x,t_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7063c8d624a28d6798322fb19c71a580adfcbd68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.289ex; height:2.843ex;" alt="{\displaystyle x\mapsto f(x,t_{0})}"> using the arrow notation. The expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto f(x,t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto f(x,t_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7063c8d624a28d6798322fb19c71a580adfcbd68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.289ex; height:2.843ex;" alt="{\displaystyle x\mapsto f(x,t_{0})}"> (read: "the map taking x to f of x comma t nought") represents this new function with just one argument, whereas the expression f(x0, t0) refers to the value of the function f at the point (x0, t0). <div class="mw-heading mw-heading3"><h3 id="Index_notation">Index notation</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=8" title="Edit section: Index notation">edit</a>]</div> Index notation may be used instead of functional notation. That is, instead of writing f (x), one writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a7e56cbda7f3fa285ea6a6500af7901cbf58d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.959ex; height:2.509ex;" alt="{\displaystyle f_{x}.}"> This is typically the case for functions whose domain is the set of the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. Such a function is called a <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a>, and, in this case the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2702450f0458a5e01a698e248af552a7fab2b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}"> is called the nth element of the sequence. The index notation can also be used for distinguishing some variables called <a href="/wiki/Parameter_(mathematics)" class="mw-redirect" title="Parameter (mathematics)">parameters</a> from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto f(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto f(x,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e82a8366e1505166fcef3257f1fec8f056140157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.235ex; height:2.843ex;" alt="{\displaystyle x\mapsto f(x,t)}"> (see above) would be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874c306411e808e8191e8aeb95e3440e1c68d6e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.965ex; height:2.509ex;" alt="{\displaystyle f_{t}}"> using index notation, if we define the collection of maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/874c306411e808e8191e8aeb95e3440e1c68d6e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.965ex; height:2.509ex;" alt="{\displaystyle f_{t}}"> by the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{t}(x)=f(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{t}(x)=f(x,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee02b1f83705d1df5a9414b665d3e0eead9e786c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.494ex; height:2.843ex;" alt="{\displaystyle f_{t}(x)=f(x,t)}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,t\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,t\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b32f7879b9ed63837691bf63021ba6d450daca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.024ex; height:2.509ex;" alt="{\displaystyle x,t\in X}">. <div class="mw-heading mw-heading3"><h3 id="Dot_notation">Dot notation</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=9" title="Edit section: Dot notation">edit</a>]</div> In the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto f(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto f(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3322446e350dec66a9be262d198979af3d1c0da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.008ex; height:2.843ex;" alt="{\displaystyle x\mapsto f(x),}"> the symbol x does not represent any value; it is simply a <a href="/wiki/Placeholder_name" title="Placeholder name">placeholder</a>, meaning that, if x is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, x may be replaced by any symbol, often an <a href="/wiki/Interpunct" title="Interpunct">interpunct</a> " ⋅ ". This may be useful for distinguishing the function f (⋅) from its value f (x) at x. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(\cdot )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(\cdot )^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8095cc4a24833a2d13ab6afe89cf29873c13b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.74ex; height:3.176ex;" alt="{\displaystyle a(\cdot )^{2}}"> may stand for the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto ax^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto ax^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab922de5eb858b22ba59ddfcab0ed41db3e259d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.557ex; height:2.676ex;" alt="{\displaystyle x\mapsto ax^{2}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{a}^{\,(\cdot )}f(u)\,du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{a}^{\,(\cdot )}f(u)\,du}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9e4e970c8a3275e4c0fcceb79a3a55815f73b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.608ex; height:3.676ex;" alt="{\textstyle \int _{a}^{\,(\cdot )}f(u)\,du}"> may stand for a function defined by an integral with variable upper bound: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\mapsto \int _{a}^{x}f(u)\,du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\mapsto \int _{a}^{x}f(u)\,du}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fca324c6d96cbfbeb58d21833ee4321420a712c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.368ex; height:3.176ex;" alt="{\textstyle x\mapsto \int _{a}^{x}f(u)\,du}">. <div class="mw-heading mw-heading3"><h3 id="Specialized_notations">Specialized notations</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=10" title="Edit section: Specialized notations">edit</a>]</div> There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, <a href="/wiki/Linear_form" title="Linear form">linear forms</a> and the <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> they act upon are denoted using a <a href="/wiki/Dual_pair" class="mw-redirect" title="Dual pair">dual pair</a> to show the underlying <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">duality</a>. This is similar to the use of <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> in quantum mechanics. In <a href="/wiki/Mathematical_logic" title="Mathematical logic">logic</a> and the <a href="/wiki/Theory_of_computation" title="Theory of computation">theory of computation</a>, the function notation of <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> is used to explicitly express the basic notions of function <a href="/wiki/Abstraction_(computer_science)" title="Abstraction (computer science)">abstraction</a> and <a href="/wiki/Function_application" title="Function application">application</a>. In <a href="/wiki/Category_theory" title="Category theory">category theory</a> and <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>, networks of functions are described in terms of how they and their compositions <a href="/wiki/Commutative_property" title="Commutative property">commute</a> with each other using <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative diagrams</a> that extend and generalize the arrow notation for functions described above. <div class="mw-heading mw-heading3"><h3 id="Functions_of_more_than_one_variable">Functions of more than one variable</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=11" title="Edit section: Functions of more than one variable">edit</a>]</div> In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function f can be defined as mapping any pair of real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> to the sum of their squares, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455605b597282c27d7cf2238821bc331479a7e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.439ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}}">. Such a function is commonly written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=x^{2}+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=x^{2}+y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62cd27ae17a5f4e156905fdcee054a41d15fa36f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.144ex; height:3.176ex;" alt="{\displaystyle f(x,y)=x^{2}+y^{2}}"> and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(w,x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(w,x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55642fd8a3e6f2d566c81564b0a3481d4bfaf2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.305ex; height:2.843ex;" alt="{\displaystyle f(w,x,y)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(w,x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(w,x,y,z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580ec13f912c2810fd8d118acc67daad3924b005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.427ex; height:2.843ex;" alt="{\displaystyle f(w,x,y,z)}">. <div class="mw-heading mw-heading2"><h2 id="Other_terms">Other terms</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=12" title="Edit section: Other terms">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map (mathematics)</a>.</div> <table class="wikitable floatright" style="width: 50%"> <tbody><tr> <th>Term </th> <th>Distinction from "function" </th></tr> <tr> <td rowspan="3"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map/Mapping</a> </td> <td>None; the terms are synonymous.<a href="#cite_note-15">[12]</a> </td></tr> <tr> <td>A map can have any set as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> numbers.<a href="#cite_note-Lang87p43-16">[13]</a> </td></tr> <tr> <td>Alternatively, a map is associated with a special structure (e.g. by explicitly specifying a structured codomain in its definition). For example, a <a href="/wiki/Linear_map" title="Linear map">linear map</a>.<a href="#cite_note-Apostol81p35-17">[14]</a> </td></tr> <tr> <td><a href="/wiki/Homomorphism" title="Homomorphism">Homomorphism</a> </td> <td>A function between two <a href="/wiki/Structure_(mathematics)" class="mw-redirect" title="Structure (mathematics)">structures</a> of the same type that preserves the operations of the structure (e.g. a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a>).<a href="#cite_note-18">[15]</a> </td></tr> <tr> <td><a href="/wiki/Morphism" title="Morphism">Morphism</a> </td> <td>A generalisation of homomorphisms to any <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>, even when the objects of the category are not sets (for example, a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> defines a category with only one object, which has the elements of the group as morphisms; see <a href="/wiki/Category_(mathematics)#Examples" title="Category (mathematics)">Category (mathematics) § Examples</a> for this example and other similar ones).<a href="#cite_note-19">[16]</a> </td></tr></tbody></table> A function may also be called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. <a href="/wiki/Maps_of_manifolds" title="Maps of manifolds">maps of manifolds</a>). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., <a href="/wiki/Linear_map" title="Linear map">linear map</a> or map from G to H instead of <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from G to H). Some authors<a href="#cite_note-Apostol81p35-17">[14]</a> reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as <a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a>,<a href="#cite_note-Lang87p43-16">[13]</a> use "function" only to refer to maps for which the <a href="/wiki/Codomain" title="Codomain">codomain</a> is a subset of the <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> numbers, and use the term mapping for more general functions. In the theory of <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>, a map denotes an <a href="/wiki/Discrete-time_dynamical_system" class="mw-redirect" title="Discrete-time dynamical system">evolution function</a> used to create <a href="/wiki/Dynamical_system#Maps" title="Dynamical system">discrete dynamical systems</a>. See also <a href="/wiki/Poincar%C3%A9_map" title="Poincaré map">Poincaré map</a>. Whichever definition of map is used, related terms like <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a>, <a href="/wiki/Codomain" title="Codomain">codomain</a>, <a href="/wiki/Injective_function" title="Injective function">injective</a>, <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> have the same meaning as for a function. <div class="mw-heading mw-heading2"><h2 id="Specifying_a_function">Specifying a function</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=13" title="Edit section: Specifying a function">edit</a>]</div> Given a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, by definition, to each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> of the domain of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, there is a unique element associated to it, the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">. There are several ways to specify or describe how <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is related to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}">, both explicitly and implicitly. Sometimes, a theorem or an <a href="/wiki/Axiom" title="Axiom">axiom</a> asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. <div class="mw-heading mw-heading3"><h3 id="By_listing_function_values">By listing function values</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=14" title="Edit section: By listing function values">edit</a>]</div> On a finite set a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\{1,2,3\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75709a44930c3f020087d434e9bb260a0c46a0c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.722ex; height:2.843ex;" alt="{\displaystyle A=\{1,2,3\}}">, then one can define a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b96eb73b42abe3f9b5fc1b9a94dece18f9abf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.251ex; height:2.509ex;" alt="{\displaystyle f:A\to \mathbb {R} }"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=2,f(2)=3,f(3)=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)=2,f(2)=3,f(3)=4.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48eb562511cd44c4a8d161d225c74e1a3cc626d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.248ex; height:2.843ex;" alt="{\displaystyle f(1)=2,f(2)=3,f(3)=4.}"> <div class="mw-heading mw-heading3"><h3 id="By_a_formula">By a formula</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=15" title="Edit section: By a formula">edit</a>]</div> Functions are often defined by an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> that describes a combination of <a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">arithmetic operations</a> and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain. For example, in the above example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> can be defined by the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13080ea3fb204650e60021bcd68f35b7cb82cd50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.979ex; height:2.843ex;" alt="{\displaystyle f(n)=n+1}">, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \{1,2,3\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fd153aab97fa5d1cd50604f5b61c15bda65a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.116ex; height:2.843ex;" alt="{\displaystyle n\in \{1,2,3\}}">. When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of auxiliary functions. Similarly, if <a href="/wiki/Square_root" title="Square root">square roots</a> occur in the definition of a function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}"> the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\sqrt {1+x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {1+x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79c34c2d3762de89fa5bfe544076a6f7d8ca57e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.227ex; height:3.509ex;" alt="{\displaystyle f(x)={\sqrt {1+x^{2}}}}"> defines a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a10a3ad05781f5cf9c2d875a02227e21a8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> whose domain is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}"> because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee93a0bc45fd86054d8ac256bb58866c0ffb6336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle 1+x^{2}}"> is always positive if x is a real number. On the other hand, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {1-x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b779387bbccf8dcb6c6a52a8e052ae04529956c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.227ex; height:3.509ex;" alt="{\displaystyle f(x)={\sqrt {1-x^{2}}}}"> defines a function from the reals to the reals whose domain is reduced to the interval [−1, 1]. (In old texts, such a domain was called the domain of definition of the function.) Functions can be classified by the nature of formulas that define them: <ul><li>A <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic function</a> is a function that may be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax^{2}+bx+c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba19534f962564659c4c764c520a9b1c2583290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.791ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{2}+bx+c,}"> where a, b, c are <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constants</a>.</li> <li>More generally, a <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> is a function that can be defined by a formula involving only additions, subtractions, multiplications, and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> to nonnegative integer powers. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{3}-3x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{3}-3x-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f7a70ffe2450e80f1d0a91cb1279ecf82ffe0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.235ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{3}-3x-1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(x-1)(x^{3}+1)+2x^{2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(x-1)(x^{3}+1)+2x^{2}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329ca932140f4c467b17cf1426e063090257e515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.244ex; height:3.176ex;" alt="{\displaystyle f(x)=(x-1)(x^{3}+1)+2x^{2}-1}"> are polynomial functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">.</li> <li>A <a href="/wiki/Rational_function" title="Rational function">rational function</a> is the same, with divisions also allowed, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {x-1}{x+1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {x-1}{x+1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1175bc3324fa6a9f39bdc801c5cfa9e59e0eace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.332ex; height:5.343ex;" alt="{\displaystyle f(x)={\frac {x-1}{x+1}},}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{x+1}}+{\frac {3}{x}}-{\frac {2}{x-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mi>x</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{x+1}}+{\frac {3}{x}}-{\frac {2}{x-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4091d575d27d2f4262b6643c8cd73a0ec82547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.347ex; height:5.343ex;" alt="{\displaystyle f(x)={\frac {1}{x+1}}+{\frac {3}{x}}-{\frac {2}{x-1}}.}"></li> <li>An <a href="/wiki/Algebraic_function" title="Algebraic function">algebraic function</a> is the same, with <a href="/wiki/Nth_root" title="Nth root">nth roots</a> and <a href="/wiki/Zero_of_a_function" title="Zero of a function">roots of polynomials</a> also allowed.</li> <li>An <a href="/wiki/Elementary_function" title="Elementary function">elementary function</a><a href="#cite_note-20">[note 4]</a> is the same, with <a href="/wiki/Logarithm" title="Logarithm">logarithms</a> and <a href="/wiki/Exponential_functions" class="mw-redirect" title="Exponential functions">exponential functions</a> allowed.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Inverse_and_implicit_functions">Inverse and implicit functions</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=16" title="Edit section: Inverse and implicit functions">edit</a>]</div> A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b6b077a3059ca728f62c163fec3d93b8429769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y,}"> with domain X and codomain Y, is <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a>, if for every y in Y, there is one and only one element x in X such that y = f(x). In this case, the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> of f is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e989122f1ec4c60658a0bb4b619b1473da820393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.958ex; height:3.009ex;" alt="{\displaystyle f^{-1}:Y\to X}"> that maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"> to the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"> such that y = f(x). For example, the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, that maps the real numbers onto the positive numbers. If a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> is not bijective, it may occur that one can select subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4359fae89639fe08a113bd9154bff6b52047c7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.854ex; height:2.343ex;" alt="{\displaystyle E\subseteq X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subseteq Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊆</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subseteq Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d26bfc0333372b856bcce14daf66a1086ca46270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.613ex; height:2.343ex;" alt="{\displaystyle F\subseteq Y}"> such that the <a href="/wiki/Restriction_of_a_function" class="mw-redirect" title="Restriction of a function">restriction</a> of f to E is a bijection from E to F, and has thus an inverse. The <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse trigonometric functions</a> are defined this way. For example, the <a href="/wiki/Cosine_function" class="mw-redirect" title="Cosine function">cosine function</a> induces, by restriction, a bijection from the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> [0, π] onto the interval [−1, 1], and its inverse function, called <a href="/wiki/Arccosine" class="mw-redirect" title="Arccosine">arccosine</a>, maps [−1, 1] onto [0, π]. The other inverse trigonometric functions are defined similarly. More generally, given a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> R between two sets X and Y, let E be a subset of X such that, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac74fb0c2e4233f6f7171ba8efbc87da28a8ff02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.593ex; height:2.509ex;" alt="{\displaystyle x\in E,}"> there is some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"> such that x R y. If one has a criterion allowing selecting such a y for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac74fb0c2e4233f6f7171ba8efbc87da28a8ff02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.593ex; height:2.509ex;" alt="{\displaystyle x\in E,}"> this defines a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:E\to Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:E\to Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b5356c234a6fda8ca6a341339b83609ff327f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.026ex; height:2.509ex;" alt="{\displaystyle f:E\to Y,}"> called an <a href="/wiki/Implicit_function" title="Implicit function">implicit function</a>, because it is implicitly defined by the relation R. For example, the equation of the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=1}"> defines a relation on real numbers. If −1 < x < 1 there are two possible values of y, one positive and one negative. For x = ± 1, these two values become both equal to 0. Otherwise, there is no possible value of y. This means that the equation defines two implicit functions with domain [−1, 1] and respective codomains [0, +∞) and (−∞, 0]. In this example, the equation can be solved in y, giving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\pm {\sqrt {1-x^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\pm {\sqrt {1-x^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2200b08e3d1e0047706d8545856c4637fd3bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.419ex; height:3.509ex;" alt="{\displaystyle y=\pm {\sqrt {1-x^{2}}},}"> but, in more complicated examples, this is impossible. For example, the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{5}+y+x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{5}+y+x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d82fbc2ad8b2a28d99505ba655143535160a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.642ex; height:3.009ex;" alt="{\displaystyle y^{5}+y+x=0}"> defines y as an implicit function of x, called the <a href="/wiki/Bring_radical" title="Bring radical">Bring radical</a>, which has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and <a href="/wiki/Nth_root" title="Nth root">nth roots</a>. The <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a> provides mild <a href="/wiki/Differentiability" class="mw-redirect" title="Differentiability">differentiability</a> conditions for existence and uniqueness of an implicit function in the neighborhood of a point. <div class="mw-heading mw-heading3"><h3 id="Using_differential_calculus">Using differential calculus</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=17" title="Edit section: Using differential calculus">edit</a>]</div> Many functions can be defined as the <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> of another function. This is the case of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>, which is the antiderivative of 1/x that is 0 for x = 1. Another common example is the <a href="/wiki/Error_function" title="Error function">error function</a>. More generally, many functions, including most <a href="/wiki/Special_function" class="mw-redirect" title="Special function">special functions</a>, can be defined as solutions of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. The simplest example is probably the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, which can be defined as the unique function that is equal to its derivative and takes the value 1 for x = 0. <a href="/wiki/Power_series" title="Power series">Power series</a> can be used to define functions on the domain in which they converge. For example, the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0933c92fd40ead20121e8de3ad48c04ff6fcfb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.256ex; height:3.843ex;" alt="{\textstyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}}">. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> in some interval, this power series allows immediately enlarging the domain to a subset of the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, the <a href="/wiki/Disc_of_convergence" class="mw-redirect" title="Disc of convergence">disc of convergence</a> of the series. Then <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> allows enlarging further the domain for including almost the whole <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. This process is the method that is generally used for defining the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>, the <a href="/wiki/Exponential_function" title="Exponential function">exponential</a> and the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> of a complex number. <div class="mw-heading mw-heading3"><h3 id="By_recurrence">By recurrence</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=18" title="Edit section: By recurrence">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Recurrence_relation" title="Recurrence relation">Recurrence relation</a></div> Functions whose domain are the nonnegative integers, known as <a href="/wiki/Sequence" title="Sequence">sequences</a>, are sometimes defined by <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relations</a>. The <a href="/wiki/Factorial" title="Factorial">factorial</a> function on the nonnegative integers (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\mapsto n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">↦</mo> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\mapsto n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43615cae2b2eb2b4402bcd0b2bd7642eb855e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.05ex; height:2.176ex;" alt="{\displaystyle n\mapsto n!}">) is a basic example, as it can be defined by the recurrence relation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=n(n-1)!\quad {\text{for}}\quad n>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=n(n-1)!\quad {\text{for}}\quad n>0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12bc6067e14b24bf50711c08808b7b2b3b7387d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.122ex; height:2.843ex;" alt="{\displaystyle n!=n(n-1)!\quad {\text{for}}\quad n>0,}"></dd></dl> and the initial condition <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0!=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>!</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0!=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b6aeeba784d2992f64addf82d9cbbf7962a722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.717ex; height:2.176ex;" alt="{\displaystyle 0!=1.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Representing_a_function">Representing a function</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=19" title="Edit section: Representing a function">edit</a>]</div> A <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by <a href="/wiki/Bar_chart" title="Bar chart">bar charts</a>. <div class="mw-heading mw-heading3"><h3 id="Graphs_and_plots">Graphs and plots</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=20" title="Edit section: Graphs and plots">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Motor_vehicle_deaths_in_the_US.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Motor_vehicle_deaths_in_the_US.svg/220px-Motor_vehicle_deaths_in_the_US.svg.png" decoding="async" width="220" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Motor_vehicle_deaths_in_the_US.svg/330px-Motor_vehicle_deaths_in_the_US.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Motor_vehicle_deaths_in_the_US.svg/440px-Motor_vehicle_deaths_in_the_US.svg.png 2x" data-file-width="692" data-file-height="469" /></a><figcaption>The function mapping each year to its US motor vehicle death count, shown as a <a href="/wiki/Line_chart" title="Line chart">line chart</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Motor_vehicle_deaths_in_the_US_histogram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Motor_vehicle_deaths_in_the_US_histogram.svg/220px-Motor_vehicle_deaths_in_the_US_histogram.svg.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Motor_vehicle_deaths_in_the_US_histogram.svg/330px-Motor_vehicle_deaths_in_the_US_histogram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Motor_vehicle_deaths_in_the_US_histogram.svg/440px-Motor_vehicle_deaths_in_the_US_histogram.svg.png 2x" data-file-width="700" data-file-height="400" /></a><figcaption>The same function, shown as a bar chart</figcaption></figure> Given a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b6b077a3059ca728f62c163fec3d93b8429769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y,}"> its graph is, formally, the set <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\{(x,f(x))\mid x\in X\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\{(x,f(x))\mid x\in X\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f60a64d69fdc91d64307936f7d9970d9f3ec6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.575ex; height:2.843ex;" alt="{\displaystyle G=\{(x,f(x))\mid x\in X\}.}"></dd></dl> In the frequent case where X and Y are subsets of the <a href="/wiki/Real_number" title="Real number">real numbers</a> (or may be identified with such subsets, e.g. <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a>), an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124f43d066e62e8deeb05b6a0c75922786573fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.996ex; height:2.843ex;" alt="{\displaystyle (x,y)\in G}"> may be identified with a point having coordinates x, y in a 2-dimensional coordinate system, e.g. the <a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a>. Parts of this may create a <a href="/wiki/Plot_(graphics)" title="Plot (graphics)">plot</a> that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the graph of the function. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the <a href="/wiki/Square_function" class="mw-redirect" title="Square function">square function</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ec217e885b4ff48e302d65301cf25c6fc7c447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.975ex; height:3.009ex;" alt="{\displaystyle x\mapsto x^{2},}"></dd></dl> consisting of all points with coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,x^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,x^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f624b81d622794760acd0cd347602595fa2504f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.557ex; height:3.176ex;" alt="{\displaystyle (x,x^{2})}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc20967d7838bbfb6e9df31756dc24de3573ae3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.495ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {R} ,}"> yields, when depicted in Cartesian coordinates, the well known <a href="/wiki/Parabola" title="Parabola">parabola</a>. If the same quadratic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ec217e885b4ff48e302d65301cf25c6fc7c447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.975ex; height:3.009ex;" alt="{\displaystyle x\mapsto x^{2},}"> with the same formal graph, consisting of pairs of numbers, is plotted instead in <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\theta )=(x,x^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta )=(x,x^{2}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d4a9e5ffd2cb2f9c274ea62034dc8e17d454e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.284ex; height:3.176ex;" alt="{\displaystyle (r,\theta )=(x,x^{2}),}"> the plot obtained is <a href="/wiki/Fermat%27s_spiral" title="Fermat's spiral">Fermat's spiral</a>. <div class="mw-heading mw-heading3"><h3 id="Tables">Tables</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=21" title="Edit section: Tables">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematical_table" title="Mathematical table">Mathematical table</a></div> A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\{1,\ldots ,5\}^{2}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>5</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\{1,\ldots ,5\}^{2}\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c6d1fa74d7c5be5592115baba6e7311dc3fcc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.39ex; height:3.176ex;" alt="{\displaystyle f:\{1,\ldots ,5\}^{2}\to \mathbb {R} }"> defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ee958d3d5aa49d9e3954ebf344a2bddd4f4485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.191ex; height:2.843ex;" alt="{\displaystyle f(x,y)=xy}"> can be represented by the familiar <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a> <table class="wikitable" style="text-align: center;"> <tbody><tr> <th style="background:var(--background-color-neutral,#eaecf0);color:inherit;background:linear-gradient(to top right,var(--background-color-neutral,#eaecf0) 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,var(--background-color-neutral,#eaecf0) 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">y</div><div style="margin-right:2em;text-align:left">x</div> </th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5 </th></tr> <tr> <th>1 </th> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5 </td></tr> <tr> <th>2 </th> <td>2</td> <td>4</td> <td>6</td> <td>8</td> <td>10 </td></tr> <tr> <th>3 </th> <td>3</td> <td>6</td> <td>9</td> <td>12</td> <td>15 </td></tr> <tr> <th>4 </th> <td>4</td> <td>8</td> <td>12</td> <td>16</td> <td>20 </td></tr> <tr> <th>5 </th> <td>5</td> <td>10</td> <td>15</td> <td>20</td> <td>25 </td></tr></tbody></table> On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places: <table class="wikitable" style="text-align: center;"> <tbody><tr> <th>x</th> <th>sin x </th></tr> <tr> <td>1.289</td> <td>0.960557 </td></tr> <tr> <td>1.290</td> <td>0.960835 </td></tr> <tr> <td>1.291</td> <td>0.961112 </td></tr> <tr> <td>1.292</td> <td>0.961387 </td></tr> <tr> <td>1.293</td> <td>0.961662 </td></tr></tbody></table> Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions. <div class="mw-heading mw-heading3"><h3 id="Bar_chart">Bar chart</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=22" title="Edit section: Bar chart">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></div> A bar chart can represent a function whose domain is a finite set, the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, or the <a href="/wiki/Integer" title="Integer">integers</a>. In this case, an element x of the domain is represented by an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> of the x-axis, and the corresponding value of the function, f(x), is represented by a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis). <div class="mw-heading mw-heading2"><h2 id="General_properties">General properties</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=23" title="Edit section: General properties">edit</a>]</div> This section describes general properties of functions, that are independent of specific properties of the domain and the codomain. <div class="mw-heading mw-heading3"><h3 id="Standard_functions">Standard functions</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=24" title="Edit section: Standard functions">edit</a>]</div> There are a number of standard functions that occur frequently: <ul><li>For every set X, there is a unique function, called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">empty function, or empty map, from the <a href="/wiki/Empty_set" title="Empty set">empty set</a> to X. The graph of an empty function is the empty set.<a href="#cite_note-21">[note 5]</a> The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. Under the usual set-theoretic definition of a function as an <a href="/wiki/Tuple" title="Tuple">ordered triplet</a> (or equivalent ones), there is exactly one empty function for each set, thus the empty function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing \to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing \to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cbc0203001f6a0e2648aa18ed1500c6a7637ab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.402ex; height:2.176ex;" alt="{\displaystyle \varnothing \to X}"> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing \to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing \to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9dbe760456d1e87721560f1fa4a96e63db2407b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.196ex; height:2.176ex;" alt="{\displaystyle \varnothing \to Y}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\neq Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>≠</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\neq Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be749f23597a5705f52b2c13f89cde4389ff7f0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.852ex; height:2.676ex;" alt="{\displaystyle X\neq Y}">, although their graphs are both the <a href="/wiki/Empty_set" title="Empty set">empty set</a>.</li> <li>For every set X and every <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> {s}, there is a unique function from X to {s}, which maps every element of X to s. This is a surjection (see below) unless X is the empty set.</li> <li>Given a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b6b077a3059ca728f62c163fec3d93b8429769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y,}"> the canonical surjection of f onto its image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)=\{f(x)\mid x\in X\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)=\{f(x)\mid x\in X\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3316d91530b49bc3304209486939bad604be85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.996ex; height:2.843ex;" alt="{\displaystyle f(X)=\{f(x)\mid x\in X\}}"> is the function from X to f(X) that maps x to f(x).</li> <li>For every <a href="/wiki/Subset" title="Subset">subset</a> A of a set X, the <a href="/wiki/Inclusion_map" title="Inclusion map">inclusion map</a> of A into X is the injective (see below) function that maps every element of A to itself.</li> <li>The <a href="/wiki/Identity_function" title="Identity function">identity function</a> on a set X, often denoted by idX, is the inclusion of X into itself.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Function_composition">Function composition</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=25" title="Edit section: Function composition">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Function_composition" title="Function composition">Function composition</a></div> Given two functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:Y\to Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e7e6d2116baaa7aa88d1adbf796ade5997a237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.121ex; height:2.509ex;" alt="{\displaystyle g:Y\to Z}"> such that the domain of g is the codomain of f, their composition is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f:X\rightarrow Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f:X\rightarrow Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68801eb2856ac23fc685a20fc4a4761873d9d9eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.801ex; height:2.509ex;" alt="{\displaystyle g\circ f:X\rightarrow Z}"> defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g\circ f)(x)=g(f(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\circ f)(x)=g(f(x)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50410f65ec5a8654714720d54c21fcaa3d2d2916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.626ex; height:2.843ex;" alt="{\displaystyle (g\circ f)(x)=g(f(x)).}"></dd></dl> That is, the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"> is obtained by first applying f to x to obtain y = f(x) and then applying g to the result y to obtain g(y) = g(f(x)). In this notation, the function that is applied first is always written on the right. The composition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"> is an <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}"> satisfy these conditions, the composition is not necessarily <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>, that is, the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}"> need not be equal, but may deliver different values for the same argument. For example, let f(x) = x2 and g(x) = x + 1, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(f(x))=x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(f(x))=x^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9050d0a90f81df4de49249892dcbf3bffca33fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.828ex; height:3.176ex;" alt="{\displaystyle g(f(x))=x^{2}+1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(x))=(x+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(x))=(x+1)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea84d7ee047fd301242a46fbdf5c994605eb660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.637ex; height:3.176ex;" alt="{\displaystyle f(g(x))=(x+1)^{2}}"> agree just for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71320d636c7a43546bf4e94edb94649c5b2e82b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle x=0.}"> The function composition is <a href="/wiki/Associative_property" title="Associative property">associative</a> in the sense that, if one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h\circ g)\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h\circ g)\circ f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c227490c73a1fd0ed1ef9a3f9d989398e2aca797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.932ex; height:2.843ex;" alt="{\displaystyle (h\circ g)\circ f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\circ (g\circ f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\circ (g\circ f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e05ca5e3aeb39bf0b062d3f4ebcfc2e339c16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.932ex; height:2.843ex;" alt="{\displaystyle h\circ (g\circ f)}"> is defined, then the other is also defined, and they are equal, that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h\circ g)\circ f=h\circ (g\circ f).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <mi>h</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h\circ g)\circ f=h\circ (g\circ f).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463113c07156c96f462e121f73e238e185ee0134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.61ex; height:2.843ex;" alt="{\displaystyle (h\circ g)\circ f=h\circ (g\circ f).}"> Therefore, it is usual to just write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\circ g\circ f.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\circ g\circ f.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f903279fd25d031541feb82152f1592e2086598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.77ex; height:2.509ex;" alt="{\displaystyle h\circ g\circ f.}"> The <a href="/wiki/Identity_function" title="Identity function">identity functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/623b238853010f9319f926446bc6cb2253e53e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.572ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{X}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {id} _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {id} _{Y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f280c0425786426a2cac3d1df83e73a5cdb4d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.426ex; height:2.509ex;" alt="{\displaystyle \operatorname {id} _{Y}}"> are respectively a <a href="/wiki/Right_identity" class="mw-redirect" title="Right identity">right identity</a> and a <a href="/wiki/Left_identity" class="mw-redirect" title="Left identity">left identity</a> for functions from X to Y. That is, if f is a function with domain X, and codomain Y, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae14fa7c63c808ffaf93f6df9b16baf3f87f6cff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.421ex; height:2.509ex;" alt="{\displaystyle f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.}"> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 285px"> <div class="thumb" style="width: 280px; height: 330px;"><a href="/wiki/File:Function_machine5.svg" class="mw-file-description" title="A composite function g(f(x)) can be visualized as the combination of two "machines"."><img alt="A composite function g(f(x)) can be visualized as the combination of two "machines"." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Function_machine5.svg/201px-Function_machine5.svg.png" decoding="async" width="201" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Function_machine5.svg/302px-Function_machine5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Function_machine5.svg/402px-Function_machine5.svg.png 2x" data-file-width="257" data-file-height="383" /></a></div> <div class="gallerytext">A composite function g(f(x)) can be visualized as the combination of two "machines".</div> </li> <li class="gallerybox" style="width: 285px"> <div class="thumb" style="width: 280px; height: 330px;"><a href="/wiki/File:Example_for_a_composition_of_two_functions.svg" class="mw-file-description" title="A simple example of a function composition"><img alt="A simple example of a function composition" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Example_for_a_composition_of_two_functions.svg/249px-Example_for_a_composition_of_two_functions.svg.png" decoding="async" width="249" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Example_for_a_composition_of_two_functions.svg/374px-Example_for_a_composition_of_two_functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Example_for_a_composition_of_two_functions.svg/499px-Example_for_a_composition_of_two_functions.svg.png 2x" data-file-width="450" data-file-height="541" /></a></div> <div class="gallerytext">A simple example of a function composition</div> </li> <li class="gallerybox" style="width: 285px"> <div class="thumb" style="width: 280px; height: 330px;"><a href="/wiki/File:Compfun.svg" class="mw-file-description" title="Another composition. In this example, (g ∘ f )(c) = #."><img alt="Another composition. In this example, (g ∘ f )(c) = #." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Compfun.svg/250px-Compfun.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Compfun.svg/375px-Compfun.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Compfun.svg/500px-Compfun.svg.png 2x" data-file-width="600" data-file-height="300" /></a></div> <div class="gallerytext">Another composition. In this example, (g ∘ f )(c) = #.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Image_and_preimage">Image and preimage</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=26" title="Edit section: Image and preimage">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">Image (mathematics)</a></div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b918aeefba8721a6732102a5848bd4238615ec55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.23ex; height:2.509ex;" alt="{\displaystyle f:X\to Y.}"> The image under f of an element x of the domain X is f(x).<a href="#cite_note-EOM_Function-7">[6]</a> If A is any subset of X, then the image of A under f, denoted f(A), is the subset of the codomain Y consisting of all images of elements of A,<a href="#cite_note-EOM_Function-7">[6]</a> that is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A)=\{f(x)\mid x\in A\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(A)=\{f(x)\mid x\in A\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95793cb619ae7bd33c59c5d5de5fd02ee1b092d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.169ex; height:2.843ex;" alt="{\displaystyle f(A)=\{f(x)\mid x\in A\}.}"></dd></dl> The image of f is the image of the whole domain, that is, f(X).<a href="#cite_note-PCM_p.11-22">[17]</a> It is also called the <a href="/wiki/Range_of_a_function" title="Range of a function">range</a> of f,<a href="#cite_note-EOM_Function-7">[6]</a><a href="#cite_note-T&K_Calc_p.3-8">[7]</a><a href="#cite_note-Trench_RA_pp.30-32-9">[8]</a><a href="#cite_note-TBB_RA_pp.A4-A5-10">[9]</a> although the term range may also refer to the codomain.<a href="#cite_note-TBB_RA_pp.A4-A5-10">[9]</a><a href="#cite_note-PCM_p.11-22">[17]</a><a href="#cite_note-standard-23">[18]</a> On the other hand, the <a href="/wiki/Inverse_image" class="mw-redirect" title="Inverse image">inverse image</a> or <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimage</a> under f of an element y of the codomain Y is the set of all elements of the domain X whose images under f equal y.<a href="#cite_note-EOM_Function-7">[6]</a> In symbols, the preimage of y is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b357745fa4a2178733a502b4432072be8222fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}"> and is given by the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)=\{x\in X\mid f(x)=y\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)=\{x\in X\mid f(x)=y\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7e4068e7e4675fb278d2098f3eeb0ec67c2881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.447ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)=\{x\in X\mid f(x)=y\}.}"></dd></dl> Likewise, the preimage of a subset B of the codomain Y is the set of the preimages of the elements of B, that is, it is the subset of the domain X consisting of all elements of X whose images belong to B.<a href="#cite_note-EOM_Function-7">[6]</a> It is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/572ddad8cd0a0758fb98e1c94c432dc2f7a06636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.227ex; height:3.176ex;" alt="{\displaystyle f^{-1}(B)}"> and is given by the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(B)=\{x\in X\mid f(x)\in B\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(B)=\{x\in X\mid f(x)\in B\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/485f68954316eb2c28391474d3df7c40fd589413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.407ex; height:3.176ex;" alt="{\displaystyle f^{-1}(B)=\{x\in X\mid f(x)\in B\}.}"></dd></dl> For example, the preimage of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{4,9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{4,9\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b235effd45a537b68a200facce95137756ab04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{4,9\}}"> under the <a href="/wiki/Square_function" class="mw-redirect" title="Square function">square function</a> is the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-3,-2,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{-3,-2,2,3\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132441f34cc733c4645d0fa80d8544fc90262d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.693ex; height:2.843ex;" alt="{\displaystyle \{-3,-2,2,3\}}">. By definition of a function, the image of an element x of the domain is always a single element of the codomain. However, the preimage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b357745fa4a2178733a502b4432072be8222fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}"> of an element y of the codomain may be <a href="/wiki/Empty_set" title="Empty set">empty</a> or contain any number of elements. For example, if f is the function from the integers to themselves that maps every integer to 0, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(0)=\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(0)=\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32720d89ba91ec2b5d5b706db20ee12420f187da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.274ex; height:3.176ex;" alt="{\displaystyle f^{-1}(0)=\mathbb {Z} }">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> is a function, A and B are subsets of X, and C and D are subsets of Y, then one has the following properties: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq B\Longrightarrow f(A)\subseteq f(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>B</mi> <mo stretchy="false">⟹</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq B\Longrightarrow f(A)\subseteq f(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74aab63eb9c503ca1d6168cba3d4a956f1aaaf45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.483ex; height:2.843ex;" alt="{\displaystyle A\subseteq B\Longrightarrow f(A)\subseteq f(B)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\subseteq D\Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>⊆</mo> <mi>D</mi> <mo stretchy="false">⟹</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\subseteq D\Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e57c217e011000e3e367cf58c8f8911cec1fb7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.599ex; height:3.176ex;" alt="{\displaystyle C\subseteq D\Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq f^{-1}(f(A))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq f^{-1}(f(A))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7745c86e833c00204c441b3420ecdf3a4e86bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.135ex; height:3.176ex;" alt="{\displaystyle A\subseteq f^{-1}(f(A))}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\supseteq f(f^{-1}(C))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>⊇</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\supseteq f(f^{-1}(C))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f0c665d4fd411c57fc15878a0407b5e74ec8c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.182ex; height:3.176ex;" alt="{\displaystyle C\supseteq f(f^{-1}(C))}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(f^{-1}(f(A)))=f(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(f^{-1}(f(A)))=f(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57dc6b4af65497aea87a51de455949fb31fdf6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.311ex; height:3.176ex;" alt="{\displaystyle f(f^{-1}(f(A)))=f(A)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(f(f^{-1}(C)))=f^{-1}(C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(f(f^{-1}(C)))=f^{-1}(C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d05d156017b8ab2055e098ee863bc393efd2373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.107ex; height:3.176ex;" alt="{\displaystyle f^{-1}(f(f^{-1}(C)))=f^{-1}(C)}"></li></ul> The preimage by f of an element y of the codomain is sometimes called, in some contexts, the <a href="/wiki/Fiber_(mathematics)" title="Fiber (mathematics)">fiber</a> of y under f. If a function f has an inverse (see below), this inverse is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d2b575826d75cfdbc1bea2f34ccfa71f1c59b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.3ex; height:3.009ex;" alt="{\displaystyle f^{-1}.}"> In this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc65ae260d9fcb0f29161fe1d5a8301c7616721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.229ex; height:3.176ex;" alt="{\displaystyle f^{-1}(C)}"> may denote either the image by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"> or the preimage by f of C. This is not a problem, as these sets are equal. The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2711c647e5397c7016ee21bbcea53565480bd5e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.831ex; height:2.843ex;" alt="{\displaystyle f(A)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc65ae260d9fcb0f29161fe1d5a8301c7616721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.229ex; height:3.176ex;" alt="{\displaystyle f^{-1}(C)}"> may be ambiguous in the case of sets that contain some subsets as elements, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,\{x\}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x,\{x\}\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/532d1022eef8f5f427703de9c6dd5b8d38d86316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.99ex; height:2.843ex;" alt="{\displaystyle \{x,\{x\}\}.}"> In this case, some care may be needed, for example, by using square brackets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[A],f^{-1}[C]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">]</mo> <mo>,</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>C</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[A],f^{-1}[C]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d728b72b3681c1a33529ac867bc49952dc812a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.063ex; height:3.176ex;" alt="{\displaystyle f[A],f^{-1}[C]}"> for images and preimages of subsets and ordinary parentheses for images and preimages of elements. <div class="mw-heading mw-heading3"><h3 id="Injective,_surjective_and_bijective_functions">Injective, surjective and bijective functions</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=27" title="Edit section: Injective, surjective and bijective functions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bijection,_injection_and_surjection" title="Bijection, injection and surjection">Bijection, injection and surjection</a></div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> be a function. The function f is <a href="/wiki/Injective_function" title="Injective function">injective</a> (or one-to-one, or is an injection) if f(a) ≠ f(b) for every two different elements a and b of X.<a href="#cite_note-PCM_p.11-22">[17]</a><a href="#cite_note-EOM_Injection-24">[19]</a> Equivalently, f is injective if and only if, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e1353f0febe9bcf693d849ec82ce8d94e5f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle y\in Y,}"> the preimage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b357745fa4a2178733a502b4432072be8222fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}"> contains at most one element. An empty function is always injective. If X is not the empty set, then f is injective if and only if there exists a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10a9844999dbfd6d1ba60a6d5d37779df277a74f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.421ex; height:2.509ex;" alt="{\displaystyle g:Y\to X}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f=\operatorname {id} _{X},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f=\operatorname {id} _{X},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5645c27403915e3d21d6eca5fad99cdcd5561b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.906ex; height:2.509ex;" alt="{\displaystyle g\circ f=\operatorname {id} _{X},}"> that is, if f has a <a href="/wiki/Left_inverse_function" class="mw-redirect" title="Left inverse function">left inverse</a>.<a href="#cite_note-EOM_Injection-24">[19]</a> Proof: If f is injective, for defining g, one chooses an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> in X (which exists as X is supposed to be nonempty),<a href="#cite_note-25">[note 6]</a> and one defines g by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(y)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(y)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b60bb15cd66fb9d9f721d9892b67c01cddd153be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.509ex; height:2.843ex;" alt="{\displaystyle g(y)=x}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(y)=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(y)=x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c58233f69bc7034d7b59fc99d4ada7ddc50243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.563ex; height:2.843ex;" alt="{\displaystyle g(y)=x_{0}}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\not \in f(X).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∉</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\not \in f(X).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d355db1cec99772b7d696063b1ef77d712d48f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.711ex; height:2.843ex;" alt="{\displaystyle y\not \in f(X).}"> Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f=\operatorname {id} _{X},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f=\operatorname {id} _{X},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5645c27403915e3d21d6eca5fad99cdcd5561b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.906ex; height:2.509ex;" alt="{\displaystyle g\circ f=\operatorname {id} _{X},}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9867a6ecb3cc19e19e0af39fb46523e69e616c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle y=f(x),}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=g(y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=g(y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2f479eab87e0bc784a0ed4246fa429c6164392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.156ex; height:2.843ex;" alt="{\displaystyle x=g(y),}"> and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)=\{x\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)=\{x\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6684307f77a0709f6a253a27a719b9d8bf8a5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.018ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)=\{x\}.}"> The function f is <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> (or onto, or is a surjection) if its range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b884e2d65b3356219702968b6751485fb8f38570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}"> equals its codomain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}">, that is, if, for each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> of the codomain, there exists some element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> of the domain such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5080a8b0a963407ea74ffa50702563771518d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle f(x)=y}"> (in other words, the preimage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b357745fa4a2178733a502b4432072be8222fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}"> of every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"> is nonempty).<a href="#cite_note-PCM_p.11-22">[17]</a><a href="#cite_note-EOM_Surjection-26">[20]</a> If, as usual in modern mathematics, the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> is assumed, then f is surjective if and only if there exists a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10a9844999dbfd6d1ba60a6d5d37779df277a74f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.421ex; height:2.509ex;" alt="{\displaystyle g:Y\to X}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=\operatorname {id} _{Y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g=\operatorname {id} _{Y},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/307ae8225c2aa1e6d08f0bc76bf698729263f9da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.76ex; height:2.509ex;" alt="{\displaystyle f\circ g=\operatorname {id} _{Y},}"> that is, if f has a <a href="/wiki/Right_inverse_function" class="mw-redirect" title="Right inverse function">right inverse</a>.<a href="#cite_note-EOM_Surjection-26">[20]</a> The axiom of choice is needed, because, if f is surjective, one defines g by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(y)=x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(y)=x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68c9ed4e962731675c415d078cad57281ae86b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.156ex; height:2.843ex;" alt="{\displaystyle g(y)=x,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is an arbitrarily chosen element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9041b1fe9d0edbeee6a144fc971a4d51b9e5dda9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.265ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y).}"> The function f is <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> (or is a bijection or a one-to-one correspondence) if it is both injective and surjective.<a href="#cite_note-PCM_p.11-22">[17]</a><a href="#cite_note-EOM_Bijection-27">[21]</a> That is, f is bijective if, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e1353f0febe9bcf693d849ec82ce8d94e5f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle y\in Y,}"> the preimage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b357745fa4a2178733a502b4432072be8222fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}"> contains exactly one element. The function f is bijective if and only if it admits an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>, that is, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10a9844999dbfd6d1ba60a6d5d37779df277a74f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.421ex; height:2.509ex;" alt="{\displaystyle g:Y\to X}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f=\operatorname {id} _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f=\operatorname {id} _{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2935e82106a6aeb3c354d02246be16fe2ac7e4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.259ex; height:2.509ex;" alt="{\displaystyle g\circ f=\operatorname {id} _{X}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=\operatorname {id} _{Y}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g=\operatorname {id} _{Y}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340d648bc681ff51cbcd847e207a453baccd313b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.76ex; height:2.509ex;" alt="{\displaystyle f\circ g=\operatorname {id} _{Y}.}"><a href="#cite_note-EOM_Bijection-27">[21]</a> (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward). Every function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> may be <a href="/wiki/Factorization" title="Factorization">factorized</a> as the composition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\circ s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∘</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\circ s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c1d5fd76a33e0efee96d289ac9422fcc23ef43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.088ex; height:2.176ex;" alt="{\displaystyle i\circ s}"> of a surjection followed by an injection, where s is the canonical surjection of X onto f(X) and i is the canonical injection of f(X) into Y. This is the canonical factorization of f. "One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki group</a> and imported into English.<a href="#cite_note-28">[22]</a> As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "f maps X onto Y" differs from "f maps X into B", in that the former implies that f is surjective, while the latter makes no assertion about the nature of f. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical. <div class="mw-heading mw-heading3"><h3 id="Restriction_and_extension">Restriction and extension </h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=28" title="Edit section: Restriction and extension">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">Restriction (mathematics)</a></div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> is a function and S is a subset of X, then the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> to S, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{S}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd8f943a75adb4c382a271541c010eb0d60b3b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.218ex; height:3.009ex;" alt="{\displaystyle f|_{S}}">, is the function from S to Y defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{S}(x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{S}(x)=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dda9855e14e326136e924a70d0b0218a736bb7c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.873ex; height:3.009ex;" alt="{\displaystyle f|_{S}(x)=f(x)}"></dd></dl> for all x in S. Restrictions can be used to define partial <a href="/wiki/Inverse_function" title="Inverse function">inverse functions</a>: if there is a <a href="/wiki/Subset" title="Subset">subset</a> S of the domain of a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{S}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd8f943a75adb4c382a271541c010eb0d60b3b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.218ex; height:3.009ex;" alt="{\displaystyle f|_{S}}"> is injective, then the canonical surjection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{S}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd8f943a75adb4c382a271541c010eb0d60b3b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.218ex; height:3.009ex;" alt="{\displaystyle f|_{S}}"> onto its image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{S}(S)=f(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{S}(S)=f(S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bac2730187f424f13f20378bbff79347e40dcea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.212ex; height:3.009ex;" alt="{\displaystyle f|_{S}(S)=f(S)}"> is a bijection, and thus has an inverse function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a865d36b32ea1d60f15dc3093f5b28093f192b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.587ex; height:2.843ex;" alt="{\displaystyle f(S)}"> to S. One application is the definition of <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse trigonometric functions</a>. For example, the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> function is injective when restricted to the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> [0, π]. The image of this restriction is the interval [−1, 1], and thus the restriction has an inverse function from [−1, 1] to [0, π], which is called <a href="/wiki/Arccosine" class="mw-redirect" title="Arccosine">arccosine</a> and is denoted arccos. Function restriction may also be used for "gluing" functions together. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X=\bigcup _{i\in I}U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X=\bigcup _{i\in I}U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87cdc2074a4c87a94344669b3d5d69c8228252a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.513ex; height:3.009ex;" alt="{\textstyle X=\bigcup _{i\in I}U_{i}}"> be the decomposition of X as a <a href="/wiki/Set_union" class="mw-redirect" title="Set union">union</a> of subsets, and suppose that a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}:U_{i}\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}:U_{i}\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/767ac1e7aabcf452776449785782fd0f36fec9c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.651ex; height:2.509ex;" alt="{\displaystyle f_{i}:U_{i}\to Y}"> is defined on each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> such that for each pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cbf8bbc622154cda8208d6e339495fe16a1f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}"> of indices, the restrictions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acc195ab3f9d65994b47774eb013601d09217aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.843ex;" alt="{\displaystyle f_{j}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}\cap U_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}\cap U_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be592e07a07b4ce92cc30774ae3a8d984eca426e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.843ex;" alt="{\displaystyle U_{i}\cap U_{j}}"> are equal. Then this defines a unique function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{U_{i}}=f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{U_{i}}=f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92463d8f349ef5f9731495571ea43612abc5efd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.942ex; height:3.176ex;" alt="{\displaystyle f|_{U_{i}}=f_{i}}"> for all i. This is the way that functions on <a href="/wiki/Manifold" title="Manifold">manifolds</a> are defined. An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a>, that allows extending functions whose domain is a small part of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> to functions whose domain is almost the whole complex plane. Here is another classical example of a function extension that is encountered when studying <a href="/wiki/Homography" title="Homography">homographies</a> of the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>. A homography is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)={\frac {ax+b}{cx+d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </mrow> <mrow> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)={\frac {ax+b}{cx+d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ea2d0884deecb78aa8fcff23d688b6cc0f512d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.81ex; height:5.676ex;" alt="{\displaystyle h(x)={\frac {ax+b}{cx+d}}}"> such that ad − bc ≠ 0. Its domain is the set of all <a href="/wiki/Real_number" title="Real number">real numbers</a> different from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -d/c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -d/c,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fe216fd10ada1d567e72ed5f059bfe97f1a1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.84ex; height:2.843ex;" alt="{\displaystyle -d/c,}"> and its image is the set of all real numbers different from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a/c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6eba9778a4571d653a0173b97d104de1ef7216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.046ex; height:2.843ex;" alt="{\displaystyle a/c.}"> If one extends the real line to the <a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">projectively extended real line</a> by including ∞, one may extend h to a bijection from the extended real line to itself by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(\infty )=a/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(\infty )=a/c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993cd2447b94a4bbf667f8524f89ac64114d7fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.97ex; height:2.843ex;" alt="{\displaystyle h(\infty )=a/c}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(-d/c)=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(-d/c)=\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b4e44944640076eadf15cd9062568526a2f868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.764ex; height:2.843ex;" alt="{\displaystyle h(-d/c)=\infty }">. <div class="mw-heading mw-heading2"><h2 id="In_calculus">In calculus</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=29" title="Edit section: In calculus">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_the_function_concept" title="History of the function concept">History of the function concept</a></div> The idea of function, starting in the 17th century, was fundamental to the new <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a>. At that time, only <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued</a> functions of a <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">real variable</a> were considered, and all functions were assumed to be <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a>. But the definition was soon extended to <a href="#Multivariate_function">functions of several variables</a> and to <a href="/wiki/Functions_of_a_complex_variable" class="mw-redirect" title="Functions of a complex variable">functions of a complex variable</a>. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined. Functions are now used throughout all areas of mathematics. In introductory <a href="/wiki/Calculus" title="Calculus">calculus</a>, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with <a href="/wiki/STEM" class="mw-redirect" title="STEM">STEM</a> majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a> and <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>. <div class="mw-heading mw-heading3"><h3 id="Real_function">Real function</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=30" title="Edit section: Real function">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Gerade.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Gerade.svg/220px-Gerade.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Gerade.svg/330px-Gerade.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Gerade.svg/440px-Gerade.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Graph of a linear function</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Polynomialdeg2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/220px-Polynomialdeg2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/330px-Polynomialdeg2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/440px-Polynomialdeg2.svg.png 2x" data-file-width="320" data-file-height="320" /></a><figcaption>Graph of a polynomial function, here a quadratic function.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sine_cosine_one_period.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/220px-Sine_cosine_one_period.svg.png" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/330px-Sine_cosine_one_period.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Sine_cosine_one_period.svg/440px-Sine_cosine_one_period.svg.png 2x" data-file-width="600" data-file-height="240" /></a><figcaption>Graph of two trigonometric functions: <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>.</figcaption></figure> A real function is a <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued</a> <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">function of a real variable</a>, that is, a function whose codomain is the <a href="/wiki/Real_number" title="Real number">field of real numbers</a> and whose domain is a set of <a href="/wiki/Real_number" title="Real number">real numbers</a> that contains an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>. In this section, these functions are simply called functions. The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>, <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a>, and even <a href="/wiki/Analytic_function" title="Analytic function">analytic</a>. This regularity insures that these functions can be visualized by their <a href="#Graph_and_plots">graphs</a>. In this section, all functions are differentiable in some interval. Functions enjoy <a href="/wiki/Pointwise_operation" class="mw-redirect" title="Pointwise operation">pointwise operations</a>, that is, if f and g are functions, their sum, difference and product are functions defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(f\cdot g)(x)&=f(x)\cdot g(x)\\\end{aligned}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(f\cdot g)(x)&=f(x)\cdot g(x)\\\end{aligned}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e426d18a2bc5fb702b5ba0497477739c65f1a667" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.193ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(f\cdot g)(x)&=f(x)\cdot g(x)\\\end{aligned}}.}"></dd></dl> The domains of the resulting functions are the <a href="/wiki/Set_intersection" class="mw-redirect" title="Set intersection">intersection</a> of the domains of f and g. The quotient of two functions is defined similarly by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f}{g}}(x)={\frac {f(x)}{g(x)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>g</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f}{g}}(x)={\frac {f(x)}{g(x)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f8c6b970bb8c30650eaea1922c3921aec7d659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.253ex; height:6.509ex;" alt="{\displaystyle {\frac {f}{g}}(x)={\frac {f(x)}{g(x)}},}"></dd></dl> but the domain of the resulting function is obtained by removing the <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of g from the intersection of the domains of f and g. The <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> are defined by <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>, and their domain is the whole set of real numbers. They include <a href="/wiki/Constant_function" title="Constant function">constant functions</a>, <a href="/wiki/Linear_function" title="Linear function">linear functions</a> and <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic functions</a>. <a href="/wiki/Rational_function" title="Rational function">Rational functions</a> are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a>. The simplest rational function is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto {\frac {1}{x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\frac {1}{x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669c4e870914fd443f1c2264450695a3b98fa040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.756ex; height:5.176ex;" alt="{\displaystyle x\mapsto {\frac {1}{x}},}"> whose graph is a <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>, and whose domain is the whole <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> except for 0. The <a href="/wiki/Derivative" title="Derivative">derivative</a> of a real differentiable function is a real function. An <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> of a continuous real function is a real function that has the original function as a derivative. For example, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\mapsto {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\mapsto {\frac {1}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e512047f779008dd32f5b041baceb2c4dfa9a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.72ex; height:3.343ex;" alt="{\textstyle x\mapsto {\frac {1}{x}}}"> is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for x = 1, is a differentiable function called the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>. A real function f is <a href="/wiki/Monotonic_function" title="Monotonic function">monotonic</a> in an interval if the sign of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(x)-f(y)}{x-y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(x)-f(y)}{x-y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857bf1f8402b34ead332926a1050dff7f421bc3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.337ex; height:6.176ex;" alt="{\displaystyle {\frac {f(x)-f(y)}{x-y}}}"> does not depend of the choice of x and y in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function f is monotonic in an interval I, it has an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>, which is a real function with domain f(I) and image I. This is how <a href="/wiki/Inverse_trigonometric_functions" title="Inverse trigonometric functions">inverse trigonometric functions</a> are defined in terms of <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a <a href="/wiki/Bijection" title="Bijection">bijection</a> between the real numbers and the positive real numbers. This inverse is the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>. Many other real functions are defined either by the <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a> (the inverse function is a particular instance) or as solutions of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. For example, the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> functions are the solutions of the <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equation</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''+y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''+y=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ca5e3d2957217ee417aff328d32fbff2de8d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.554ex; height:2.843ex;" alt="{\displaystyle y''+y=0}"></dd></dl> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin 0=0,\quad \cos 0=1,\quad {\frac {\partial \sin x}{\partial x}}(0)=1,\quad {\frac {\partial \cos x}{\partial x}}(0)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin 0=0,\quad \cos 0=1,\quad {\frac {\partial \sin x}{\partial x}}(0)=1,\quad {\frac {\partial \cos x}{\partial x}}(0)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd7b575eed317bfdd3cbf64c5411ae3fcf72764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:57.252ex; height:5.509ex;" alt="{\displaystyle \sin 0=0,\quad \cos 0=1,\quad {\frac {\partial \sin x}{\partial x}}(0)=1,\quad {\frac {\partial \cos x}{\partial x}}(0)=0.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Vector-valued_function">Vector-valued function</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=31" title="Edit section: Vector-valued function">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Vector-valued_function" title="Vector-valued function">Vector-valued function</a> and <a href="/wiki/Vector_field" title="Vector field">Vector field</a></div> When the elements of the codomain of a function are <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its <a href="/wiki/Velocity_vector" class="mw-redirect" title="Velocity vector">velocity vector</a> is a vector-valued function. Some vector-valued functions are defined on a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> or other spaces that share geometric or <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">, such as <a href="/wiki/Manifolds" class="mw-redirect" title="Manifolds">manifolds</a>. These vector-valued functions are given the name vector fields. <div class="mw-heading mw-heading2"><h2 id="Function_space">Function space</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=32" title="Edit section: Function space">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Function_space" title="Function space">Function space</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></div> In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, and more specifically in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, a function space is a set of <a href="/wiki/Scalar-valued_function" class="mw-redirect" title="Scalar-valued function">scalar-valued</a> or <a href="/wiki/Vector-valued_function" title="Vector-valued function">vector-valued functions</a>, which share a specific property and form a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a>. For example, the real <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> with a <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a> (that is, they are zero outside some <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact set</a>) form a function space that is at the basis of the theory of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and <a href="/wiki/Topology" title="Topology">topological</a> properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary</a> or <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> result of the study of function spaces. <div class="mw-heading mw-heading2"><h2 id="Multi-valued_functions">Multi-valued functions</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=33" title="Edit section: Multi-valued functions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Multi-valued_function" class="mw-redirect" title="Multi-valued function">Multi-valued function</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Function_with_two_values_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Function_with_two_values_1.svg/220px-Function_with_two_values_1.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Function_with_two_values_1.svg/330px-Function_with_two_values_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Function_with_two_values_1.svg/440px-Function_with_two_values_1.svg.png 2x" data-file-width="440" data-file-height="320" /></a><figcaption>Together, the two square roots of all nonnegative real numbers form a single smooth curve.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Xto3minus3x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Xto3minus3x.svg/220px-Xto3minus3x.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Xto3minus3x.svg/330px-Xto3minus3x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Xto3minus3x.svg/440px-Xto3minus3x.svg.png 2x" data-file-width="319" data-file-height="239" /></a><figcaption></figcaption></figure> Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b35dd572e629881da4083ad1681bc7cf420304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0},}"> there are several possible starting values for the function. For example, in defining the <a href="/wiki/Square_root" title="Square root">square root</a> as the inverse function of the square function, for any positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b35dd572e629881da4083ad1681bc7cf420304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0},}"> there are two choices for the value of the square root, one of which is positive and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x_{0}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x_{0}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43d17683413882a6695d00f6e2d49ba450ab4446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.967ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x_{0}}},}"> and another which is negative and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\sqrt {x_{0}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt {x_{0}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d50109ab6da5d52c015b69ef106459324ba333e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.775ex; height:3.009ex;" alt="{\displaystyle -{\sqrt {x_{0}}}.}"> These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single <a href="/wiki/Smooth_curve" class="mw-redirect" title="Smooth curve">smooth curve</a>. It is therefore often useful to consider these two square root functions as a single function that has two values for positive x, one value for 0 and no value for negative x. In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the <a href="/wiki/Implicit_function" title="Implicit function">implicit function</a> that maps y to a <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> x of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-3x-y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-3x-y=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b2ac300ad197f4bf3deae55b4b2e95ff3a03e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.973ex; height:3.009ex;" alt="{\displaystyle x^{3}-3x-y=0}"> (see the figure on the right). For y = 0 one may choose either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,{\sqrt {3}},{\text{ or }}-{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,{\sqrt {3}},{\text{ or }}-{\sqrt {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03884332a438b88000c08a603f510e1cb1afb7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.503ex; height:2.843ex;" alt="{\displaystyle 0,{\sqrt {3}},{\text{ or }}-{\sqrt {3}}}"> for x. By the <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a>, each choice defines a function; for the first one, the (maximal) domain is the interval [−2, 2] and the image is [−1, 1]; for the second one, the domain is [−2, ∞) and the image is [1, ∞); for the last one, the domain is (−∞, 2] and the image is (−∞, −1]. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of y that has three values for −2 < y < 2, and only one value for y ≤ −2 and y ≥ −2. Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>. The domain to which a complex function may be extended by <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> generally consists of almost the whole <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets i for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets −i. There are generally two ways of solving the problem. One may define a function that is not <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> along some curve, called a <a href="/wiki/Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a>. Such a function is called the <a href="/wiki/Principal_value" title="Principal value">principal value</a> of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the <a href="/wiki/Monodromy" title="Monodromy">monodromy</a>. <div class="mw-heading mw-heading2"><h2 id="In_the_foundations_of_mathematics">In the foundations of mathematics</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=34" title="Edit section: In the foundations of mathematics">edit</a>]</div> The definition of a function that is given in this article requires the concept of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions. For example, the <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> may be considered as a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \{x\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto \{x\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b393766c20d0e835b81a0f3fd5007f5746eb1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.245ex; height:2.843ex;" alt="{\displaystyle x\mapsto \{x\}.}"> Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definitions for these weakly specified functions.<a href="#cite_note-29">[23]</a> These generalized functions may be critical in the development of a formalization of the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>. For example, <a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel set theory</a>, is an extension of the set theory in which the collection of all sets is a <a href="/wiki/Class_(set_theory)" title="Class (set theory)">class</a>. This theory includes the <a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#NBG's_axiom_of_replacement" title="Von Neumann–Bernays–Gödel set theory">replacement axiom</a>, which may be stated as: If X is a set and F is a function, then F[X] is a set. In alternative formulations of the foundations of mathematics using <a href="/wiki/Type_theory" title="Type theory">type theory</a> rather than set theory, functions are taken as <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notions</a> rather than defined from other kinds of object. They are the inhabitants of <a href="/wiki/Function_type" title="Function type">function types</a>, and may be constructed using expressions in the <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a>.<a href="#cite_note-30">[24]</a> <div class="mw-heading mw-heading2"><h2 id="In_computer_science">In computer science</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=35" title="Edit section: In computer science">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Function_(computer_programming)" title="Function (computer programming)">Function (computer programming)</a> and <a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></div> In <a href="/wiki/Computer_programming" title="Computer programming">computer programming</a>, a <a href="/wiki/Function_(programming)" class="mw-redirect" title="Function (programming)">function</a> is, in general, a piece of a <a href="/wiki/Computer_program" title="Computer program">computer program</a>, which <a href="/wiki/Implementation" title="Implementation">implements</a> the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many <a href="/wiki/Programming_language" title="Programming language">programming languages</a> every <a href="/wiki/Subroutine" class="mw-redirect" title="Subroutine">subroutine</a> is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the <a href="/wiki/Computer_memory" title="Computer memory">computer memory</a>. <a href="/wiki/Functional_programming" title="Functional programming">Functional programming</a> is the <a href="/wiki/Programming_paradigm" title="Programming paradigm">programming paradigm</a> consisting of building programs by using only subroutines that behave like mathematical functions. For example, <code>if_then_else</code> is a function that takes three functions as arguments, and, depending on the result of the first function (true or false), returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier <a href="/wiki/Program_proof" class="mw-redirect" title="Program proof">program proofs</a>, as being based on a well founded theory, the <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> (see below). Except for computer-language terminology, "function" has the usual mathematical meaning in <a href="/wiki/Computer_science" title="Computer science">computer science</a>. In this area, a property of major interest is the <a href="/wiki/Computable_function" title="Computable function">computability</a> of a function. For giving a precise meaning to this concept, and to the related concept of <a href="/wiki/Algorithm" title="Algorithm">algorithm</a>, several <a href="/wiki/Models_of_computation" class="mw-redirect" title="Models of computation">models of computation</a> have been introduced, the old ones being <a href="/wiki/%CE%9C-recursive_function" class="mw-redirect" title="Μ-recursive function">general recursive functions</a>, <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> and <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>. The fundamental theorem of <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a> is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a> is the claim that every philosophically acceptable definition of a computable function defines also the same functions. General recursive functions are <a href="/wiki/Partial_function" title="Partial function">partial functions</a> from integers to integers that can be defined from <ul><li><a href="/wiki/Constant_function" title="Constant function">constant functions</a>,</li> <li><a href="/wiki/Successor_function" title="Successor function">successor</a>, and</li> <li><a href="/wiki/Projection_function" class="mw-redirect" title="Projection function">projection</a> functions</li></ul> via the operators <ul><li><a href="#Function_composition">composition</a>,</li> <li><a href="/wiki/Primitive_recursion" class="mw-redirect" title="Primitive recursion">primitive recursion</a>, and</li> <li><a href="/wiki/%CE%9C_operator" title="Μ operator">minimization</a>.</li></ul> Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties: <ul><li>a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...),</li> <li>every sequence of symbols may be coded as a sequence of <a href="/wiki/Bit" title="Bit">bits</a>,</li> <li>a bit sequence can be interpreted as the <a href="/wiki/Binary_representation" class="mw-redirect" title="Binary representation">binary representation</a> of an integer.</li></ul> <a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a> is a theory that defines computable functions without using <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and is the theoretical background of functional programming. It consists of terms that are either variables, function definitions (𝜆-terms), or applications of functions to terms. Terms are manipulated through some rules, (the α-equivalence, the β-reduction, and the η-conversion), which are the <a href="/wiki/Axiom" title="Axiom">axioms</a> of the theory and may be interpreted as rules of computation. In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of type in <a href="/wiki/Typed_lambda_calculus" title="Typed lambda calculus">typed lambda calculus</a>. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=36" title="Edit section: See also">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Subpages">Subpages</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=37" title="Edit section: Subpages">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">History of the function concept</a></li> <li><a href="/wiki/List_of_types_of_functions" title="List of types of functions">List of types of functions</a></li> <li><a href="/wiki/List_of_functions" class="mw-redirect" title="List of functions">List of functions</a></li> <li><a href="/wiki/Function_fitting" class="mw-redirect" title="Function fitting">Function fitting</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit function</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Generalizations">Generalizations</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=38" title="Edit section: Generalizations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Higher-order_function" title="Higher-order function">Higher-order function</a></li> <li><a href="/wiki/Homomorphism" title="Homomorphism">Homomorphism</a></li> <li><a href="/wiki/Morphism" title="Morphism">Morphism</a></li> <li><a href="/wiki/Microfunction" class="mw-redirect" title="Microfunction">Microfunction</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a></li> <li><a href="/wiki/Functor" title="Functor">Functor</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Related_topics">Related topics</h3>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=39" title="Edit section: Related topics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Closed-form_expression" title="Closed-form expression">Closed-form expression</a></li> <li><a href="/wiki/Elementary_function" title="Elementary function">Elementary function</a></li> <li><a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">Functional</a></li> <li><a href="/wiki/Functional_decomposition" title="Functional decomposition">Functional decomposition</a></li> <li><a href="/wiki/Functional_predicate" title="Functional predicate">Functional predicate</a></li> <li><a href="/wiki/Functional_programming" title="Functional programming">Functional programming</a></li> <li><a href="/wiki/Parametric_equation" title="Parametric equation">Parametric equation</a></li> <li><a href="/wiki/Set_function" title="Set function">Set function</a></li> <li><a href="/wiki/Simple_function" title="Simple function">Simple function</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=40" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-4"><a href="#cite_ref-4">^</a> This definition of "graph" refers to a set of pairs of objects. Graphs, in the sense of diagrams, are most applicable to functions from the real numbers to themselves. All functions can be described by sets of pairs but it may not be practical to construct a diagram for functions between other sets (such as sets of matrices). </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> The true domain of such a function is often called the domain of definition of the function. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> n may also be 1, thus subsuming functions as defined above. For n = 0, each <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> is a special case of a multivariate function, too. </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> By definition, the graph of the empty function to X is a subset of the Cartesian product ∅ × X, and this product is empty. </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> The <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> is not needed here, as the choice is done in a single set. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=41" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-halmos-1"><a href="#cite_ref-halmos_1-0">^</a> <a href="#CITEREFHalmos1970">Halmos 1970</a>, p. 30; the words map, mapping, transformation, correspondence, and operator are sometimes used synonymously. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFHalmos1970">Halmos 1970</a> </li> <li id="cite_note-codomain-3"><a href="#cite_ref-codomain_3-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Mapping&oldid=37940">"Mapping"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mapping&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMapping%26oldid%3D37940&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/function-mathematics">"function | Definition, Types, Examples, & Facts"</a>. Encyclopedia Britannica. Retrieved 2020-08-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+Britannica&rft.atitle=function+%7C+Definition%2C+Types%2C+Examples%2C+%26+Facts&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Ffunction-mathematics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTESpivak200839-6"><a href="#cite_ref-FOOTNOTESpivak200839_6-0">^</a> <a href="#CITEREFSpivak2008">Spivak 2008</a>, p. 39. </li> <li id="cite_note-EOM_Function-7">^ <a href="#cite_ref-EOM_Function_7-0">a</a> <a href="#cite_ref-EOM_Function_7-1">b</a> <a href="#cite_ref-EOM_Function_7-2">c</a> <a href="#cite_ref-EOM_Function_7-3">d</a> <a href="#cite_ref-EOM_Function_7-4">e</a> <a href="#cite_ref-EOM_Function_7-5">f</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKudryavtsev2001" class="citation cs1">Kudryavtsev, L.D. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Function&oldid=36823">"Function"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Kudryavtsev&rft.aufirst=L.D.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFunction%26oldid%3D36823&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-T&K_Calc_p.3-8">^ <a href="#cite_ref-T&K_Calc_p.3_8-0">a</a> <a href="#cite_ref-T&K_Calc_p.3_8-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaalmanKohn2014" class="citation book cs1"><a href="/wiki/Laura_Taalman" title="Laura Taalman">Taalman, Laura</a>; Kohn, Peter (2014). Calculus. <a href="/wiki/New_York_City" title="New York City">New York City</a>: <a href="/wiki/W._H._Freeman_and_Company" title="W. H. Freeman and Company">W. H. Freeman and Company</a>. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4292-4186-1" title="Special:BookSources/978-1-4292-4186-1"><bdi>978-1-4292-4186-1</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2012947365">2012947365</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/856545590">856545590</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL27544563M">27544563M</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.place=New+York+City&rft.pages=3&rft.pub=W.+H.+Freeman+and+Company&rft.date=2014&rft_id=info%3Aoclcnum%2F856545590&rft_id=info%3Alccn%2F2012947365&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL27544563M%23id-name%3DOL&rft.isbn=978-1-4292-4186-1&rft.aulast=Taalman&rft.aufirst=Laura&rft.au=Kohn%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Trench_RA_pp.30-32-9">^ <a href="#cite_ref-Trench_RA_pp.30-32_9-0">a</a> <a href="#cite_ref-Trench_RA_pp.30-32_9-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrench2013" class="citation book cs1">Trench, William F. (2013) [2003]. <a rel="nofollow" class="external text" href="https://digitalcommons.trinity.edu/mono/7/">Introduction to Real Analysis</a> (2.04th ed.). <a href="/wiki/Pearson_Education" title="Pearson Education">Pearson Education</a> (originally; self-republished by the author). pp. 30–32. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-045786-8" title="Special:BookSources/0-13-045786-8"><bdi>0-13-045786-8</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2002032369">2002032369</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/953799815">953799815</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1204.00023">1204.00023</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Real+Analysis&rft.pages=%3Cspan+class%3D%22nowrap%22%3E30-%3C%2Fspan%3E32&rft.edition=2.04th&rft.pub=Pearson+Education+%28originally%3B+self-republished+by+the+author%29&rft.date=2013&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1204.00023%23id-name%3DZbl&rft_id=info%3Alccn%2F2002032369&rft_id=info%3Aoclcnum%2F953799815&rft.isbn=0-13-045786-8&rft.aulast=Trench&rft.aufirst=William+F.&rft_id=https%3A%2F%2Fdigitalcommons.trinity.edu%2Fmono%2F7%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-TBB_RA_pp.A4-A5-10">^ <a href="#cite_ref-TBB_RA_pp.A4-A5_10-0">a</a> <a href="#cite_ref-TBB_RA_pp.A4-A5_10-1">b</a> <a href="#cite_ref-TBB_RA_pp.A4-A5_10-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomsonBrucknerBruckner2008" class="citation book cs1">Thomson, Brian S.; Bruckner, Judith B.; <a href="/wiki/Andrew_M._Bruckner" title="Andrew M. Bruckner">Bruckner, Andrew M.</a> (2008) [2001]. <a rel="nofollow" class="external text" href="https://www.classicalrealanalysis.info/com/documents/TBB-AllChapters-Portrait.pdf">Elementary Real Analysis</a> (PDF) (2nd ed.). <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a> (originally; 2nd ed. self-republished by the authors). pp. A-4 – A-5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4348-4367-8" title="Special:BookSources/978-1-4348-4367-8"><bdi>978-1-4348-4367-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1105855173">1105855173</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL31844948M">31844948M</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0872.26001">0872.26001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Real+Analysis&rft.pages=%3Cspan+class%3D%22nowrap%22%3EA-4+-%3C%2Fspan%3E+%3Cspan+class%3D%22nowrap%22%3EA-5%3C%2Fspan%3E&rft.edition=2nd&rft.pub=Prentice+Hall+%28originally%3B+2nd+ed.+self-republished+by+the+authors%29&rft.date=2008&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0872.26001%23id-name%3DZbl&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL31844948M%23id-name%3DOL&rft_id=info%3Aoclcnum%2F1105855173&rft.isbn=978-1-4348-4367-8&rft.aulast=Thomson&rft.aufirst=Brian+S.&rft.au=Bruckner%2C+Judith+B.&rft.au=Bruckner%2C+Andrew+M.&rft_id=https%3A%2F%2Fwww.classicalrealanalysis.info%2Fcom%2Fdocuments%2FTBB-AllChapters-Portrait.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1974" class="citation book cs1">Halmos, Paul R. (1974). Naive Set Theory. Springer. pp. 30–33.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E30-%3C%2Fspan%3E33&rft.pub=Springer&rft.date=1974&rft.aulast=Halmos&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarsonEdwards2010" class="citation book cs1">Larson, Ron; Edwards, Bruce H. (2010). Calculus of a Single Variable. Cengage Learning. p. 19. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-538-73552-0" title="Special:BookSources/978-0-538-73552-0"><bdi>978-0-538-73552-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+of+a+Single+Variable&rft.pages=19&rft.pub=Cengage+Learning&rft.date=2010&rft.isbn=978-0-538-73552-0&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Map.html">"Map"</a>. mathworld.wolfram.com. Retrieved 2019-06-12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Map&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FMap.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Lang87p43-16">^ <a href="#cite_ref-Lang87p43_16-0">a</a> <a href="#cite_ref-Lang87p43_16-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1987" class="citation book cs1">Lang, Serge (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0DUXym7QWfYC&pg=PA43">"III §1. Mappings"</a>. Linear Algebra (3rd ed.). Springer. p. 43. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96412-6" title="Special:BookSources/978-0-387-96412-6"><bdi>978-0-387-96412-6</bdi></a>. <q>A function is a special type of mapping, namely it is a mapping from a set into the set of numbers, i.e. into, R, or C or into a field K.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=III+%C2%A71.+Mappings&rft.btitle=Linear+Algebra&rft.pages=43&rft.edition=3rd&rft.pub=Springer&rft.date=1987&rft.isbn=978-0-387-96412-6&rft.aulast=Lang&rft.aufirst=Serge&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0DUXym7QWfYC%26pg%3DPA43&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Apostol81p35-17">^ <a href="#cite_ref-Apostol81p35_17-0">a</a> <a href="#cite_ref-Apostol81p35_17-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1981" class="citation book cs1">Apostol, T.M. (1981). Mathematical Analysis (2nd ed.). Addison-Wesley. p. 35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-00288-1" title="Special:BookSources/978-0-201-00288-1"><bdi>978-0-201-00288-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/928947543">928947543</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Analysis&rft.pages=35&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1981&rft_id=info%3Aoclcnum%2F928947543&rft.isbn=978-0-201-00288-1&rft.aulast=Apostol&rft.aufirst=T.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJamesJames1992" class="citation book cs1"><a href="/wiki/Robert_C._James" title="Robert C. James">James, Robert C.</a>; James, Glenn (1992). Mathematics dictionary (5th ed.). Van Nostrand Reinhold. p. 202. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-442-00741-8" title="Special:BookSources/0-442-00741-8"><bdi>0-442-00741-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/25409557">25409557</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+dictionary&rft.pages=202&rft.edition=5th&rft.pub=Van+Nostrand+Reinhold&rft.date=1992&rft_id=info%3Aoclcnum%2F25409557&rft.isbn=0-442-00741-8&rft.aulast=James&rft.aufirst=Robert+C.&rft.au=James%2C+Glenn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFJamesJames1992">James & James 1992</a>, p. 48 </li> <li id="cite_note-PCM_p.11-22">^ <a href="#cite_ref-PCM_p.11_22-0">a</a> <a href="#cite_ref-PCM_p.11_22-1">b</a> <a href="#cite_ref-PCM_p.11_22-2">c</a> <a href="#cite_ref-PCM_p.11_22-3">d</a> <a href="#cite_ref-PCM_p.11_22-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGowersBarrow-GreenLeader2008" class="citation book cs1"><a href="/wiki/Timothy_Gowers" title="Timothy Gowers">Gowers, Timothy</a>; <a href="/wiki/June_Barrow-Green" title="June Barrow-Green">Barrow-Green, June</a>; <a href="/wiki/Imre_Leader" title="Imre Leader">Leader, Imre</a>, eds. (2008). <a href="/wiki/The_Princeton_Companion_to_Mathematics" title="The Princeton Companion to Mathematics">The Princeton Companion to Mathematics</a>. <a href="/wiki/Princeton,_New_Jersey" title="Princeton, New Jersey">Princeton, New Jersey</a>: <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. p. 11. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400830398">10.1515/9781400830398</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11880-2" title="Special:BookSources/978-0-691-11880-2"><bdi>978-0-691-11880-2</bdi></a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/j.ctt7sd01">j.ctt7sd01</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2008020450">2008020450</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2467561">2467561</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/227205932">227205932</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL19327100M">19327100M</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1242.00016">1242.00016</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Princeton+Companion+to+Mathematics&rft.place=Princeton%2C+New+Jersey&rft.pages=11&rft.pub=Princeton+University+Press&rft.date=2008&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1242.00016%23id-name%3DZbl&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2Fj.ctt7sd01%23id-name%3DJSTOR&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL19327100M%23id-name%3DOL&rft_id=info%3Adoi%2F10.1515%2F9781400830398&rft_id=info%3Aoclcnum%2F227205932&rft_id=info%3Alccn%2F2008020450&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2467561%23id-name%3DMR&rft.isbn=978-0-691-11880-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-standard-23"><a href="#cite_ref-standard_23-0">^</a> Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, p. 15. ISO 80000-2 (ISO/IEC 2009-12-01) </li> <li id="cite_note-EOM_Injection-24">^ <a href="#cite_ref-EOM_Injection_24-0">a</a> <a href="#cite_ref-EOM_Injection_24-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvanova2001" class="citation cs1">Ivanova, O.A. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Injection&oldid=30986">"Injection"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Injection&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Ivanova&rft.aufirst=O.A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DInjection%26oldid%3D30986&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-EOM_Surjection-26">^ <a href="#cite_ref-EOM_Surjection_26-0">a</a> <a href="#cite_ref-EOM_Surjection_26-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvanova2001" class="citation cs1">Ivanova, O.A. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Surjection&oldid=35689">"Surjection"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Surjection&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Ivanova&rft.aufirst=O.A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DSurjection%26oldid%3D35689&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-EOM_Bijection-27">^ <a href="#cite_ref-EOM_Bijection_27-0">a</a> <a href="#cite_ref-EOM_Bijection_27-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvanova2001" class="citation cs1">Ivanova, O.A. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bijection&oldid=30987">"Bijection"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bijection&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Ivanova&rft.aufirst=O.A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DBijection%26oldid%3D30987&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartnett2020" class="citation web cs1">Hartnett, Kevin (9 November 2020). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/inside-the-secret-math-society-known-as-nicolas-bourbaki-20201109/">"Inside the Secret Math Society Known Simply as Nicolas Bourbaki"</a>. Quanta Magazine. Retrieved 2024-06-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=Inside+the+Secret+Math+Society+Known+Simply+as+Nicolas+Bourbaki&rft.date=2020-11-09&rft.aulast=Hartnett&rft.aufirst=Kevin&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Finside-the-secret-math-society-known-as-nicolas-bourbaki-20201109%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="#CITEREFGödel1940">Gödel 1940</a>, p. 16; <a href="#CITEREFJech2003">Jech 2003</a>, p. 11; <a href="#CITEREFCunningham2016">Cunningham 2016</a>, p. 57 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlev2019" class="citation book cs1">Klev, Ansten (2019). "A comparison of type theory with set theory". In Centrone, Stefania; Kant, Deborah; Sarikaya, Deniz (eds.). Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Synthese Library. Vol. 407. Cham: Springer. pp. 271–292. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-15655-8_12">10.1007/978-3-030-15655-8_12</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-15654-1" title="Special:BookSources/978-3-030-15654-1"><bdi>978-3-030-15654-1</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=4352345">4352345</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+comparison+of+type+theory+with+set+theory&rft.btitle=Reflections+on+the+Foundations+of+Mathematics%3A+Univalent+Foundations%2C+Set+Theory+and+General+Thoughts&rft.place=Cham&rft.series=Synthese+Library&rft.pages=%3Cspan+class%3D%22nowrap%22%3E271-%3C%2Fspan%3E292&rft.pub=Springer&rft.date=2019&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D4352345%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-030-15655-8_12&rft.isbn=978-3-030-15654-1&rft.aulast=Klev&rft.aufirst=Ansten&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=42" title="Edit section: Sources">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBartle1976" class="citation book cs1"><a href="/wiki/Robert_G._Bartle" title="Robert G. Bartle">Bartle, Robert</a> (1976). The Elements of Real Analysis (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-05465-8" title="Special:BookSources/978-0-471-05465-8"><bdi>978-0-471-05465-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/465115030">465115030</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Elements+of+Real+Analysis&rft.edition=2nd&rft.pub=Wiley&rft.date=1976&rft_id=info%3Aoclcnum%2F465115030&rft.isbn=978-0-471-05465-8&rft.aulast=Bartle&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBloch2011" class="citation book cs1">Bloch, Ethan D. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QJ_537n8zKYC">Proofs and Fundamentals: A First Course in Abstract Mathematics</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7126-5" title="Special:BookSources/978-1-4419-7126-5"><bdi>978-1-4419-7126-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+and+Fundamentals%3A+A+First+Course+in+Abstract+Mathematics&rft.pub=Springer&rft.date=2011&rft.isbn=978-1-4419-7126-5&rft.aulast=Bloch&rft.aufirst=Ethan+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQJ_537n8zKYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCunningham2016" class="citation book cs1">Cunningham, Daniel W. (2016). Set theory: A First Course. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-12032-7" title="Special:BookSources/978-1-107-12032-7"><bdi>978-1-107-12032-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+theory%3A+A+First+Course&rft.pub=Cambridge+University+Press&rft.date=2016&rft.isbn=978-1-107-12032-7&rft.aulast=Cunningham&rft.aufirst=Daniel+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1940" class="citation book cs1"><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1940). The Consistency of the Continuum Hypothesis. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-07927-1" title="Special:BookSources/978-0-691-07927-1"><bdi>978-0-691-07927-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Consistency+of+the+Continuum+Hypothesis&rft.pub=Princeton+University+Press&rft.date=1940&rft.isbn=978-0-691-07927-1&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1970" class="citation book cs1"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul R.</a> (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x6cZBQ9qtgoC">Naive Set Theory</a>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90092-6" title="Special:BookSources/978-0-387-90092-6"><bdi>978-0-387-90092-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.pub=Springer-Verlag&rft.date=1970&rft.isbn=978-0-387-90092-6&rft.aulast=Halmos&rft.aufirst=Paul+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx6cZBQ9qtgoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJech2003" class="citation book cs1"><a href="/wiki/Thomas_Jech" title="Thomas Jech">Jech, Thomas</a> (2003). Set theory (3rd ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-44085-7" title="Special:BookSources/978-3-540-44085-7"><bdi>978-3-540-44085-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+theory&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2003&rft.isbn=978-3-540-44085-7&rft.aulast=Jech&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak2008" class="citation book cs1"><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7JKVu_9InRUC">Calculus</a> (4th ed.). Publish or Perish. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-914098-91-1" title="Special:BookSources/978-0-914098-91-1"><bdi>978-0-914098-91-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=4th&rft.pub=Publish+or+Perish&rft.date=2008&rft.isbn=978-0-914098-91-1&rft.aulast=Spivak&rft.aufirst=Michael&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7JKVu_9InRUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Function_(mathematics)&action=edit&section=43" title="Edit section: Further reading">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton1980" class="citation book cs1">Anton, Howard (1980). <a rel="nofollow" class="external text" href="https://archive.org/details/studentssolution00anto">Calculus with Analytical Geometry</a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-03248-9" title="Special:BookSources/978-0-471-03248-9"><bdi>978-0-471-03248-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+with+Analytical+Geometry&rft.pub=Wiley&rft.date=1980&rft.isbn=978-0-471-03248-9&rft.aulast=Anton&rft.aufirst=Howard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstudentssolution00anto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBartle1976" class="citation book cs1">Bartle, Robert G. (1976). The Elements of Real Analysis (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-05464-1" title="Special:BookSources/978-0-471-05464-1"><bdi>978-0-471-05464-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Elements+of+Real+Analysis&rft.edition=2nd&rft.pub=Wiley&rft.date=1976&rft.isbn=978-0-471-05464-1&rft.aulast=Bartle&rft.aufirst=Robert+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDubinskyHarel1992" class="citation book cs1">Dubinsky, Ed; Harel, Guershon (1992). The Concept of Function: Aspects of Epistemology and Pedagogy. Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-081-7" title="Special:BookSources/978-0-88385-081-7"><bdi>978-0-88385-081-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Concept+of+Function%3A+Aspects+of+Epistemology+and+Pedagogy&rft.pub=Mathematical+Association+of+America&rft.date=1992&rft.isbn=978-0-88385-081-7&rft.aulast=Dubinsky&rft.aufirst=Ed&rft.au=Harel%2C+Guershon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHammack2009" class="citation book cs1">Hammack, Richard (2009). <a rel="nofollow" class="external text" href="https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf#page=235">"12. Functions"</a> (PDF). <a rel="nofollow" class="external text" href="https://www.people.vcu.edu/~rhammack/BookOfProof/">Book of Proof</a>. <a href="/wiki/Virginia_Commonwealth_University" title="Virginia Commonwealth University">Virginia Commonwealth University</a>. Retrieved 2012-08-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=12.+Functions&rft.btitle=Book+of+Proof&rft.pub=Virginia+Commonwealth+University&rft.date=2009&rft.aulast=Hammack&rft.aufirst=Richard&rft_id=https%3A%2F%2Fwww.people.vcu.edu%2F~rhammack%2FBookOfProof%2FMain.pdf%23page%3D235&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHusch2001" class="citation book cs1">Husch, Lawrence S. (2001). <a rel="nofollow" class="external text" href="http://archives.math.utk.edu/visual.calculus/">Visual Calculus</a>. <a href="/wiki/University_of_Tennessee" title="University of Tennessee">University of Tennessee</a>. Retrieved 2007-09-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Visual+Calculus&rft.pub=University+of+Tennessee&rft.date=2001&rft.aulast=Husch&rft.aufirst=Lawrence+S.&rft_id=http%3A%2F%2Farchives.math.utk.edu%2Fvisual.calculus%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz1964" class="citation book cs1">Katz, Robert (1964). Axiomatic Analysis. <a href="/wiki/D._C._Heath_and_Company" title="D. C. Heath and Company">D. C. Heath and Company</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Axiomatic+Analysis&rft.pub=D.+C.+Heath+and+Company&rft.date=1964&rft.aulast=Katz&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner1989" class="citation journal cs1">Kleiner, Israel (1989). "Evolution of the Function Concept: A Brief Survey". The College Mathematics Journal. 20 (4): 282–300. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.6352">10.1.1.113.6352</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2686848">10.2307/2686848</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2686848">2686848</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+College+Mathematics+Journal&rft.atitle=Evolution+of+the+Function+Concept%3A+A+Brief+Survey&rft.volume=20&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E282-%3C%2Fspan%3E300&rft.date=1989&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.113.6352%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2686848%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2686848&rft.aulast=Kleiner&rft.aufirst=Israel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLützen2003" class="citation book cs1">Lützen, Jesper (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B3WvWhJTTX8C&pg=PA468">"Between rigor and applications: Developments in the concept of function in mathematical analysis"</a>. In Porter, Roy (ed.). The Cambridge History of Science: The modern physical and mathematical sciences. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57199-9" title="Special:BookSources/978-0-521-57199-9"><bdi>978-0-521-57199-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Between+rigor+and+applications%3A+Developments+in+the+concept+of+function+in+mathematical+analysis&rft.btitle=The+Cambridge+History+of+Science%3A+The+modern+physical+and+mathematical+sciences&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-57199-9&rft.aulast=L%C3%BCtzen&rft.aufirst=Jesper&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DB3WvWhJTTX8C%26pg%3DPA468&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"> An approachable and diverting historical presentation.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMalik1980" class="citation journal cs1">Malik, M. A. (1980). "Historical and pedagogical aspects of the definition of function". International Journal of Mathematical Education in Science and Technology. 11 (4): 489–492. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0020739800110404">10.1080/0020739800110404</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Mathematical+Education+in+Science+and+Technology&rft.atitle=Historical+and+pedagogical+aspects+of+the+definition+of+function&rft.volume=11&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E489-%3C%2Fspan%3E492&rft.date=1980&rft_id=info%3Adoi%2F10.1080%2F0020739800110404&rft.aulast=Malik&rft.aufirst=M.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReichenbach1947" class="citation book cs1">Reichenbach, Hans (1947). Elements of Symbolic Logic. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-24004-5" title="Special:BookSources/0-486-24004-5"><bdi>0-486-24004-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Symbolic+Logic&rft.pub=Dover&rft.date=1947&rft.isbn=0-486-24004-5&rft.aulast=Reichenbach&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuthing1984" class="citation journal cs1">Ruthing, D. (1984). "Old Intelligencer: Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N.". Mathematical Intelligencer. 6 (4): 71–78. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03026743">10.1007/BF03026743</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:189883712">189883712</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Intelligencer&rft.atitle=Old+Intelligencer%3A+Some+definitions+of+the+concept+of+function+from+Bernoulli%2C+Joh.+to+Bourbaki%2C+N.&rft.volume=6&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E71-%3C%2Fspan%3E78&rft.date=1984&rft_id=info%3Adoi%2F10.1007%2FBF03026743&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189883712%23id-name%3DS2CID&rft.aulast=Ruthing&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomasFinney1995" class="citation book cs1">Thomas, George B.; Finney, Ross L. (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusanalytic00geor_0">Calculus and Analytic Geometry</a> (9th ed.). <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-53174-9" title="Special:BookSources/978-0-201-53174-9"><bdi>978-0-201-53174-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+and+Analytic+Geometry&rft.edition=9th&rft.pub=Addison-Wesley&rft.date=1995&rft.isbn=978-0-201-53174-9&rft.aulast=Thomas&rft.aufirst=George+B.&rft.au=Finney%2C+Ross+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcalculusanalytic00geor_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunction+%28mathematics%29" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a 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