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class="vector-toc-numb">1</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forskrift" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Forskrift"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Forskrift</span> </div> </a> <ul id="toc-Forskrift-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definitions-,_dispositions-_og_værdimængde" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitions-,_dispositions-_og_værdimængde"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Definitions-, dispositions- og værdimængde</span> </div> </a> <button aria-controls="toc-Definitions-,_dispositions-_og_værdimængde-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vis/skjul underafsnit Definitions-, dispositions- og værdimængde</span> </button> <ul id="toc-Definitions-,_dispositions-_og_værdimængde-sublist" class="vector-toc-list"> <li id="toc-Definitionsmængde" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitionsmængde"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Definitionsmængde</span> </div> </a> <ul id="toc-Definitionsmængde-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dispositionsmængde" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dispositionsmængde"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Dispositionsmængde</span> </div> </a> <ul id="toc-Dispositionsmængde-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Værdimængde" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Værdimængde"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Værdimængde</span> </div> </a> <ul id="toc-Værdimængde-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In-,_sur-_og_bijektivitet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In-,_sur-_og_bijektivitet"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In-, sur- og bijektivitet</span> </div> </a> <ul id="toc-In-,_sur-_og_bijektivitet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monotoni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Monotoni"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Monotoni</span> </div> </a> <ul id="toc-Monotoni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graf_for_en_funktion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Graf_for_en_funktion"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Graf for en funktion</span> </div> </a> <ul id="toc-Graf_for_en_funktion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Invers_funktion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Invers_funktion"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Invers funktion</span> </div> </a> <ul id="toc-Invers_funktion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regning_med_funktioner" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Regning_med_funktioner"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Regning med funktioner</span> </div> </a> <ul id="toc-Regning_med_funktioner-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Særlige_funktioner" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Særlige_funktioner"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Særlige funktioner</span> </div> </a> <ul id="toc-Særlige_funktioner-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_også" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_også"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Se også</span> </div> </a> <ul id="toc-Se_også-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bog" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bog"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Bog</span> </div> </a> <ul id="toc-Bog-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kildehenvisninger" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kildehenvisninger"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Kildehenvisninger</span> </div> </a> <ul id="toc-Kildehenvisninger-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="Indhold" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Vis/skjul indholdsfortegnelsen" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Vis/skjul indholdsfortegnelsen</span> </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"><span class="mw-page-title-main">Funktion (matematik)</span></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="Gå til en artikel på et andet sprog. Tilgængelig på 120 sprog" > <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-120" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">120 sprog</span> </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"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik) – schweizertysk" lang="gsw" hreflang="gsw" data-title="Funktion (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="schweizertysk" class="interlanguage-link-target"><span>Alemannisch</span></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 – amharisk" lang="am" hreflang="am" data-title="አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="amharisk" class="interlanguage-link-target"><span>አማርኛ</span></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 – aragonsk" lang="an" hreflang="an" data-title="Función matematica" data-language-autonym="Aragonés" data-language-local-name="aragonsk" class="interlanguage-link-target"><span>Aragonés</span></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 – arabisk" lang="ar" hreflang="ar" data-title="دالة" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></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"><span>الدارجة</span></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 – asturisk" lang="ast" hreflang="ast" data-title="Función matemática" data-language-autonym="Asturianu" data-language-local-name="asturisk" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Funksiya_(riyaziyyat)" title="Funksiya (riyaziyyat) – aserbajdsjansk" lang="az" hreflang="az" data-title="Funksiya (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</span></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"><span>Башҡортса</span></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"><span>Žemaitėška</span></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) – belarusisk" lang="be" hreflang="be" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="belarusisk" class="interlanguage-link-target"><span>Беларуская</span></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"><span>Беларуская (тарашкевіца)</span></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 – bulgarsk" lang="bg" hreflang="bg" data-title="Функция" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></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"><span>भोजपुरी</span></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) – bengali" lang="bn" hreflang="bn" data-title="অপেক্ষক (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – bosnisk" lang="bs" hreflang="bs" data-title="Funkcija (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosnisk" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3" title="Funció – catalansk" lang="ca" hreflang="ca" data-title="Funció" data-language-autonym="Català" data-language-local-name="catalansk" class="interlanguage-link-target"><span>Català</span></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) – sorani" lang="ckb" hreflang="ckb" data-title="فانکشن (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Funkce_(matematika)" title="Funkce (matematika) – tjekkisk" lang="cs" hreflang="cs" data-title="Funkce (matematika)" data-language-autonym="Čeština" data-language-local-name="tjekkisk" class="interlanguage-link-target"><span>Čeština</span></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) – tjuvasjisk" lang="cv" hreflang="cv" data-title="Функци (математика)" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjisk" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffwythiant" title="Ffwythiant – walisisk" lang="cy" hreflang="cy" data-title="Ffwythiant" data-language-autonym="Cymraeg" data-language-local-name="walisisk" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik) – tysk" lang="de" hreflang="de" data-title="Funktion (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></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 – græsk" lang="el" hreflang="el" data-title="Συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="græsk" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics) – engelsk" lang="en" hreflang="en" data-title="Function (mathematics)" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></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"><span>Esperanto</span></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) – spansk" lang="es" hreflang="es" data-title="Función (matemática)" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Funktsioon_(matemaatika)" title="Funktsioon (matemaatika) – estisk" lang="et" hreflang="et" data-title="Funktsioon (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estisk" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_(matematika)" title="Funtzio (matematika) – baskisk" lang="eu" hreflang="eu" data-title="Funtzio (matematika)" data-language-autonym="Euskara" data-language-local-name="baskisk" class="interlanguage-link-target"><span>Euskara</span></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 – persisk" lang="fa" hreflang="fa" data-title="تابع" data-language-autonym="فارسی" data-language-local-name="persisk" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Funktio" title="Funktio – finsk" lang="fi" hreflang="fi" data-title="Funktio" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Cakacaka_(fika)" title="Cakacaka (fika) – fijiansk" lang="fj" hreflang="fj" data-title="Cakacaka (fika)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fijiansk" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Funksj%C3%B3n" title="Funksjón – færøsk" lang="fo" hreflang="fo" data-title="Funksjón" data-language-autonym="Føroyskt" data-language-local-name="færøsk" class="interlanguage-link-target"><span>Føroyskt</span></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) – fransk" lang="fr" hreflang="fr" data-title="Fonction (mathématiques)" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Funksion" title="Funksion – nordfrisisk" lang="frr" hreflang="frr" data-title="Funksion" data-language-autonym="Nordfriisk" data-language-local-name="nordfrisisk" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Feidhm_(matamaitic)" title="Feidhm (matamaitic) – irsk" lang="ga" hreflang="ga" data-title="Feidhm (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="irsk" class="interlanguage-link-target"><span>Gaeilge</span></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-kinesisk" lang="gan" hreflang="gan" data-title="函數" data-language-autonym="贛語" data-language-local-name="gan-kinesisk" class="interlanguage-link-target"><span>贛語</span></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"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n" title="Función – galicisk" lang="gl" hreflang="gl" data-title="Función" data-language-autonym="Galego" data-language-local-name="galicisk" class="interlanguage-link-target"><span>Galego</span></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 – hebraisk" lang="he" hreflang="he" data-title="פונקציה" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></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"><span>हिन्दी</span></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"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – kroatisk" lang="hr" hreflang="hr" data-title="Funkcija (matematika)" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" class="interlanguage-link-target"><span>Hrvatski</span></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) – ungarsk" lang="hu" hreflang="hu" data-title="Függvény (matematika)" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</span></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) – armensk" lang="hy" hreflang="hy" data-title="Ֆունկցիա (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armensk" class="interlanguage-link-target"><span>Հայերեն</span></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"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_(matematika)" title="Fungsi (matematika) – indonesisk" lang="id" hreflang="id" data-title="Fungsi (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></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"><span>Ido</span></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) – islandsk" lang="is" hreflang="is" data-title="Fall (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islandsk" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_(matematica)" title="Funzione (matematica) – italiensk" lang="it" hreflang="it" data-title="Funzione (matematica)" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></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) – japansk" lang="ja" hreflang="ja" data-title="関数 (数学)" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></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"><span>Patois</span></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"><span>La .lojban.</span></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) – georgisk" lang="ka" hreflang="ka" data-title="ფუნქცია (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="georgisk" class="interlanguage-link-target"><span>ქართული</span></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) – kabylsk" lang="kab" hreflang="kab" data-title="Tasɣent (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="kabylsk" class="interlanguage-link-target"><span>Taqbaylit</span></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"><span>Kabɩyɛ</span></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) – kasakhisk" lang="kk" hreflang="kk" data-title="Функция (математика)" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" class="interlanguage-link-target"><span>Қазақша</span></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 – koreansk" lang="ko" hreflang="ko" data-title="함수" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></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"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Funktioun_(Mathematik)" title="Funktioun (Mathematik) – luxembourgsk" lang="lb" hreflang="lb" data-title="Funktioun (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxembourgsk" class="interlanguage-link-target"><span>Lëtzebuergesch</span></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"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%95%E0%BA%B3%E0%BA%A5%E0%BA%B2_(%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94)" title="ຕຳລາ (ຄະນິດສາດ) – lao" lang="lo" hreflang="lo" data-title="ຕຳລາ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – litauisk" lang="lt" hreflang="lt" data-title="Funkcija (matematika)" data-language-autonym="Lietuvių" data-language-local-name="litauisk" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Funkcija" title="Funkcija – lettisk" lang="lv" hreflang="lv" data-title="Funkcija" data-language-autonym="Latviešu" data-language-local-name="lettisk" class="interlanguage-link-target"><span>Latviešu</span></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) – makedonsk" lang="mk" hreflang="mk" data-title="Функција (математика)" data-language-autonym="Македонски" data-language-local-name="makedonsk" class="interlanguage-link-target"><span>Македонски</span></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"><span>മലയാളം</span></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) – mongolsk" lang="mn" hreflang="mn" data-title="Функц (математик)" data-language-autonym="Монгол" data-language-local-name="mongolsk" class="interlanguage-link-target"><span>Монгол</span></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) – marathisk" lang="mr" hreflang="mr" data-title="फल (गणित)" data-language-autonym="मराठी" data-language-local-name="marathisk" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi" title="Fungsi – malajisk" lang="ms" hreflang="ms" data-title="Fungsi" data-language-autonym="Bahasa Melayu" data-language-local-name="malajisk" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Funzjonijiet_(matematika)" title="Funzjonijiet (matematika) – maltesisk" lang="mt" hreflang="mt" data-title="Funzjonijiet (matematika)" data-language-autonym="Malti" data-language-local-name="maltesisk" class="interlanguage-link-target"><span>Malti</span></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 – burmesisk" lang="my" hreflang="my" data-title="ဖန်ရှင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmesisk" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Afbillen_(Mathematik)" title="Afbillen (Mathematik) – nedertysk" lang="nds" hreflang="nds" data-title="Afbillen (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="nedertysk" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Functie_(wiskunde)" title="Functie (wiskunde) – nederlandsk" lang="nl" hreflang="nl" data-title="Functie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_funksjon" title="Matematisk funksjon – nynorsk" lang="nn" hreflang="nn" data-title="Matematisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Funksjon_(matematikk)" title="Funksjon (matematikk) – bokmål" lang="nb" hreflang="nb" data-title="Funksjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Aplicacion_(matematicas)" title="Aplicacion (matematicas) – occitansk" lang="oc" hreflang="oc" data-title="Aplicacion (matematicas)" data-language-autonym="Occitan" data-language-local-name="occitansk" class="interlanguage-link-target"><span>Occitan</span></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"><span>Oromoo</span></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) – punjabisk" lang="pa" hreflang="pa" data-title="ਫੰਕਸ਼ਨ (ਹਿਸਾਬ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabisk" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja" title="Funkcja – polsk" lang="pl" hreflang="pl" data-title="Funkcja" data-language-autonym="Polski" data-language-local-name="polsk" class="interlanguage-link-target"><span>Polski</span></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"><span>Piemontèis</span></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"><span>پنجابی</span></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) – portugisisk" lang="pt" hreflang="pt" data-title="Função (matemática)" data-language-autonym="Português" data-language-local-name="portugisisk" class="interlanguage-link-target"><span>Português</span></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"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie" title="Funcție – rumænsk" lang="ro" hreflang="ro" data-title="Funcție" data-language-autonym="Română" data-language-local-name="rumænsk" class="interlanguage-link-target"><span>Română</span></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) – russisk" lang="ru" hreflang="ru" data-title="Функция (математика)" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></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 – jakutisk" lang="sah" hreflang="sah" data-title="Функция. Функция чэрчитэ, суолталарын түмсээнэ" data-language-autonym="Саха тыла" data-language-local-name="jakutisk" class="interlanguage-link-target"><span>Саха тыла</span></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) – siciliansk" lang="scn" hreflang="scn" data-title="Funzioni (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="siciliansk" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics) – skotsk" lang="sco" hreflang="sco" data-title="Function (mathematics)" data-language-autonym="Scots" data-language-local-name="skotsk" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Funkcija" title="Funkcija – serbokroatisk" lang="sh" hreflang="sh" data-title="Funkcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatisk" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></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"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zobrazenie_(matematika)" title="Zobrazenie (matematika) – slovakisk" lang="sk" hreflang="sk" data-title="Zobrazenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – slovensk" lang="sl" hreflang="sl" data-title="Funkcija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Funktio" title="Funktio – enaresamisk" lang="smn" hreflang="smn" data-title="Funktio" data-language-autonym="Anarâškielâ" data-language-local-name="enaresamisk" class="interlanguage-link-target"><span>Anarâškielâ</span></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"><span>ChiShona</span></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"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni" title="Funksioni – albansk" lang="sq" hreflang="sq" data-title="Funksioni" data-language-autonym="Shqip" data-language-local-name="albansk" class="interlanguage-link-target"><span>Shqip</span></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) – serbisk" lang="sr" hreflang="sr" data-title="Функција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="serbisk" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Fungsi_(matematika)" title="Fungsi (matematika) – sundanesisk" lang="su" hreflang="su" data-title="Fungsi (matematika)" data-language-autonym="Sunda" data-language-local-name="sundanesisk" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Funktion" title="Funktion – svensk" lang="sv" hreflang="sv" data-title="Funktion" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></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"><span>Ślůnski</span></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"><span>தமிழ்</span></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"><span>ไทย</span></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"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fonksiyon" title="Fonksiyon – tyrkisk" lang="tr" hreflang="tr" data-title="Fonksiyon" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" class="interlanguage-link-target"><span>Türkçe</span></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) – tatarisk" lang="tt" hreflang="tt" data-title="Функция (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarisk" class="interlanguage-link-target"><span>Татарча / tatarça</span></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"><span>Удмурт</span></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 – uygurisk" lang="ug" hreflang="ug" data-title="فۇنكسىيە" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="uygurisk" class="interlanguage-link-target"><span>ئۇيغۇرچە / Uyghurche</span></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) – ukrainsk" lang="uk" hreflang="uk" data-title="Функція (математика)" data-language-autonym="Українська" data-language-local-name="ukrainsk" class="interlanguage-link-target"><span>Українська</span></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"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Funksiya_(matematika)" title="Funksiya (matematika) – usbekisk" lang="uz" hreflang="uz" data-title="Funksiya (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbekisk" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></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"><span>Vepsän kel’</span></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 – vietnamesisk" lang="vi" hreflang="vi" data-title="Hàm số" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></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"><span>Winaray</span></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-kinesisk" lang="wuu" hreflang="wuu" data-title="函数" data-language-autonym="吴语" data-language-local-name="wu-kinesisk" class="interlanguage-link-target"><span>吴语</span></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"><span>Хальмг</span></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 – jiddisch" lang="yi" hreflang="yi" data-title="פונקציע" data-language-autonym="ייִדיש" data-language-local-name="jiddisch" class="interlanguage-link-target"><span>ייִדיש</span></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) – tamazight" lang="zgh" hreflang="zgh" data-title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ)" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="tamazight" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</span></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 – kinesisk" lang="zh" hreflang="zh" data-title="函数" data-language-autonym="中文" data-language-local-name="kinesisk" class="interlanguage-link-target"><span>中文</span></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"><span>文言</span></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 – min-kinesisk" lang="nan" hreflang="nan" data-title="Hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min-kinesisk" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></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 – kantonesisk" lang="yue" hreflang="yue" data-title="函數" data-language-autonym="粵語" data-language-local-name="kantonesisk" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11348#sitelinks-wikipedia" title="Redigér sproglinks" class="wbc-editpage">Redigér links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Navnerum"> <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 id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Funktion_(matematik)" title="Se indholdssiden [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Diskussion:Funktion_(matematik)" 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.mbox-empty-cell{display:none}.mw-parser-output .compact-ambox table.ambox{border:none;border-collapse:collapse;background-color:transparent;margin:0 0 0 1.6em!important;padding:0!important;width:auto;display:block}body.mediawiki .mw-parser-output .compact-ambox table.mbox-small-left{font-size:100%;width:auto;margin:0}.mw-parser-output .compact-ambox table .mbox-text{padding:0;margin:0}.mw-parser-output .compact-ambox table .mbox-text-span{display:list-item;line-height:1.5em;list-style-type:square}.mw-parser-output .skin-vector .compact-ambox table .mbox-text-span{list-style-type:disc}.mw-parser-output .compact-ambox .hide-when-compact{display:none}.mw-parser-output .template-documentation{clear:both;margin:1em 0 0 0;border:1px solid #aaa;background-color:#ecfcf4;padding:1em}</style> <dl><dd><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Disambig_bordered_fade.svg/19px-Disambig_bordered_fade.svg.png" decoding="async" width="19" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Disambig_bordered_fade.svg/29px-Disambig_bordered_fade.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Disambig_bordered_fade.svg/38px-Disambig_bordered_fade.svg.png 2x" data-file-width="236" data-file-height="185" /></span></span><i> For alternative betydninger, se <a href="/wiki/Funktion" class="mw-disambig" title="Funktion">Funktion</a>.</i> <small>(<a href="/wiki/Speciel:Pr%C3%A6fiksindeks/Funktion" title="Speciel:Præfiksindeks/Funktion">Se også artikler, som begynder med Funktion</a>)</small></dd></dl> <p>En <b>funktion</b> eller <b>afbildning</b> er i <a href="/wiki/Matematik" title="Matematik">matematisk</a> forstand et redskab, der beskriver sammenhængen mellem en såkaldt <i>uafhængig</i> variabel og en anden, såkaldt <i>afhængig</i> variabel. Et hverdagseksempel på en funktion, er sammenhængen mellem hvor meget man bruger sin <a href="/wiki/Telefon" title="Telefon">telefon</a> i løbet af en måned, og hvad man betaler for det: Her er den forbrugte taletid den uafhængige variabel, mens prisen er den afhængige variabel – »afhængig« fordi den afhænger af forbruget. <b>Operator</b> og <b>transformation</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Halmos1960_2-0" class="reference"><a href="#cite_note-Halmos1960-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> er andre navne anvendt for funktion. </p><p>Funktioner er specialtilfælde af det matematiske begreb <i><a href="/wiki/Relation_(matematik)" title="Relation (matematik)">relation</a></i>: Det særlige ved en funktion er, at der til en bestemt værdi af den uafhængige variabel hører <i>én og kun én værdi</i> for den afhængige variabel<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> – andre relationer "har lov til" at knytte mere end én værdi for den afhængige variabel, til hver mulig værdi af den uafhængige variabel. Set i forhold til eksemplet med telefonen betyder det, at der kun er knyttet én pris (værdi af den afhængige variabel) til et bestemt forbrug (værdi af den uafhængige variabel); hvis man brugte den samme mængde taletid i løbet af hver måned, ville man få en telefonregning på det samme beløb måned efter måned (her ses bort fra udlands-takster, prisændringer m.v.). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=1" title="Redigér afsnit: Notation" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=1" title="Edit section's source code: Notation"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kalder man den <a href="/wiki/Uafh%C3%A6ngige_variabel" class="mw-redirect" title="Uafhængige variabel">uafhængige variabel</a> for <i>x</i> og den <a href="/wiki/Afh%C3%A6ngige_variabel" title="Afhængige variabel">afhængige variabel</a> for <i>y</i>, skrives en funktion helt kort som </p> <dl><dd><i>y</i> = <i>f</i>(<i>x</i>)</dd></dl> <p>Denne notation blev første gang brugt af matematikeren <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. Bogstaverne <i>x</i> og <i>y</i> skal blot betragtes som "pladsholdere" for <a href="/wiki/Tal" title="Tal">tal</a>, mens <i>f</i> er et "navn", der gør det muligt at kende flere funktioner fra hinanden; navnet kan være et eller flere bogstaver. Nogle funktioner har "vedtagne" navne (som f.eks. cos for <a href="/wiki/Cosinus" title="Cosinus">cosinus</a>) I mere generelle situationer, f.eks. opgaver og eksempler i lærebøger, bruges oftest <i>f</i>, <i>g</i>, <i>h</i> osv., på samme måde som generelle eksempler på <a href="/wiki/Ligning" title="Ligning">ligninger</a> traditionelt bruger bogstavet <i>x</i> for den ubekendte størrelse. </p> <div class="mw-heading mw-heading2"><h2 id="Forskrift">Forskrift</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=2" title="Redigér afsnit: Forskrift" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=2" title="Edit section's source code: Forskrift"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Den nøjere sammenhæng mellem den uafhængige og afhængige variabel kan være givet ved det der kaldes funktionens <i>forskrift</i>: Det er regneudtryk, der direkte beregner værdien af den afhængige variabel ud fra en given værdi af den uafhængige. Hvis telefonen i ovenstående eksempel koster 25 kr. om måneden i abonnement, plus 50 øre (½ krone) pr. samtaleminut, kan man opstille en forskrift for en funktion <i>f</i>, der beregner den månedlige telefonregning:<br /> </p> <dl><dd><i>f</i>(<i>x</i>) = ½·<i>x</i> + 25<br /></dd></dl> <p>Her står <i>x</i> for det antal minutter man har talt i telefon i en given måned: Hvis man erstatter <i>x'</i>et i regneudtrykket til højre for lighedstegnet med antallet af samtaleminutter, får man et ganske almindeligt regnestykke – resultatet af det regnestykke er det beløb der står på telefonregningen for den måned. En funktion kan have flere forskellige forskrifter som giver de samme værdier for funktionen og det er derfor i den sammenhæng bedre at kalde forskrifterne for repræsentationer af funktionen. En repræsentation af en funktion kan fx være en sum af uendelig mange tal, som ofte er det eneste man kan finde. Det er ikke nødvendigvis sådan at summen er et endeligt tal for alle x værdier af interesse, summen kan blive uendelig. Summen skal så omformes til en anden repræsentation, et endeligt udtryk eller en sum der summeres op til et endeligt tal. Dette er meget ofte muligt. </p> <div class="mw-heading mw-heading2"><h2 id="Definitions-,_dispositions-_og_værdimængde"><span id="Definitions-.2C_dispositions-_og_v.C3.A6rdim.C3.A6ngde"></span>Definitions-, dispositions- og værdimængde</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=3" title="Redigér afsnit: Definitions-, dispositions- og værdimængde" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=3" title="Edit section's source code: Definitions-, dispositions- og værdimængde"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For en given funktion findes der tre <a href="/wiki/M%C3%A6ngde_(matematik)" class="mw-redirect" title="Mængde (matematik)">mængder</a> med særlig relevans: <a href="/wiki/Definitionsm%C3%A6ngde" title="Definitionsmængde">Definitionsmængden</a> til en funktion <i>f</i>, der ofte skrives som Dm(<i>f</i>), <a href="/wiki/Dispositionsm%C3%A6ngde" class="mw-redirect" title="Dispositionsmængde">dispositionsmængden</a>, og <a href="/wiki/V%C3%A6rdim%C3%A6ngde" title="Værdimængde">værdimængden</a> til <i>f</i>, der tilsvarende skrives som Vm(<i>f</i>). Funktionen <i>f</i> siges at være "defineret på mængden Dm(<i>f</i>)". </p> <div class="mw-heading mw-heading3"><h3 id="Definitionsmængde"><span id="Definitionsm.C3.A6ngde"></span>Definitionsmængde</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=4" title="Redigér afsnit: Definitionsmængde" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=4" title="Edit section's source code: Definitionsmængde"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Definitionsmængden til en funktion er mængden af <i>samtlige de værdier</i> for den uafhængige variabel <i>x</i>, som kan bruges med den pågældende funktion. Eksempelvis giver det i ovenstående eksempel med telefonregningen ikke mening at bruge negative tal for antallet af samtaleminutter <i>x</i>. Af den grund må man forlange, at <i>x</i> ikke er et negativt tal, når man bruger telefonregnings-funktionen <i>f</i>; dette gøres ved at begrænse definitionsmængden til alle <a href="/wiki/Reelle_tal" title="Reelle tal">reelle tal</a> større end eller lig med nul. I dette tilfælde skyldes begrænsningen, at visse tal ikke giver mening i forhold til den virkelighed, funktionen bruges til at beskrive: Selv om man ikke kan tale i telefon i negativ tid, er der ingen aritmetiske hindringer for at erstatte <i>x</i> med −5 i føromtalte forskrift for funktionen <i>f</i>. </p><p>Andre funktioner benytter sig af <a href="/w/index.php?title=Operation_(matematik)&action=edit&redlink=1" class="new" title="Operation (matematik) (ikke skrevet endnu)">operationer</a> med begrænsede definitionsmængder, som f.eks. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec93ed8b8435119433e03c7dc50de2848202776a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.519ex; height:5.176ex;" alt="{\displaystyle g(x)={\frac {1}{x}}}"></span></dd></dl> <p>som er defineret på ℝ\{0}, eller </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)={\sqrt {x}}}"> <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"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)={\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1910f357edaffddc6460dea8fbb72a02d6ca58a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.842ex; height:3.009ex;" alt="{\displaystyle h(x)={\sqrt {x}}}"></span></dd></dl> <p>som ikke er defineret for negative tal, hvis der ses bort fra <a href="/wiki/Komplekse_tal" title="Komplekse tal">komplekse tal</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Dispositionsmængde"><span id="Dispositionsm.C3.A6ngde"></span>Dispositionsmængde</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=5" title="Redigér afsnit: Dispositionsmængde" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=5" title="Edit section's source code: Dispositionsmængde"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dispositionsmængden er den mængde, som funktionen afbilder definitionsmængden ind i, og er dermed altid en overmængde til værdimængden. Givet en funktions forskrift og definitionsmængde, men ikke dens dispositionsmængde, er det ikke nødvendigvis muligt at bestemme denne. </p> <div class="mw-heading mw-heading3"><h3 id="Værdimængde"><span id="V.C3.A6rdim.C3.A6ngde"></span>Værdimængde</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=6" title="Redigér afsnit: Værdimængde" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=6" title="Edit section's source code: Værdimængde"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Værdimængden til en funktion er mængden af <i>samtlige værdier</i> som den afhængige variabel kan antage for den funktion. I eksemplet med telefonregningen betaler man som minimum et vist beløb (det faste abonnement) hver måned for at have telefonen – uanset hvor meget eller lidt man bruger telefonen, kommer regningerne aldrig ned under denne minimumsgrænse. Værdimængden Vm(<i>f</i>) til telefonregningsfunktionen <i>f</i> er dermed alle reelle tal, der er større end eller lig med abonnementsprisen. </p><p>Som for definitionsmængden kan der også være aritmetiske årsager til, at visse tal ikke optræder i en funktions værdimængde. I tilfældet med <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec93ed8b8435119433e03c7dc50de2848202776a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.519ex; height:5.176ex;" alt="{\displaystyle g(x)={\frac {1}{x}}}"></span> findes der ikke noget tal <i>x</i>, som i den viste forskrift giver resultatet nul – af den grund hører nul ikke til værdimængden for funktionen <i>g</i>. </p> <div class="mw-heading mw-heading2"><h2 id="In-,_sur-_og_bijektivitet"><span id="In-.2C_sur-_og_bijektivitet"></span>In-, sur- og bijektivitet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=7" title="Redigér afsnit: In-, sur- og bijektivitet" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=7" title="Edit section's source code: In-, sur- og bijektivitet"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Fil:Surjection_Injection_Bijection-fr.svg" class="mw-file-description" title="Idet X og Y er definitions- og dispositionsmængder, er f, g og h henholdsvis surjektiv, injektiv og bijektiv."><img alt="Idet X og Y er definitions- og dispositionsmængder, er f, g og h henholdsvis surjektiv, injektiv og bijektiv." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Surjection_Injection_Bijection-fr.svg/700px-Surjection_Injection_Bijection-fr.svg.png" decoding="async" width="700" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Surjection_Injection_Bijection-fr.svg/1050px-Surjection_Injection_Bijection-fr.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Surjection_Injection_Bijection-fr.svg/1400px-Surjection_Injection_Bijection-fr.svg.png 2x" data-file-width="700" data-file-height="250" /></a><figcaption>Idet X og Y er definitions- og dispositionsmængder, er f, g og h henholdsvis surjektiv, injektiv og bijektiv.</figcaption></figure> <p>En funktion kaldes <a href="/wiki/Injektiv" title="Injektiv">injektiv</a>, hvis hvert element i dispositionsmængden bliver afbildet af højst ét element i definitionsmængden, dvs. hvis </p><p>f(x) = f(x') ⇒ x = x' </p><p>En funktion kaldes <a href="/wiki/Surjektiv" title="Surjektiv">surjektiv</a>, hvis der til ethvert y i dispositionsmængden eksisterer mindst ét element x i definitionsmængden, der opfylder f(x) = y. For en surjektiv funktion er dispositionsmængde og værdimængde en og samme ting. En funktion, der er både injektiv og surjektiv, kaldes <a href="/wiki/Bijektiv" title="Bijektiv">bijektiv</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Monotoni">Monotoni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=8" title="Redigér afsnit: Monotoni" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=8" title="Edit section's source code: Monotoni"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Visse funktioner, som eksemplet med <i>f</i>(<i>x</i>) = ½·<i>x</i> + 25, har den egenskab, at <i>uanset hvilken værdi man vælger</i> for den uafhængige variabel <i>x</i>, så vil en lille stigning i <i>x'</i>s værdi medføre en stigning i den afhængige størrelse <i>y</i> = <i>f</i>(<i>x</i>): Sådan en funktion siges at være <i>monotont voksende</i> – »monotont« eftersom den afhængige variabel aldrig "laver andet" end at stige, uanset hvilken værdi af den uafhængige variabel <i>x</i> man går ud fra.<br /> Med andre funktioner, f.eks. <i>y</i> = <i>g</i>(<i>x</i>) = 1/<i>x</i> vil <i>y</i> konsekvent falde hvis man forøger <i>x</i> en kende: Sådanne funktioner omtaler matematikerne tilsvarende som <i>monotont aftagende</i>. Atter andre funktioner, som f.eks. sinus og cosinus er voksende indenfor bestemte intervaller for <i>x</i>, og aftagende når <i>x</i> falder imellem disse intervaller. </p><p>I forbindelse med <a href="/wiki/Funktionsanalyse_(matematik)" title="Funktionsanalyse (matematik)">funktionsanalyse</a> gør man ofte rede for i hvilke intervaller den analyserede funktion er hhv. voksende og aftagende; denne redegørelse kaldes almindeligvis for funktionens <i><a href="/w/index.php?title=Monotoniforhold&action=edit&redlink=1" class="new" title="Monotoniforhold (ikke skrevet endnu)">monotoniforhold</a></i>. Disse monotoniforhold bestemmes ofte ved hjælp af differentialregning, idet den <a href="/w/index.php?title=Afledt_funktion&action=edit&redlink=1" class="new" title="Afledt funktion (ikke skrevet endnu)">afledede funktions</a> fortegn er positivt, nul og negativt, når den oprindelige funktion er hhv. voksende, konstant og aftagende. </p> <div class="mw-heading mw-heading2"><h2 id="Graf_for_en_funktion">Graf for en funktion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=9" title="Redigér afsnit: Graf for en funktion" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=9" title="Edit section's source code: Graf for en funktion"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:X_cubed_plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/X_cubed_plot.svg/300px-X_cubed_plot.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/X_cubed_plot.svg/450px-X_cubed_plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/X_cubed_plot.svg/600px-X_cubed_plot.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>Eksempel på graf for en funktion, i dette tilfælde:<br /> <i>f</i>(<i>x</i>) = <i>x</i><sup>3</sup></figcaption></figure> <p>Ofte kan man skabe et hurtigt, visuelt overblik over en funktion, ved at tegne en <i>graf</i> (ofte kaldet en <i>kurve</i>) over, hvordan den afhængige variabel udvikler sig hen over et <a href="/wiki/Interval_(matematik)" title="Interval (matematik)">interval</a>. </p><p>Til højre er vist et eksempel på en graf, i dette tilfælde over en funktion <i>f</i>(<i>x</i>) = <i>x</i>³ Her kan man med et enkelt blik se hvordan den afhængige variabel generelt stiger ret kraftigt – undtagen i området omkring <i>x</i>=0, hvor "væksten" tilsyneladende tager en kort "pause". </p> <div class="mw-heading mw-heading2"><h2 id="Invers_funktion">Invers funktion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=10" title="Redigér afsnit: Invers funktion" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=10" title="Edit section's source code: Invers funktion"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hvis en funktion <i>f</i>(<i>x</i>) er bijektiv, findes der en <i>invers</i> eller <i><a href="/wiki/Omvendt_funktion" class="mw-redirect" title="Omvendt funktion">omvendt funktion</a></i> til den pågældende funktion, som skrives <i>f</i><sup>-1</sup>(<i>x</i>): Denne inverse funktion kan tage imod et tal der er beregnet med <i>f</i>(<i>x</i>), og så at sige "regne baglæns" for at finde tilbage til <i>x</i>; Hvis <i>f</i>(x) = y, så vil <i>f</i><sup> -1</sup>(y) være lig med x. Heraf ses, at definitionsmængden til <i>f</i>, bliver værdimængden til <i>f</i><sup> -1</sup>, og vice versa. Altså:<br /> Dm(<i>f</i>) = Vm(<i>f</i><sup> -1</sup>), og Vm(<i>f</i>) = Dm(<i>f</i><sup> -1</sup>). </p><p>Hvis en funktion ikke er <a href="/wiki/Injektiv" title="Injektiv">injektiv</a>, kan den ikke umiddelbart inverteres, idet nogle elementer i dens værdimængde vil blive afbildet af mere end ét element; en <a href="/wiki/Afbildning" class="mw-redirect" title="Afbildning">afbildning</a> af en mængde, derimod, har altid en modsat afbildning. Dog kan man ved at begrænse funktionens definitionsmængde gøre funktionen invertibel, som det er tilfældet med <a href="/wiki/Sinus_(matematisk_funktion)" class="mw-redirect" title="Sinus (matematisk funktion)">sinusfunktionen</a>. Sinusfunktionen er monotont tiltagende i <span class="texhtml">[-π/2, π/2]</span>, som den afbilder i <span class="texhtml">[-1, 1]</span>; derfor kan den her inverteres, hvorved <a href="/wiki/Arcussinus" class="mw-redirect" title="Arcussinus">arcsin</a>-funktionen er defineret. sin<sup>-1</sup> er en udbredt måde på hvilken at udtrykke det samme. </p> <div class="mw-heading mw-heading2"><h2 id="Regning_med_funktioner">Regning med funktioner</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=11" title="Redigér afsnit: Regning med funktioner" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=11" title="Edit section's source code: Regning med funktioner"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Når <i>f</i> og <i>g</i> er funktioner med samme (eller i det mindste overlappende) definitionsmængder kan man indføre nye funktioner <i>f</i>+<i>g</i>, <i>f</i>-<i>g</i>, <i>f</i>·<i>g</i> og <i>f</i>/<i>g</i> ved <i>punktvise</i> operationer. Det vil sige at de nye funktioner er <i>defineret</i> ved </p> <ul><li>(<i>f</i>+<i>g</i>)(x) = <i>f</i>(x) + <i>g</i>(x)</li> <li>(<i>f</i>−<i>g</i>)(x) = <i>f</i>(x) − <i>g</i>(x)</li> <li>(<i>f</i>·<i>g</i>)(x) = <i>f</i>(x)·<i>g</i>(x)</li> <li>(<i>f</i>/<i>g</i>)(x) = <i>f</i>(x)/<i>g</i>(x)</li></ul> <p>Bemærk at den sidste funktion ikke er defineret for <i>x</i> sådan at <i>g(x)=0</i>, også selvom både <i>f</i> og <i>g</i> <i>er</i>; et forhold mellem funktioner er ikke nødvendigvis defineret alle de steder de funktioner den består af er. </p><p>Særlig agtpågivenhed må udvises når skrivemåder som <i>f</i><sup>3</sup> anvendes, altså <a href="/wiki/Potens_(matematik)" title="Potens (matematik)">potenser</a> af en funktion. Notationen kan nemlig fortolkes på to måder: enten som punktvis <a href="/wiki/Multiplikation" title="Multiplikation">multiplikation</a> (som ovenfor) eller som <a href="/wiki/Sammensat_funktion" title="Sammensat funktion">funktionssammensætning</a>. Eksplicit kan <i>f</i><sup>3</sup> betyde </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{3}(x)=(f\cdot f\cdot f)(x)=f(x)\cdot f(x)\cdot f(x)=(f(x))^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <mi>f</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">)</mo> <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> <mo>⋅</mo> <mi>f</mi> <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> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{3}(x)=(f\cdot f\cdot f)(x)=f(x)\cdot f(x)\cdot f(x)=(f(x))^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c2da5ac0134881009ac957caca834d1bf979de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.843ex; height:3.176ex;" alt="{\displaystyle f^{3}(x)=(f\cdot f\cdot f)(x)=f(x)\cdot f(x)\cdot f(x)=(f(x))^{3}}"></span></li></ul> <p>eller </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{3}(x)=(f\circ f\circ f)(x)=f(f(f(x)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>f</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{3}(x)=(f\circ f\circ f)(x)=f(f(f(x)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cb8fa49f6852292fc5ec87160e8585c309e3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.477ex; height:3.176ex;" alt="{\displaystyle f^{3}(x)=(f\circ f\circ f)(x)=f(f(f(x)))}"></span></li></ul> <p>Hvilken betydning der bruges må forstås af sammenhængen. </p> <div class="mw-heading mw-heading2"><h2 id="Særlige_funktioner"><span id="S.C3.A6rlige_funktioner"></span>Særlige funktioner</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=12" title="Redigér afsnit: Særlige funktioner" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=12" title="Edit section's source code: Særlige funktioner"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I det følgende nævnes en række funktioner og kategorier af funktioner, som behandles i særskilte artikler: </p> <ul><li><a href="/wiki/Trigonometrisk_funktion" title="Trigonometrisk funktion">Trigonometriske funktioner</a>, herunder <a href="/wiki/Sinus_(matematik)" title="Sinus (matematik)">sinus</a>, <a href="/wiki/Cosinus" title="Cosinus">cosinus</a> og <a href="/wiki/Tangens" title="Tangens">tangens</a>.</li> <li><a href="/wiki/Logaritme" title="Logaritme">Logaritmer</a></li> <li><a href="/wiki/Polynomium" title="Polynomium">Polynomier</a></li> <li><a href="/w/index.php?title=Potensfunktion&action=edit&redlink=1" class="new" title="Potensfunktion (ikke skrevet endnu)">Potensfunktioner</a></li> <li><a href="/wiki/Eksponentialfunktion" class="mw-redirect" title="Eksponentialfunktion">Eksponentialfunktioner</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Se_også"><span id="Se_ogs.C3.A5"></span>Se også</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=13" title="Redigér afsnit: Se også" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=13" title="Edit section's source code: Se også"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Logisk_operator" title="Logisk operator">Logisk operator</a></li> <li><a href="/wiki/Line%C3%A6r_transformation" class="mw-redirect" title="Lineær transformation">Lineær transformation</a></li> <li><a href="/wiki/Homomorfi" title="Homomorfi">Homomorfi</a></li> <li><a href="/wiki/Sammensat_funktion" title="Sammensat funktion">Sammensat funktion</a></li> <li><a href="/wiki/Vektorfunktion" title="Vektorfunktion">Vektorfunktion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Bog">Bog</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=14" title="Redigér afsnit: Bog" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=14" title="Edit section's source code: Bog"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Holth, Klaus m.fl. (1987): <i>Matematik Grundbog 1</i>. Forlaget Trip, Vejle. <a href="/wiki/Internationalt_Standardbognummer" title="Internationalt Standardbognummer">ISBN</a> <a href="/wiki/Speciel:ISBN-s%C3%B8gning/87-88049-18-3" title="Speciel:ISBN-søgning/87-88049-18-3">87-88049-18-3</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Kildehenvisninger">Kildehenvisninger</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funktion_(matematik)&veaction=edit&section=15" title="Redigér afsnit: Kildehenvisninger" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funktion_(matematik)&action=edit&section=15" title="Edit section's source code: Kildehenvisninger"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r11780210">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.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-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">^</a></span> <span class="reference-text">Fra <a href="https://en.wikipedia.org/wiki/Transformation_(function)" class="extiw" title="en:Transformation (function)">en:Transformation (function)</a> Citat: "...In other areas of mathematics, a transformation may simply be any function, regardless of domain and codomain...."</span> </li> <li id="cite_note-Halmos1960-2"><span class="mw-cite-backlink"><a href="#cite_ref-Halmos1960_2-0">^</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r11752076">.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 a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),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 a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),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:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.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}</style><cite id="CITEREFP._R._Halmos1960" class="citation book cs1">P. R. Halmos (1960). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x6cZBQ9qtgoC&pg=PA30"><i>Naive Set Theory</i></a>. Springer Science & Business Media. s. 30–. <a href="/wiki/Internationalt_Standardbognummer" title="Internationalt Standardbognummer">ISBN</a> <a href="/wiki/Speciel:ISBN-s%C3%B8gning/978-0-387-90092-6" title="Speciel:ISBN-søgning/978-0-387-90092-6"><bdi>978-0-387-90092-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200721165639/https://books.google.com/books?id=x6cZBQ9qtgoC">Arkiveret</a> fra originalen 21. juli 2020<span class="reference-accessdate">. Hentet 21. september 2019</span>.</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=30-&rft.pub=Springer+Science+%26+Business+Media&rft.date=1960&rft.isbn=978-0-387-90092-6&rft.au=P.+R.+Halmos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx6cZBQ9qtgoC%26pg%3DPA30&rfr_id=info%3Asid%2Fda.wikipedia.org%3AFunktion+%28matematik%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">^</a></span> <span class="reference-text">Holth (1987) s. 71</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">^</a></span> <span class="reference-text">Peter Bollerslev: <i>Matematik i læreruddannelsen: Kultur, Kundskab og Kompetence 2</i>, ss. 69-71. Nordisk Forlag A.S. 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