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id="toc-Descrizione-sublist" class="vector-toc-list"> <li id="toc-Esempi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Esempi"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Esempi</span> </div> </a> <ul id="toc-Esempi-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definizione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definizione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definizione</span> </div> </a> <button aria-controls="toc-Definizione-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>Attiva/disattiva la sottosezione Definizione</span> </button> <ul id="toc-Definizione-sublist" class="vector-toc-list"> <li id="toc-Immagine_e_controimmagine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Immagine_e_controimmagine"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Immagine e controimmagine</span> </div> </a> <ul id="toc-Immagine_e_controimmagine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altre_notazioni_per_le_funzioni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Altre_notazioni_per_le_funzioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Altre notazioni per le funzioni</span> </div> </a> <ul id="toc-Altre_notazioni_per_le_funzioni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estensione_e_restrizione_di_una_funzione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estensione_e_restrizione_di_una_funzione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Estensione e restrizione di una funzione</span> </div> </a> <ul id="toc-Estensione_e_restrizione_di_una_funzione-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Funzioni_di_due_o_più_variabili" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funzioni_di_due_o_più_variabili"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Funzioni di due o più variabili</span> </div> </a> <button aria-controls="toc-Funzioni_di_due_o_più_variabili-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>Attiva/disattiva la sottosezione Funzioni di due o più variabili</span> </button> <ul id="toc-Funzioni_di_due_o_più_variabili-sublist" class="vector-toc-list"> <li id="toc-Operazioni_binarie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operazioni_binarie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Operazioni binarie</span> </div> </a> <ul id="toc-Operazioni_binarie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Funzioni_a_più_valori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funzioni_a_più_valori"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Funzioni a più valori</span> </div> </a> <ul id="toc-Funzioni_a_più_valori-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tipologia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tipologia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Tipologia</span> </div> </a> <button aria-controls="toc-Tipologia-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>Attiva/disattiva la sottosezione Tipologia</span> </button> <ul id="toc-Tipologia-sublist" class="vector-toc-list"> <li id="toc-Classificazione_puramente_insiemistica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classificazione_puramente_insiemistica"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Classificazione puramente insiemistica</span> </div> </a> <ul id="toc-Classificazione_puramente_insiemistica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classificazione_delle_funzioni_nell'ambito_della_teoria_della_calcolabilità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classificazione_delle_funzioni_nell'ambito_della_teoria_della_calcolabilità"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Classificazione delle funzioni nell'ambito della teoria della calcolabilità</span> </div> </a> <ul id="toc-Classificazione_delle_funzioni_nell'ambito_della_teoria_della_calcolabilità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classificazione_delle_funzioni_nell'ambito_dell'analisi_matematica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classificazione_delle_funzioni_nell'ambito_dell'analisi_matematica"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Classificazione delle funzioni nell'ambito dell'analisi matematica</span> </div> </a> <ul id="toc-Classificazione_delle_funzioni_nell'ambito_dell'analisi_matematica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alcune_funzioni_notevoli" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alcune_funzioni_notevoli"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Alcune funzioni notevoli</span> </div> </a> <ul id="toc-Alcune_funzioni_notevoli-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funzioni_di_interesse_probabilistico_e_statistico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Funzioni_di_interesse_probabilistico_e_statistico"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Funzioni di interesse probabilistico e statistico</span> </div> </a> <ul id="toc-Funzioni_di_interesse_probabilistico_e_statistico-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operazioni_elementari_su_funzioni_di_variabile_reale_a_valori_reali" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operazioni_elementari_su_funzioni_di_variabile_reale_a_valori_reali"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Operazioni elementari su funzioni di variabile reale a valori reali</span> </div> </a> <button aria-controls="toc-Operazioni_elementari_su_funzioni_di_variabile_reale_a_valori_reali-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>Attiva/disattiva la sottosezione Operazioni elementari su funzioni di variabile reale a valori reali</span> </button> <ul id="toc-Operazioni_elementari_su_funzioni_di_variabile_reale_a_valori_reali-sublist" class="vector-toc-list"> <li id="toc-Composizione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composizione"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Composizione</span> </div> </a> <ul id="toc-Composizione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Traslazione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Traslazione"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Traslazione</span> </div> </a> <ul id="toc-Traslazione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simmetria" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simmetria"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Simmetria</span> </div> </a> <ul id="toc-Simmetria-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Funzione (matematica)</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="Vai a una voce in un'altra lingua. Disponibile in 120 lingue" > <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 lingue</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) - tedesco svizzero" lang="gsw" hreflang="gsw" data-title="Funktion (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="tedesco svizzero" 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 - amarico" lang="am" hreflang="am" data-title="አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="amarico" 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 - aragonese" lang="an" hreflang="an" data-title="Función matematica" data-language-autonym="Aragonés" data-language-local-name="aragonese" 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 - arabo" lang="ar" hreflang="ar" data-title="دالة" data-language-autonym="العربية" data-language-local-name="arabo" 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 - arabo marocchino" lang="ary" hreflang="ary" data-title="دالة" data-language-autonym="الدارجة" data-language-local-name="arabo marocchino" 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 - asturiano" lang="ast" hreflang="ast" data-title="Función matemática" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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) - azerbaigiano" lang="az" hreflang="az" data-title="Funksiya (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" 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) - baschiro" lang="ba" hreflang="ba" data-title="Функция (математика)" data-language-autonym="Башҡортса" data-language-local-name="baschiro" 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 - samogitico" lang="sgs" hreflang="sgs" data-title="Funkcėjė" data-language-autonym="Žemaitėška" data-language-local-name="samogitico" 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) - bielorusso" lang="be" hreflang="be" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="bielorusso" 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 - bulgaro" lang="bg" hreflang="bg" data-title="Функция" data-language-autonym="Български" data-language-local-name="bulgaro" 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) - bengalese" lang="bn" hreflang="bn" data-title="অপেক্ষক (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengalese" 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) - bosniaco" lang="bs" hreflang="bs" data-title="Funkcija (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosniaco" 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 - catalano" lang="ca" hreflang="ca" data-title="Funció" data-language-autonym="Català" data-language-local-name="catalano" 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) - curdo centrale" lang="ckb" hreflang="ckb" data-title="فانکشن (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="curdo centrale" 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) - ceco" lang="cs" hreflang="cs" data-title="Funkce (matematika)" data-language-autonym="Čeština" data-language-local-name="ceco" 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) - ciuvascio" lang="cv" hreflang="cv" data-title="Функци (математика)" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" 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 - gallese" lang="cy" hreflang="cy" data-title="Ffwythiant" data-language-autonym="Cymraeg" data-language-local-name="gallese" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Funktion_(matematik)" title="Funktion (matematik) - danese" lang="da" hreflang="da" data-title="Funktion (matematik)" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik) - tedesco" lang="de" hreflang="de" data-title="Funktion (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="tedesco" 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 - greco" lang="el" hreflang="el" data-title="Συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="greco" 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) - inglese" lang="en" hreflang="en" data-title="Function (mathematics)" data-language-autonym="English" data-language-local-name="inglese" 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) - spagnolo" lang="es" hreflang="es" data-title="Función (matemática)" data-language-autonym="Español" data-language-local-name="spagnolo" 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) - estone" lang="et" hreflang="et" data-title="Funktsioon (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estone" 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) - basco" lang="eu" hreflang="eu" data-title="Funtzio (matematika)" data-language-autonym="Euskara" data-language-local-name="basco" 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 - persiano" lang="fa" hreflang="fa" data-title="تابع" data-language-autonym="فارسی" data-language-local-name="persiano" 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 - finlandese" lang="fi" hreflang="fi" data-title="Funktio" data-language-autonym="Suomi" data-language-local-name="finlandese" 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) - figiano" lang="fj" hreflang="fj" data-title="Cakacaka (fika)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="figiano" 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 - faroese" lang="fo" hreflang="fo" data-title="Funksjón" data-language-autonym="Føroyskt" data-language-local-name="faroese" 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) - francese" lang="fr" hreflang="fr" data-title="Fonction (mathématiques)" data-language-autonym="Français" data-language-local-name="francese" 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 - frisone settentrionale" lang="frr" hreflang="frr" data-title="Funksion" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" 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) - irlandese" lang="ga" hreflang="ga" data-title="Feidhm (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="irlandese" 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" lang="gan" hreflang="gan" data-title="函數" data-language-autonym="贛語" data-language-local-name="gan" 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 - galiziano" lang="gl" hreflang="gl" data-title="Función" data-language-autonym="Galego" data-language-local-name="galiziano" 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 - ebraico" lang="he" hreflang="he" data-title="פונקציה" data-language-autonym="עברית" data-language-local-name="ebraico" 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 - hindi figiano" lang="hif" hreflang="hif" data-title="Function" data-language-autonym="Fiji Hindi" data-language-local-name="hindi figiano" 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) - croato" lang="hr" hreflang="hr" data-title="Funkcija (matematika)" data-language-autonym="Hrvatski" data-language-local-name="croato" 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) - ungherese" lang="hu" hreflang="hu" data-title="Függvény (matematika)" data-language-autonym="Magyar" data-language-local-name="ungherese" 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) - armeno" lang="hy" hreflang="hy" data-title="Ֆունկցիա (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeno" 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) - indonesiano" lang="id" hreflang="id" data-title="Fungsi (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" 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) - islandese" lang="is" hreflang="is" data-title="Fall (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</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) - giapponese" lang="ja" hreflang="ja" data-title="関数 (数学)" data-language-autonym="日本語" data-language-local-name="giapponese" 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) - creolo giamaicano" lang="jam" hreflang="jam" data-title="Fongshan (matimatix)" data-language-autonym="Patois" data-language-local-name="creolo giamaicano" 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) - georgiano" lang="ka" hreflang="ka" data-title="ფუნქცია (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="georgiano" 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) - cabilo" lang="kab" hreflang="kab" data-title="Tasɣent (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="cabilo" 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) - kazako" lang="kk" hreflang="kk" data-title="Функция (математика)" data-language-autonym="Қазақша" data-language-local-name="kazako" 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 - coreano" lang="ko" hreflang="ko" data-title="함수" data-language-autonym="한국어" data-language-local-name="coreano" 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 - latino" lang="la" hreflang="la" data-title="Functio" data-language-autonym="Latina" data-language-local-name="latino" 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) - lussemburghese" lang="lb" hreflang="lb" data-title="Funktioun (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="lussemburghese" 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) - lombardo" lang="lmo" hreflang="lmo" data-title="Fonzion (matematega)" data-language-autonym="Lombard" data-language-local-name="lombardo" 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) - lituano" lang="lt" hreflang="lt" data-title="Funkcija (matematika)" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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 - lettone" lang="lv" hreflang="lv" data-title="Funkcija" data-language-autonym="Latviešu" data-language-local-name="lettone" 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) - macedone" lang="mk" hreflang="mk" data-title="Функција (математика)" data-language-autonym="Македонски" data-language-local-name="macedone" 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) - mongolo" lang="mn" hreflang="mn" data-title="Функц (математик)" data-language-autonym="Монгол" data-language-local-name="mongolo" 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) - marathi" lang="mr" hreflang="mr" data-title="फल (गणित)" data-language-autonym="मराठी" data-language-local-name="marathi" 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 - malese" lang="ms" hreflang="ms" data-title="Fungsi" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" 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) - maltese" lang="mt" hreflang="mt" data-title="Funzjonijiet (matematika)" data-language-autonym="Malti" data-language-local-name="maltese" 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 - birmano" lang="my" hreflang="my" data-title="ဖန်ရှင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" 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) - basso tedesco" lang="nds" hreflang="nds" data-title="Afbillen (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="basso tedesco" 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) - olandese" lang="nl" hreflang="nl" data-title="Functie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="olandese" 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 - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Matematisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese 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) - norvegese bokmål" lang="nb" hreflang="nb" data-title="Funksjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese 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) - occitano" lang="oc" hreflang="oc" data-title="Aplicacion (matematicas)" data-language-autonym="Occitan" data-language-local-name="occitano" 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) - punjabi" lang="pa" hreflang="pa" data-title="ਫੰਕਸ਼ਨ (ਹਿਸਾਬ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" 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 - polacco" lang="pl" hreflang="pl" data-title="Funkcja" data-language-autonym="Polski" data-language-local-name="polacco" 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 - piemontese" lang="pms" hreflang="pms" data-title="Fonsion" data-language-autonym="Piemontèis" data-language-local-name="piemontese" 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) - portoghese" lang="pt" hreflang="pt" data-title="Função (matemática)" data-language-autonym="Português" data-language-local-name="portoghese" 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 - rumeno" lang="ro" hreflang="ro" data-title="Funcție" data-language-autonym="Română" data-language-local-name="rumeno" 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) - russo" lang="ru" hreflang="ru" data-title="Функция (математика)" data-language-autonym="Русский" data-language-local-name="russo" 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 - sacha" lang="sah" hreflang="sah" data-title="Функция. Функция чэрчитэ, суолталарын түмсээнэ" data-language-autonym="Саха тыла" data-language-local-name="sacha" 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) - siciliano" lang="scn" hreflang="scn" data-title="Funzioni (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="siciliano" 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) - scozzese" lang="sco" hreflang="sco" data-title="Function (mathematics)" data-language-autonym="Scots" data-language-local-name="scozzese" 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 - serbo-croato" lang="sh" hreflang="sh" data-title="Funkcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" 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) - slovacco" lang="sk" hreflang="sk" data-title="Zobrazenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovacco" 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) - sloveno" lang="sl" hreflang="sl" data-title="Funkcija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="sloveno" 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 - sami di Inari" lang="smn" hreflang="smn" data-title="Funktio" data-language-autonym="Anarâškielâ" data-language-local-name="sami di Inari" 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) - somalo" lang="so" hreflang="so" data-title="Shaqada (xisaabta)" data-language-autonym="Soomaaliga" data-language-local-name="somalo" 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 - albanese" lang="sq" hreflang="sq" data-title="Funksioni" data-language-autonym="Shqip" data-language-local-name="albanese" 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) - serbo" lang="sr" hreflang="sr" data-title="Функција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="serbo" 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) - sundanese" lang="su" hreflang="su" data-title="Fungsi (matematika)" data-language-autonym="Sunda" data-language-local-name="sundanese" 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 - svedese" lang="sv" hreflang="sv" data-title="Funktion" data-language-autonym="Svenska" data-language-local-name="svedese" 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 - slesiano" lang="szl" hreflang="szl" data-title="Funkcyjo" data-language-autonym="Ślůnski" data-language-local-name="slesiano" 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) - thailandese" lang="th" hreflang="th" data-title="ฟังก์ชัน (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="thailandese" 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 - turco" lang="tr" hreflang="tr" data-title="Fonksiyon" data-language-autonym="Türkçe" data-language-local-name="turco" 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) - tataro" lang="tt" hreflang="tt" data-title="Функция (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="tataro" 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 - uiguro" lang="ug" hreflang="ug" data-title="فۇنكسىيە" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="uiguro" 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) - ucraino" lang="uk" hreflang="uk" data-title="Функція (математика)" data-language-autonym="Українська" data-language-local-name="ucraino" 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) - uzbeco" lang="uz" hreflang="uz" data-title="Funksiya (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" 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) - vepso" lang="vep" hreflang="vep" data-title="Funkcii (matematik)" data-language-autonym="Vepsän kel’" data-language-local-name="vepso" 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 - vietnamita" lang="vi" hreflang="vi" data-title="Hàm số" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" 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" lang="wuu" hreflang="wuu" data-title="函数" data-language-autonym="吴语" data-language-local-name="wu" 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 - yiddish" lang="yi" hreflang="yi" 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class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Funcao_venn.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Funcao_venn.svg/220px-Funcao_venn.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Funcao_venn.svg/330px-Funcao_venn.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Funcao_venn.svg/440px-Funcao_venn.svg.png 2x" data-file-width="400" data-file-height="300" /></a><figcaption>Rappresentazione di una funzione che associa a ogni elemento del dominio X un elemento del codominio Y</figcaption></figure> <p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, una <b>funzione</b> è una <a href="/wiki/Relazione_(matematica)" title="Relazione (matematica)">relazione</a> tra due <a href="/wiki/Insieme" title="Insieme">insiemi</a>, chiamati <a href="/wiki/Dominio_e_codominio" title="Dominio e codominio">dominio e codominio</a> della funzione, che associa a ogni elemento del dominio uno e un solo elemento del codominio. </p><p>Se il dominio e il codominio della funzione <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}"> <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></span><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}"></span> sono rispettivamente indicati con <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 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></span><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}"></span> e <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 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></span><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}"></span>, la relazione si indica con <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\colon 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\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> e l’elemento che <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}"> <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></span><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}"></span> associa a <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 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></span><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}"></span> si indica con <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(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></span><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)}"></span> (si pronuncia "effe di x"). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Descrizione">Descrizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=1" title="Modifica la sezione Descrizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=1" title="Edit section's source code: Descrizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La parola "funzione" non si riferisce alla sola associazione di elementi, ma alla terna: associazione di elementi, dominio e codominio. Specificare un'associazione non definisce una funzione: occorre specificare anche dominio e codominio. Infatti, due funzioni che hanno una "stessa" associazione di elementi ma diverso dominio o diverso codominio sono funzioni diverse. Per esempio, la funzione che associa a ogni <a href="/wiki/Numero_naturale" title="Numero naturale">numero <i>naturale</i></a> il suo quadrato (<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\colon \mathbb {N} \to \mathbb {N} ,f(n)=n^{2}}"> <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">N</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {N} \to \mathbb {N} ,f(n)=n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501b3a272de3ca7dde212cd11c6f6faef82289c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.347ex; height:3.176ex;" alt="{\displaystyle f\colon \mathbb {N} \to \mathbb {N} ,f(n)=n^{2}}"></span>) è diversa dalla funzione che associa a un <a href="/wiki/Numero_intero" title="Numero intero">numero <i>intero</i></a> il quadrato di quel numero (<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\colon \mathbb {Z} \to \mathbb {N} ,f(z)=z^{2}}"> <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">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {Z} \to \mathbb {N} ,f(z)=z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd11cc758be2aac7d9aee451f4ca3b59601b328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.608ex; height:3.176ex;" alt="{\displaystyle f\colon \mathbb {Z} \to \mathbb {N} ,f(z)=z^{2}}"></span>); infatti la prima è <a href="/wiki/Funzione_iniettiva" title="Funzione iniettiva">iniettiva</a> la seconda no. In molti casi, quando il dominio e il codominio sono chiari dal contesto, una funzione viene espressa indicando solamente la relazione e sottintendendo dominio e codominio. </p><p>Si dice che <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 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></span><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}"></span> è l'argomento della funzione, oppure un valore della <a href="/wiki/Variabile_(matematica)" title="Variabile (matematica)">variabile</a> indipendente, mentre <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 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></span><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)}"></span> è un valore della <a href="/wiki/Variabili_dipendenti_e_indipendenti" title="Variabili dipendenti e indipendenti">variabile dipendente</a> della funzione. </p><p>Sinonimi del termine "funzione" sono <i>applicazione</i> e <i>mappa</i>. Il termine <i><a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione</a></i> viene utilizzato spesso in ambito geometrico per indicare una funzione <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\colon X\rightarrow 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>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\rightarrow X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc989b42bdfd5a6cbf6f901340156fce82a8fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.887ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow X}"></span> invertibile e che conserva le proprietà geometriche di <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 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></span><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}"></span>, mentre <i>operatore</i> è talvolta utilizzato nella trattazione di <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">funzioni lineari</a> tra <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazi vettoriali</a>. </p><p>Le funzioni hanno un ruolo molto importante in tutte le <a href="/wiki/Scienze_esatte" class="mw-redirect" title="Scienze esatte">scienze esatte</a>. Il concetto di <a href="/wiki/Dipendenza_funzionale" title="Dipendenza funzionale">dipendenza funzionale</a> tra due grandezze sostituisce infatti, all'interno delle teorie fisiche e matematiche, quello di causa-effetto, che, al contrario del precedente, non riguarda enti teorici ma direttamente gli elementi della realtà concreta. Se si afferma, ad esempio, che la <a href="/wiki/Pressione" title="Pressione">pressione</a> di una certa quantità di <a href="/wiki/Gas_perfetto" class="mw-redirect" title="Gas perfetto">gas perfetto</a> è funzione della sua <a href="/wiki/Temperatura" title="Temperatura">temperatura</a> e del suo <a href="/wiki/Volume" title="Volume">volume</a> si sta facendo un'affermazione interna a un modello <a href="/wiki/Termodinamica" title="Termodinamica">termodinamico</a>, mentre il rapporto di causa-effetto che viene individuato fra le tre grandezze dipende in modo sostanziale dalle possibilità di intervento concreto su di esse. Rimanendo a questo esempio, il valore della pressione viene visto più spesso come conseguenza del valore degli altri due parametri, poiché è generalmente molto più facile intervenire sul volume e sulla temperatura che direttamente sulla pressione. </p> <div class="mw-heading mw-heading3"><h3 id="Esempi">Esempi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=2" title="Modifica la sezione Esempi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=2" title="Edit section's source code: Esempi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gli esempi più semplici di funzione sono quelli per cui sia il dominio che il codominio sono <a href="/wiki/Numero" title="Numero">insiemi numerici</a>. Per esempio, se a ogni numero naturale si associa il doppio di tale numero, si ha una funzione, il cui dominio è l'insieme dei numeri naturali e il cui codominio è l'insieme dei numeri naturali pari. </p><p>Tuttavia si parla di funzione anche quando il dominio o il codominio, o entrambi, non sono insiemi numerici. Se, per esempio, a ogni triangolo del piano si associa il cerchio in esso inscritto, si ha ugualmente una funzione, in quanto per ogni triangolo esiste uno e un solo cerchio in esso inscritto. </p><p>Inoltre spesso si parla di funzioni con più argomenti, o con più valori: per esempio la funzione che alle coordinate <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 x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span> di un punto nello spazio fa corrispondere temperatura <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> e pressione <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> dell'aria. In tal caso, la funzione ha in realtà sempre un solo argomento, che è la terna <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 (x,y,z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd88635b67420fb82863e226c5e6f772a58fd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.097ex; height:2.843ex;" alt="{\displaystyle (x,y,z),}"></span> e ha sempre un solo valore, che è la coppia <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 (T,P).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T,P).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea296a8062d2fe8de04831a06b3cddf0823a3b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.872ex; height:2.843ex;" alt="{\displaystyle (T,P).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=3" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=3" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dati due insiemi non vuoti <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 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></span><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}"></span> e <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 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></span><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}"></span>, si chiama funzione da <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 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></span><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}"></span> in <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 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></span><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}"></span> una relazione <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}"> <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></span><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}"></span> tale che per ogni <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 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></span><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}"></span> esiste uno ed un solo elemento <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 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></span><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}"></span> tale che <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 (x,y)\in f}"> <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>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e505597712f896aa68749192e4462a3b9a813a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.448ex; height:2.843ex;" alt="{\displaystyle (x,y)\in f}"></span>. Tale elemento tradizionalmente si denota con <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(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></span><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)}"></span>: in altre parole, invece di scrivere <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 (x,y)\in f}"> <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>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e505597712f896aa68749192e4462a3b9a813a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.448ex; height:2.843ex;" alt="{\displaystyle (x,y)\in f}"></span> si può usare la scrittura più tradizionale: </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 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></span><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).}"></span></dd></dl> <p>Il fatto che <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}"> <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></span><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}"></span> sia una funzione da <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 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></span><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}"></span> in <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 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></span><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}"></span> che associa a <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 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></span><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}"></span> l’elemento <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(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></span><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)}"></span> si può esprimere con la scrittura: </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 {\begin{matrix}f\colon &X&\longrightarrow &Y\\&x&\longmapsto &f(x)\end{matrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>f</mi> <mo>:</mo> </mtd> <mtd> <mi>X</mi> </mtd> <mtd> <mo stretchy="false">⟶</mo> </mtd> <mtd> <mi>Y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>x</mi> </mtd> <mtd> <mo stretchy="false">⟼</mo> </mtd> <mtd> <mi>f</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{matrix}f\colon &X&\longrightarrow &Y\\&x&\longmapsto &f(x)\end{matrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b6b451d953d69ab7c1d9a9cfdbd75117b44352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.495ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}f\colon &X&\longrightarrow &Y\\&x&\longmapsto &f(x)\end{matrix}}.}"></span></dd></dl> <p>L’insieme <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 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></span><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}"></span> (da cui la funzione <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}"> <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></span><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}"></span> “prende” i valori) è il <a href="/wiki/Dominio_(matematica)" class="mw-redirect" title="Dominio (matematica)">dominio</a> della funzione <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}"> <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></span><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}"></span>, mentre l’insieme <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 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></span><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}"></span> (in cui si trovano i valori “restituiti” dalla funzione <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}"> <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></span><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}"></span>) è il <a href="/wiki/Codominio" class="mw-redirect" title="Codominio">codominio</a> della funzione <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}"> <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></span><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}"></span>.<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> </p><p>Le espressioni “prendere un valore” e “restituire un valore” fanno riferimento a un modello meccanico delle funzioni, rappresentate come meccanismi che, fornito loro un elemento del dominio, lo “trasformano” nel corrispondente elemento del codominio. </p> <div class="mw-heading mw-heading3"><h3 id="Immagine_e_controimmagine">Immagine e controimmagine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=4" title="Modifica la sezione Immagine e controimmagine" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=4" title="Edit section's source code: Immagine e controimmagine"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Immagine_(matematica)" title="Immagine (matematica)">Immagine (matematica)</a></b> e <b><a href="/wiki/Controimmagine" title="Controimmagine">Controimmagine</a></b>.</span></div> </div> <p>Data una funzione <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}"> <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></span><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}"></span> di dominio <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 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></span><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}"></span> e codominio <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 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></span><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,}"></span> comunque scelto un elemento <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 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></span><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}"></span> del dominio, si chiama <i>immagine</i> di <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 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></span><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}"></span> il corrispondente elemento del codominio, indicato con <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(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e889a7f3ee0216318cbf9f3478208fa1404cc51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x).}"></span> Analogamente, se <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 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></span><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}"></span> è un elemento del codominio che sia immagine di un elemento <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 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></span><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}"></span> del dominio, cioè se <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 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></span><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)}"></span>, si dice che <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 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></span><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}"></span> è una <a href="/wiki/Controimmagine" title="Controimmagine">controimmagine</a> di <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 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}"></span> Mentre a ogni elemento del dominio di <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}"> <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></span><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}"></span> è assegnata una e una sola immagine, è possibile che un elemento nel codominio possegga diverse controimmagini, o che non ne possieda affatto. Si definisce quindi “controimmagine” dell’elemento <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 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></span><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}"></span> l’insieme </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 f^{-1}(y)=\{x\in X\,|\,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> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <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\,|\,f(x)=y\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c34066a7c7a404991958a4d6bb1840240ce355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.931ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)=\{x\in X\,|\,f(x)=y\}.}"></span></dd></dl> <p>Se <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^{-1}(y)\neq \varnothing }"> <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> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(y)\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675117b599f5ef0a02fcc6a6677248bef278fc1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.525ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)\neq \varnothing }"></span> per ogni <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 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></span><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}"></span> si dice che <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}"> <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></span><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}"></span> è <i><a href="/wiki/Funzione_suriettiva" title="Funzione suriettiva">suriettiva</a></i>, mentre se <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^{-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></span><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)}"></span> contiene al più un elemento per ogni <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 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></span><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}"></span> si dice che <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}"> <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></span><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}"></span> è <i><a href="/wiki/Funzione_iniettiva" title="Funzione iniettiva">iniettiva</a></i>. Se valgono entrambe le condizioni, <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}"> <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></span><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}"></span> è detta <i><a href="/wiki/Funzione_biunivoca" class="mw-redirect" title="Funzione biunivoca">biiettiva</a></i> o <i>biunivoca</i>. </p><p>L’insieme </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 \{y\in Y\,|\,\exists x\in X:y=f(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{y\in Y\,|\,\exists x\in X:y=f(x)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b239f920980665e1da164c841b6b8f3ca2393cc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.567ex; height:2.843ex;" alt="{\displaystyle \{y\in Y\,|\,\exists x\in X:y=f(x)\}}"></span></dd></dl> <p>degli elementi <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 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></span><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}"></span> del codominio per i quali esiste almeno un <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 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></span><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}"></span> nel dominio che ha <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 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></span><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}"></span> come immagine è detto <i>immagine</i> di <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}"> <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></span><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}"></span> e si denota con <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 {\textrm {Im}}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Im</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textrm {Im}}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a3f3c0799f8b50c0b1c40b27cc645d54cfb831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.863ex; height:2.843ex;" alt="{\displaystyle {\textrm {Im}}(f)}"></span> o con <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(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></span><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)}"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Altre_notazioni_per_le_funzioni">Altre notazioni per le funzioni</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=5" title="Modifica la sezione Altre notazioni per le funzioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=5" title="Edit section's source code: Altre notazioni per le funzioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per il valore di una funzione <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}"> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> corrispondente a un elemento <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 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></span><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}"></span>, denotabile con la notazione tradizionale <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(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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span>, vengono usate anche altre due scritture. </p><p>Per quella che chiamiamo <b>notazione a funzione prefissa</b> si pone </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 Fx:=F(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>x</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 Fx:=F(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c48ccbfc8b20f597f733cd8341395d573ab5d324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.342ex; height:2.843ex;" alt="{\displaystyle Fx:=F(x).}"></span></dd></dl> <p>Per quella che chiamiamo <b>notazione a funzione suffissale</b> si pone </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 xF:=F(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>F</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 xF:=F(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe0941ab103a2d6047f17e352f349ec6e978482" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.342ex; height:2.843ex;" alt="{\displaystyle xF:=F(x).}"></span></dd></dl> <p>A volte al posto delle parentesi tonde si usano parentesi quadrate: </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 F[x]=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> <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 F[x]=F(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee333382b34d62d42908d557b0eaa9cd206be23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.989ex; height:2.843ex;" alt="{\displaystyle F[x]=F(x).}"></span></dd></dl> <p>In questo modo si evitano confusioni con le parentesi che indicano l’<a href="/wiki/Ordine_delle_operazioni" title="Ordine delle operazioni">ordine delle operazioni</a>. Questa notazione è usata da alcuni programmi di calcolo simbolico. </p><p>Nelle funzioni di due variabili si usa talvolta la <b>notazione infissa</b>, ossia </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 xFy:=F(x,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>F</mi> <mi>y</mi> <mo>:=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xFy:=F(x,y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e37783f862928fe0635a0c6181f04bfb4eaadccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.687ex; height:2.843ex;" alt="{\displaystyle xFy:=F(x,y),}"></span></dd></dl> <p>ad esempio, nelle usuali operazioni di addizione e sottrazione si usa scrivere <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 x+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+y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb4441dccedb5ede51a213408b17cf83eec9a27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x+y}"></span> e <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 x-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-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3129cb3620bd9f38d0304a0fca719644d7d2d265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x-y}"></span> invece di <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 +(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e47edc6f342061601fb9ee87fb5ba79106583e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.137ex; height:2.843ex;" alt="{\displaystyle +(x,y)}"></span> e <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 -(x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(x,y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2f366eb69899440a2c3496bfea7868121776c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.783ex; height:2.843ex;" alt="{\displaystyle -(x,y).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Estensione_e_restrizione_di_una_funzione">Estensione e restrizione di una funzione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=6" title="Modifica la sezione Estensione e restrizione di una funzione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=6" title="Edit section's source code: Estensione e restrizione di una funzione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data una funzione <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\colon A\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon A\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420253f5ee7cb1af8f70fcc5afc79096f3a66acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.443ex; height:2.509ex;" alt="{\displaystyle f\colon A\to Y}"></span> e un insieme <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 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></span><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}"></span> tale che <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 A\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826569be03f873b81cdc6f12637ef5520c369d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.822ex; height:2.176ex;" alt="{\displaystyle A\subset X}"></span>, si dice che la funzione <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 {\tilde {f}}\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af66d88a03b53f943ceb852f9c886422dca6b424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.1ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}\colon X\to Y}"></span> è un’<i>estensione di <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}"> <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></span><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}"></span> all’insieme <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 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></span><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}"></span></i> se </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 {\tilde {f}}\circ i=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>∘</mo> <mi>i</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}\circ i=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb40e3731337862b6b93ba3ecb156f45443f792a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.073ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}\circ i=f}"></span></dd></dl> <p>dove <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 i\colon A\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\colon A\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a64c55f7fe66fe8e1eb4b91766d3de6a11809f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.174ex; height:2.176ex;" alt="{\displaystyle i\colon A\to X}"></span> è l’<a href="/wiki/Funzione_inclusione#Inclusione" class="mw-redirect" title="Funzione inclusione">inclusione</a> di <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> in <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 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></span><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}"></span>, data da <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 i(a)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i(a)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ee292ca5a72e7f2c54707487af2d45e26f4b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.17ex; height:2.843ex;" alt="{\displaystyle i(a)=a}"></span>. Si dice viceversa che <i><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}"> <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></span><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}"></span> è la <a href="/wiki/Restrizione_di_una_funzione" title="Restrizione di una funzione">restrizione</a> di </i><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 {\tilde {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6cb99679a4b79cb5ca3c242811bd91220c91f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}}"></span><i> all’insieme <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></i>. </p><p>La restrizione di una funzione <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}"> <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></span><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}"></span> a un insieme <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> contenuto nel suo dominio è abitualmente indicata con <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{\big |}_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f{\big |}_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/909818853a73257700c48945df8c50565b8f5518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.39ex; height:3.343ex;" alt="{\displaystyle f{\big |}_{A}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Funzioni_di_due_o_più_variabili"><span id="Funzioni_di_due_o_pi.C3.B9_variabili"></span>Funzioni di due o più variabili</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=7" title="Modifica la sezione Funzioni di due o più variabili" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=7" title="Edit section's source code: Funzioni di due o più variabili"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quando il dominio di una funzione <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}"> <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></span><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}"></span> è il <a href="/wiki/Prodotto_cartesiano" title="Prodotto cartesiano">prodotto cartesiano</a> di due o più <a href="/wiki/Insieme" title="Insieme">insiemi</a>, e dunque la funzione agisce su <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-uple di elementi di insiemi, allora l'immagine del vettore di questi elementi <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 \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> viene indicata con la notazione </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 f(x_{i}).}"> <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"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{i}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f41f5fd1e6520b9f1d2ffeb8504a11b13bbf32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.864ex; height:2.843ex;" alt="{\displaystyle f(x_{i}).}"></span></dd></dl> <p>In questo caso la funzione viene anche chiamata <b>funzione di <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a></b>. A tal proposito in fisica si parla di <a href="/wiki/Campo_(fisica)" title="Campo (fisica)">campo</a>. </p><p>Per esempio, si consideri la funzione di <a href="/wiki/Moltiplicazione" title="Moltiplicazione">moltiplicazione</a> che associa un vettore di due <a href="/wiki/Numeri_naturali" class="mw-redirect" title="Numeri naturali">numeri naturali</a> <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 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></span><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}"></span> e <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 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></span><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}"></span> al loro prodotto: <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(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></span><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}"></span>. Questa funzione può essere definita formalmente come avente per dominio <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 \mathbb {N} \times \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \times \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b25f8c4219e47350185be7ebdc5250c8d59ab35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} \times \mathbb {N} }"></span>, l'insieme di tutte le coppie di numeri naturali; si noti inoltre che in questo caso la funzione è simmetrica rispetto alle componenti del vettore: <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(x,y)=f(y,x)}"> <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>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=f(y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dff528696cdd74cda2d1590ed7c21f7fb57e705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.312ex; height:2.843ex;" alt="{\displaystyle f(x,y)=f(y,x)}"></span> e quindi si tratta di una <b>funzione di un <a href="/wiki/Insieme" title="Insieme">insieme</a></b> <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\{x,y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\{x,y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c3048cc14a3b17c7bf3b3d0b6c1d605c027fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.123ex; height:2.843ex;" alt="{\displaystyle f\{x,y\}}"></span> in cui non importa cioè l'ordine degli elementi. Sono inoltre possibili anche altri raggruppamenti delle variabili: per esempio risulta estremamente importante nello studio dei <a href="/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">sistemi</a> di <a href="/wiki/Equazione_differenziale" title="Equazione differenziale">equazioni differenziali</a> la teoria della <b><a href="/wiki/Funzione_di_matrice" title="Funzione di matrice">funzione di matrice</a></b>: </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 f(x_{ij}).}"> <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"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{ij}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575132b5d550166d328bf3a3de27c9042c4f3cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.542ex; height:3.009ex;" alt="{\displaystyle f(x_{ij}).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Operazioni_binarie">Operazioni binarie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=8" title="Modifica la sezione Operazioni binarie" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=8" title="Edit section's source code: Operazioni binarie"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Molte <a href="/wiki/Operazione_binaria" title="Operazione binaria">operazioni binarie</a> dell'<a href="/wiki/Aritmetica" title="Aritmetica">aritmetica</a>, come l'<a href="/wiki/Addizione" title="Addizione">addizione</a> e la <a href="/wiki/Moltiplicazione" title="Moltiplicazione">moltiplicazione</a>, sono funzioni dal prodotto cartesiano <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 \mathbb {Z} \times \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \times \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463bb631669181cf8a8950ac822ada5eb0542ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} \times \mathbb {Z} }"></span> a valori in <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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>, e vengono descritte tramite la <a href="/wiki/Notazione_infissa" title="Notazione infissa">notazione infissa</a>: si scrive cioè <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 x+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+y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb4441dccedb5ede51a213408b17cf83eec9a27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x+y}"></span> (e non <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 +(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e47edc6f342061601fb9ee87fb5ba79106583e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.137ex; height:2.843ex;" alt="{\displaystyle +(x,y)}"></span>) per descrivere l'immagine della coppia <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 (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></span><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)}"></span> tramite l'operazione <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span>.<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> </p><p>Questa notazione è stata generalizzata dall'<a href="/wiki/Algebra" title="Algebra">algebra</a> moderna, per definire <a href="/wiki/Struttura_algebrica" title="Struttura algebrica">strutture algebriche</a> come ad esempio quella di <a href="/wiki/Gruppo_(matematica)" title="Gruppo (matematica)">gruppo</a>, come un insieme <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 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></span><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}"></span> dotato di alcune operazioni binarie aventi determinate proprietà. </p> <div class="mw-heading mw-heading2"><h2 id="Funzioni_a_più_valori"><span id="Funzioni_a_pi.C3.B9_valori"></span>Funzioni a più valori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=9" title="Modifica la sezione Funzioni a più valori" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=9" title="Edit section's source code: Funzioni a più valori"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se il <i>codominio</i> di una funzione <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}"> <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></span><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}"></span> è il <a href="/wiki/Prodotto_cartesiano" title="Prodotto cartesiano">prodotto cartesiano</a> di due o più insiemi, questa può essere indicata come <b><a href="/wiki/Funzione_vettoriale" title="Funzione vettoriale">funzione vettoriale</a></b>. Tali variabili spesso vengono aggregate in un <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a>; a tal proposito in <a href="/wiki/Fisica" title="Fisica">fisica</a> si parla di <a href="/wiki/Campo_vettoriale" title="Campo vettoriale">campo vettoriale</a>. </p><p>Un esempio tipico è dato da una <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione lineare</a> del <a href="/wiki/Piano_(geometria)" title="Piano (geometria)">piano</a>, ad esempio: </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 (x,y)\to (y,x).}"> <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 stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\to (y,x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214b1f1a854e0723890a4909305bd166634e5640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.918ex; height:2.843ex;" alt="{\displaystyle (x,y)\to (y,x).}"></span></dd></dl> <p>Una funzione è invece detta <b><a href="/wiki/Funzione_polidroma" title="Funzione polidroma">polidroma</a></b> nel caso in cui esista almeno un elemento del dominio cui corrisponde più di un elemento del codominio. In effetti tali funzioni non rientrano nella definizione data inizialmente, ma in alcuni campi (ad esempio in <a href="/wiki/Analisi_complessa" title="Analisi complessa">analisi complessa</a>) essa viene estesa proprio in questo senso. Un esempio di funzione polidroma è la <a href="/wiki/Radice_quadrata" title="Radice quadrata">radice quadrata</a> di un <a href="/wiki/Numero_reale" title="Numero reale">numero reale</a> positivo, che può essere descritta come una funzione </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 \mathbb {R} _{+}\to \mathbb {P} (\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}\to \mathbb {P} (\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391feec867dc1092fed5cf8799b2aaad4b44208a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.711ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{+}\to \mathbb {P} (\mathbb {R} )}"></span></dd></dl> <p>che associa a ogni numero reale positivo l'<a href="/wiki/Insieme" title="Insieme">insieme</a> delle sue due radici quadrate. Un esempio analogo è il <a href="/wiki/Logaritmo" title="Logaritmo">logaritmo</a> definito sull'insieme dei <a href="/wiki/Numeri_complessi" class="mw-redirect" title="Numeri complessi">numeri complessi</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="Tipologia">Tipologia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=10" title="Modifica la sezione Tipologia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=10" title="Edit section's source code: Tipologia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nella matematica e sostanzialmente in tutte le sue applicazioni si incontrano numerosi tipi di funzioni, che si presentano anche con caratteristiche molto diverse, e che vengono classificate seguendo diversi criteri. </p> <div class="mw-heading mw-heading3"><h3 id="Classificazione_puramente_insiemistica">Classificazione puramente insiemistica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=11" title="Modifica la sezione Classificazione puramente insiemistica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=11" title="Edit section's source code: Classificazione puramente insiemistica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Funzione_iniettiva" title="Funzione iniettiva">Funzione iniettiva</a></li> <li><a href="/wiki/Funzione_suriettiva" title="Funzione suriettiva">Funzione suriettiva</a></li> <li><a href="/wiki/Corrispondenza_biunivoca" title="Corrispondenza biunivoca">Funzione biunivoca</a></li> <li><a href="/wiki/Endofunzione" title="Endofunzione">Endofunzione</a></li> <li><a href="/wiki/Permutazione" title="Permutazione">Permutazione</a></li> <li><a href="/wiki/Involuzione_(teoria_degli_insiemi)" title="Involuzione (teoria degli insiemi)">Involuzione</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Classificazione_delle_funzioni_nell'ambito_della_teoria_della_calcolabilità"><span id="Classificazione_delle_funzioni_nell.27ambito_della_teoria_della_calcolabilit.C3.A0"></span>Classificazione delle funzioni nell'ambito della teoria della calcolabilità</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=12" title="Modifica la sezione Classificazione delle funzioni nell'ambito della teoria della calcolabilità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=12" title="Edit section's source code: Classificazione delle funzioni nell'ambito della teoria della calcolabilità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Teoria_della_calcolabilit%C3%A0" title="Teoria della calcolabilità">Teoria della calcolabilità</a></b>.</span></div> </div> <ul><li><a href="/wiki/Funzione_ricorsiva_primitiva" title="Funzione ricorsiva primitiva">Funzione ricorsiva primitiva</a></li> <li><a href="/wiki/Funzione_calcolabile" title="Funzione calcolabile">Funzione calcolabile</a></li> <li><a href="/wiki/Funzione_ricorsiva" title="Funzione ricorsiva">Funzione ricorsiva</a> (secondo la <a href="/wiki/Tesi_di_Church-Turing" title="Tesi di Church-Turing">Tesi di Church-Turing</a>, funzioni ricorsive e funzioni calcolabili sono la stessa cosa)</li> <li><a href="/w/index.php?title=Funzione_ricorsiva_totale&action=edit&redlink=1" class="new" title="Funzione ricorsiva totale (la pagina non esiste)">Funzione ricorsiva totale</a></li> <li><a href="/w/index.php?title=Funzione_enumerativa&action=edit&redlink=1" class="new" title="Funzione enumerativa (la pagina non esiste)">Funzione enumerativa</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Classificazione_delle_funzioni_nell'ambito_dell'analisi_matematica"><span id="Classificazione_delle_funzioni_nell.27ambito_dell.27analisi_matematica"></span>Classificazione delle funzioni nell'ambito dell'analisi matematica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=13" title="Modifica la sezione Classificazione delle funzioni nell'ambito dell'analisi matematica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=13" title="Edit section's source code: Classificazione delle funzioni nell'ambito dell'analisi matematica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Analisi_matematica" title="Analisi matematica">Analisi matematica</a></b>.</span></div> </div> <ul><li><a href="/wiki/Funzione_di_variabile_reale#Funzioni_algebriche" title="Funzione di variabile reale">Funzione algebrica</a> e <a href="/wiki/Funzione_di_variabile_reale#Funzioni_trascendenti" title="Funzione di variabile reale">funzione trascendente</a></li> <li><a href="/wiki/Funzione_analitica" title="Funzione analitica">Funzione analitica</a></li> <li><a href="/wiki/Funzione_antiolomorfa" title="Funzione antiolomorfa">Funzione antiolomorfa</a></li> <li><a href="/wiki/Funzione_armonica" title="Funzione armonica">Funzione armonica</a></li> <li><a href="/wiki/Funzione_cilindrica" title="Funzione cilindrica">Funzione cilindrica</a></li> <li><a href="/wiki/Funzione_concava" title="Funzione concava">Funzione concava</a></li> <li><a href="/wiki/Funzione_continua" title="Funzione continua">Funzione continua</a></li> <li><a href="/wiki/Funzione_convessa" title="Funzione convessa">Funzione convessa</a></li> <li><a href="/wiki/Funzione_crescente" class="mw-redirect" title="Funzione crescente">Funzione crescente</a>, <a href="/wiki/Funzione_decrescente" class="mw-redirect" title="Funzione decrescente">funzione decrescente</a> e <a href="/wiki/Funzione_monotona" title="Funzione monotona">funzione monotona</a></li> <li><a href="/wiki/Funzione_differenziabile" title="Funzione differenziabile">Funzione differenziabile</a></li> <li><a href="/wiki/Funzione_integrabile" title="Funzione integrabile">Funzione integrabile</a></li> <li><a href="/wiki/Integrale" title="Integrale">Funzione integrale</a></li> <li><a href="/wiki/Funzione_lipschitziana" title="Funzione lipschitziana">Funzione lipschitziana</a></li> <li><a href="/wiki/Funzione_liscia" title="Funzione liscia">Funzione liscia</a></li> <li><a href="/wiki/Funzione_meromorfa" title="Funzione meromorfa">Funzione meromorfa</a></li> <li><a href="/wiki/Funzione_olomorfa" title="Funzione olomorfa">Funzione olomorfa</a></li> <li><a href="/wiki/Funzione_pari" class="mw-redirect" title="Funzione pari">Funzione pari</a> e <a href="/wiki/Funzione_dispari" class="mw-redirect" title="Funzione dispari">funzione dispari</a></li> <li><a href="/wiki/Funzione_polinomiale" class="mw-redirect" title="Funzione polinomiale">Funzione polinomiale</a></li> <li><a href="/wiki/Funzione_razionale_fratta" class="mw-redirect" title="Funzione razionale fratta">Funzione razionale fratta</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Alcune_funzioni_notevoli">Alcune funzioni notevoli</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=14" title="Modifica la sezione Alcune funzioni notevoli" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=14" title="Edit section's source code: Alcune funzioni notevoli"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Funzione_beta_di_Eulero" title="Funzione beta di Eulero">Funzione Beta</a>, <a href="/wiki/Funzione_Gamma" title="Funzione Gamma">funzione Gamma</a>, <a href="/wiki/Funzione_zeta_di_Riemann" title="Funzione zeta di Riemann">funzione zeta di Riemann</a></li> <li><a href="/wiki/Funzione_trigonometrica" title="Funzione trigonometrica">Funzioni trigonometriche</a>: <a href="/wiki/Seno_(trigonometria)" class="mw-redirect" title="Seno (trigonometria)">seno</a>, <a href="/wiki/Coseno" title="Coseno">coseno</a>, <a href="/wiki/Tangente_(matematica)" title="Tangente (matematica)">tangente</a>, <a href="/wiki/Cotangente" title="Cotangente">cotangente</a>, <a href="/wiki/Secante_(trigonometria)" title="Secante (trigonometria)">secante</a>, <a href="/wiki/Cosecante" title="Cosecante">cosecante</a></li> <li><a href="/wiki/Funzione_identit%C3%A0" title="Funzione identità">Funzione identità</a></li> <li><a href="/wiki/Funzione_esponenziale" title="Funzione esponenziale">Funzione esponenziale</a>, <a href="/wiki/Logaritmo" title="Logaritmo">logaritmo</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Funzioni_di_interesse_probabilistico_e_statistico">Funzioni di interesse probabilistico e statistico</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=15" title="Modifica la sezione Funzioni di interesse probabilistico e statistico" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=15" title="Edit section's source code: Funzioni di interesse probabilistico e statistico"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Funzione_di_ripartizione" title="Funzione di ripartizione">Funzione di ripartizione</a></li> <li><a href="/wiki/Funzione_di_probabilit%C3%A0" title="Funzione di probabilità">Funzione di probabilità</a></li> <li><a href="/wiki/Funzione_di_densit%C3%A0_di_probabilit%C3%A0" title="Funzione di densità di probabilità">Funzione di densità</a></li> <li><a href="/wiki/Funzione_generatrice_dei_momenti" title="Funzione generatrice dei momenti">Funzione generatrice dei momenti</a></li> <li><a href="/wiki/Funzione_caratteristica_(teoria_della_probabilit%C3%A0)" title="Funzione caratteristica (teoria della probabilità)">Funzione caratteristica</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Operazioni_elementari_su_funzioni_di_variabile_reale_a_valori_reali">Operazioni elementari su funzioni di variabile reale a valori reali</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=16" title="Modifica la sezione Operazioni elementari su funzioni di variabile reale a valori reali" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=16" title="Edit section's source code: Operazioni elementari su funzioni di variabile reale a valori reali"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data una funzione <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(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></span><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)}"></span> di variabile reale a valori reali e una costante <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 c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47ef490c028656282fd8b18c44c4939bbfff750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {R} }"></span>, su di essa sono applicabili le operazioni aritmetiche elementari ovvero <a href="/wiki/Somma_algebrica" title="Somma algebrica">somma</a>, <a href="/wiki/Sottrazione" title="Sottrazione">sottrazione</a>, <a href="/wiki/Moltiplicazione" title="Moltiplicazione">moltiplicazione</a>, <a href="/wiki/Divisione_(matematica)" title="Divisione (matematica)">divisione</a>, <a href="/wiki/Elevamento_a_potenza" class="mw-redirect" title="Elevamento a potenza">elevamento a potenza</a>, <a href="/wiki/Radicale_(matematica)" title="Radicale (matematica)">radice <i>n</i>-esima</a> ovvero: </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 z(x)=f(x)+c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</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>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)+c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175ee2e720f96fc13d2dc7980a8b2a7bdfcf1da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.237ex; height:2.843ex;" alt="{\displaystyle z(x)=f(x)+c,}"></span></dd></dl> <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 z(x)=f(x)-c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</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>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)-c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c61ac09f922725f06ec7f751ba6bcb783b5fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.237ex; height:2.843ex;" alt="{\displaystyle z(x)=f(x)-c,}"></span></dd></dl> <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 z(x)=f(x)\cdot c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</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>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)\cdot c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f88331f5639c4c83bb7f2cd9e89fe94e07da9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.076ex; height:2.843ex;" alt="{\displaystyle z(x)=f(x)\cdot c,}"></span></dd></dl> <p>se <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 c\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be27396bd0e62003728d08329a8767eee94409e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.268ex; height:2.676ex;" alt="{\displaystyle c\neq 0}"></span> si ha anche </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 z(x)={\frac {f(x)}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <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> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)={\frac {f(x)}{c}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e3c789366e3b3c9021e0d5d3278c92a6343b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.226ex; height:5.676ex;" alt="{\displaystyle z(x)={\frac {f(x)}{c}},}"></span></dd></dl> <p>se <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(x)>0}"> <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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af29e26aaac6c969d32c8afac1f33ae703f442c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)>0}"></span> si ha anche </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 z(x)=f(x)^{c},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)^{c},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/467aef2eb846b6433885ac15058574fe4a315c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.334ex; height:2.843ex;" alt="{\displaystyle z(x)=f(x)^{c},}"></span></dd></dl> <p>e se <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> intero maggiore di 1, e se <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> pari si deve avere anche <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(x)\geq 0}"> <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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2635ab62e1e5dc91ebf9789bb2e8d636415df57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)\geq 0}"></span>, si ha anche </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 z(x)={\sqrt[{c}]{f(x)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)={\sqrt[{c}]{f(x)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd218d7d0f3fd3ad68ba0a2f5583160fddc5068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.714ex; height:4.843ex;" alt="{\displaystyle z(x)={\sqrt[{c}]{f(x)}}.}"></span></dd></dl> <p>Date due funzioni <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(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></span><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)}"></span> e <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"></span> di variabile reale a valori reali sono applicabili le operazioni aritmetiche elementari di cui sopra ossia: </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 z(x)=f(x)+g(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</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>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)+g(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/216dbd2ea451bd5fd2e2120908b7ff36a5114bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.485ex; height:2.843ex;" alt="{\displaystyle z(x)=f(x)+g(x),}"></span></dd></dl> <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 z(x)=f(x)-g(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</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>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)-g(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07317d824b3ce33c7b314bafdac337dbbece3890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.485ex; height:2.843ex;" alt="{\displaystyle z(x)=f(x)-g(x),}"></span></dd></dl> <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 z(x)=f(x)\cdot g(x);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</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>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)\cdot g(x);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4fc6edea3205f992f5cdeefda4842d5b557bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.324ex; height:2.843ex;" alt="{\displaystyle z(x)=f(x)\cdot g(x);}"></span></dd></dl> <p>se <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)\neq 0}"> <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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304c689556e38a33dde280823338d0ef90735d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.516ex; height:2.843ex;" alt="{\displaystyle g(x)\neq 0}"></span> si ha anche </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 z(x)={\frac {f(x)}{g(x)}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <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 z(x)={\frac {f(x)}{g(x)}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36ed05a9f4ab90606a1604479d254b74d0a0ba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.226ex; height:6.509ex;" alt="{\displaystyle z(x)={\frac {f(x)}{g(x)}};}"></span></dd></dl> <p>se <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(x)>0}"> <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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af29e26aaac6c969d32c8afac1f33ae703f442c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)>0}"></span> (o <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(x)\geq 0}"> <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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2635ab62e1e5dc91ebf9789bb2e8d636415df57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)\geq 0}"></span> nel caso in cui <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(x)=0\land g(x)>0}"> <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> <mn>0</mn> <mo>∧</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=0\land g(x)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c5559e6ae374cf3e73b55c7ea0754051d07919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.777ex; height:2.843ex;" alt="{\displaystyle f(x)=0\land g(x)>0}"></span>) si ha anche </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 z(x)=f(x)^{g(x)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z(x)=f(x)^{g(x)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312e0cb9d9831838690b195aa4c40e4207775b15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.631ex; height:3.343ex;" alt="{\displaystyle z(x)=f(x)^{g(x)}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Composizione">Composizione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=17" title="Modifica la sezione Composizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=17" title="Edit section's source code: Composizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Date due funzioni <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\colon 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\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> e <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\colon 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\colon Y\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6268a0bd7f3016093edf824725f1ffcba89f8064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.218ex; height:2.509ex;" alt="{\displaystyle g\colon Y\to Z}"></span> si può definire la loro <b><a href="/wiki/Composizione_di_funzioni" title="Composizione di funzioni">composizione</a></b>: questa è definita applicando prima <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}"> <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></span><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}"></span> ad <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 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></span><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}"></span> e quindi applicando <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> al risultato <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(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></span><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)}"></span>. </p><p>Questa nuova funzione viene denotata con <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\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></span><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}"></span> (si legge: "g composta f"<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> oppure "g composto f"<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>). Riconducendoci alla notazione tradizionale con le due notazioni il risultato della precedente composizione applicato all'elemento <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 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></span><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}"></span> del dominio si può scrivere<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </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\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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb826fa65348334253bb28916f43350594be15e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.11ex; height:2.843ex;" alt="{\displaystyle (g\circ f)(x)=g[f(x)].}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Traslazione">Traslazione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=18" title="Modifica la sezione Traslazione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=18" title="Edit section's source code: Traslazione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data una funzione <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(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></span><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)}"></span> di variabile reale a valori reali e una costante <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 c\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47ef490c028656282fd8b18c44c4939bbfff750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {R} }"></span>: </p> <ul><li>la sua traslata rispetto all'asse <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 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></span><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}"></span> verso destra è <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(x-c);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x-c);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8574f4da721c3754ae08c20b7d5669af061bf3ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.912ex; height:2.843ex;" alt="{\displaystyle f(x-c);}"></span></li> <li>la sua traslata rispetto all'asse <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 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></span><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}"></span> verso sinistra è <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(x+c);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+c);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3426d4b0965992dbee0b82f094b70f1c99c54f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.912ex; height:2.843ex;" alt="{\displaystyle f(x+c);}"></span></li> <li>la sua traslata rispetto all'asse <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 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></span><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}"></span> verso l'alto è <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(x)+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>c</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)+c;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb81c9e93c570afb1cf96beba34a860a03d81df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.912ex; height:2.843ex;" alt="{\displaystyle f(x)+c;}"></span></li> <li>la sua traslata rispetto all'asse <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 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></span><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}"></span> verso il basso è <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(x)-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>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)-c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26019dff06c1e191e9901601f9ecaceb5cdd838a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.912ex; height:2.843ex;" alt="{\displaystyle f(x)-c.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Simmetria">Simmetria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=19" title="Modifica la sezione Simmetria" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=19" title="Edit section's source code: Simmetria"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data una funzione <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(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></span><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)}"></span> di variabile reale a valori reali: </p> <ul><li>la simmetrica di <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(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></span><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)}"></span> rispetto all'asse <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 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></span><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}"></span> è <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(-x);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-x);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bdbb0fb395accca386ccaa78855dcae51afa4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.873ex; height:2.843ex;" alt="{\displaystyle f(-x);}"></span></li> <li>la simmetrica di <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(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></span><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)}"></span> rispetto all'asse <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 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></span><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}"></span> è <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(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 -f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802d1e230f96dd136ceef2182fd103b289e69f27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.873ex; height:2.843ex;" alt="{\displaystyle -f(x).}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=20" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=20" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Andrea Bacciotti, Fulvio Ricci, <a rel="nofollow" class="external text" href="http://books.google.it/books?id=ZBdhfJky_D4C&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false"><span style="font-style:italic;">Analisi matematica</span></a>, Liguori Editore Srl, 1994, <b>p. 63</b>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Andrea Bacciotti, Fulvio Ricci, <a rel="nofollow" class="external text" href="http://books.google.it/books?id=ZBdhfJky_D4C&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false"><span style="font-style:italic;">Analisi matematica</span></a>, Liguori Editore Srl, 1994, <b>p. 67</b>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Francesca Dalla Volta, Marco Rigoli, <a rel="nofollow" class="external text" href="http://books.google.it/books?id=LNe7WBMPAmcC&lpg=PP1&pg=PP1#v=onepage&q=&f=true"><span style="font-style:italic;">Elementi di matematica discreta e algebra lineare</span></a>, Pearson Paravia Bruno Mondad, 2007, <b>p. 169</b>.</cite></span> </li> <li id="cite_note-4"><a href="#cite_ref-4"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Gazzola Ferrero Zanotti, <a rel="nofollow" class="external text" href="http://books.google.it/books?id=ZMMSm9L3NjYC&lpg=PP1&pg=PA127#v=onepage&q=&f=true"><span style="font-style:italic;">Elementi di analisi superiore per la fisica e l'ingegneria</span></a>, Società Editrice Esculapio, 2007, <b>pp. 127-128</b>.</cite></span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFBertsch,_Dal_Passo,_Giacomelli">Bertsch, Dal Passo, Giacomelli</a>, p. 51</cite>.</span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFPagani,_Salsa">Pagani, Salsa</a>, p. 33</cite>.</span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Andrea Bacciotti, Fulvio Ricci, <a rel="nofollow" class="external text" href="http://books.google.it/books?id=ZBdhfJky_D4C&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false"><span style="font-style:italic;">Analisi matematica</span></a>, Liguori Editore Srl, 1994, <b>pp. 69-70</b>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=21" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=21" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation libro" style="font-style:normal"> Andrea Bacciotti e Fulvio Ricci, <span style="font-style:italic;">Analisi matematica</span>, Liguori Editore, 1995, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/9788820723972" title="Speciale:RicercaISBN/9788820723972">9788820723972</a>.</cite></li> <li><cite id="CITEREFPagani,_Salsa" class="citation libro" style="font-style:normal"> Carlo Domenico Pagani e Sandro Salsa, <span style="font-style:italic;">Analisi matematica, volume 1</span>, Masson, 1995, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-214-0079-4" title="Speciale:RicercaISBN/88-214-0079-4">88-214-0079-4</a>.</cite></li> <li><cite id="CITEREFBertsch,_Dal_Passo,_Giacomelli" class="citation libro" style="font-style:normal"> Michiel Bertsch, Roberta Dal Passo e Lorenzo Giacomelli, <span style="font-style:italic;">Analisi matematica</span>, McGraw-Hill, 2011, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-386-6281-2" title="Speciale:RicercaISBN/978-88-386-6281-2">978-88-386-6281-2</a>.</cite></li> <li><a href="/wiki/Nicola_Fusco_(matematico)" title="Nicola Fusco (matematico)">Nicola Fusco</a>, <a href="/wiki/Paolo_Marcellini" title="Paolo Marcellini">Paolo Marcellini</a>, <a href="/wiki/Carlo_Sbordone" title="Carlo Sbordone">Carlo Sbordone</a>, <i>Lezioni di Analisi Matematica Due</i>, Zanichelli, 2020, <a href="/wiki/Speciale:RicercaISBN/9788808520203" class="internal mw-magiclink-isbn">ISBN 9788808520203</a></li> <li><a href="/wiki/Paolo_Marcellini" title="Paolo Marcellini">Paolo Marcellini</a>, <a href="/wiki/Carlo_Sbordone" title="Carlo Sbordone">Carlo Sbordone</a>, <i>Analisi Matematica Uno</i>, Liguori Editore, 1998, <a href="/wiki/Speciale:RicercaISBN/9788820728199" class="internal mw-magiclink-isbn">ISBN 9788820728199</a></li> <li><a href="/wiki/Enrico_Giusti" title="Enrico Giusti">Enrico Giusti</a>, <i>Analisi Matematica 1</i>, Boringhieri, 2002, <a href="/wiki/Speciale:RicercaISBN/9788833956848" class="internal mw-magiclink-isbn">ISBN 9788833956848</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=22" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=22" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dominio_e_codominio" title="Dominio e codominio">Dominio e codominio</a></li> <li><a href="/wiki/Immagine_(matematica)" title="Immagine (matematica)">Immagine (matematica)</a></li> <li><a href="/wiki/Grafico_di_una_funzione" title="Grafico di una funzione">Grafico di una funzione</a></li> <li><a href="/wiki/Funzione_di_variabile_reale" title="Funzione di variabile reale">Funzione di variabile reale</a></li> <li><a href="/wiki/Funzione_definita_a_tratti" title="Funzione definita a tratti">Funzione definita a tratti</a></li> <li><a href="/wiki/Parte_positiva_e_parte_negativa_di_una_funzione" title="Parte positiva e parte negativa di una funzione">Parte positiva e parte negativa di una funzione</a></li> <li><a href="/wiki/Studio_di_funzione" title="Studio di funzione">Studio di funzione</a></li> <li><a href="/wiki/Storia_della_nozione_di_funzione_matematica" title="Storia della nozione di funzione matematica">Storia della nozione di funzione matematica</a></li> <li><a href="/wiki/Analisi_matematica" title="Analisi matematica">Analisi matematica</a>, <a href="/wiki/Integrale" title="Integrale">integrale</a>, <a href="/wiki/Derivata" title="Derivata">derivata</a></li> <li><a href="/wiki/Funzione_speciale" title="Funzione speciale">Funzione speciale</a></li> <li><a href="/wiki/Funzione_periodica" title="Funzione periodica">Funzione periodica</a></li> <li><a href="/wiki/Funzionale" title="Funzionale">Funzionale</a></li> <li><a href="/wiki/Funzione_parziale" title="Funzione parziale">Funzione parziale</a></li> <li><a href="/wiki/Applicazione_parziale" title="Applicazione parziale">Applicazione parziale</a></li> <li><a href="/wiki/Serie_di_funzioni" title="Serie di funzioni">Serie di funzioni</a></li> <li><a href="/wiki/Serie_formale_di_potenze" title="Serie formale di potenze">Serie formale di potenze</a></li> <li><a href="/wiki/Equazione_differenziale" title="Equazione differenziale">Equazione differenziale</a></li> <li><a href="/wiki/Equazione_funzionale" title="Equazione funzionale">Equazione funzionale</a></li> <li><a href="/wiki/Teoria_delle_categorie" title="Teoria delle categorie">Teoria delle categorie</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=23" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=23" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wiktionary.org/wiki/funzione" class="extiw" title="wikt:funzione">Wikizionario</a></li> <li class="" title=""><a href="https://it.wikiversity.org/wiki/Funzioni" class="extiw" title="v:Funzioni">Wikiversità</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Functions (mathematics)"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Functions_(mathematics)?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/funzione" class="extiw" title="wikt:funzione">funzione</a></b>»</li> <li><span typeof="mw:File"><a href="https://it.wikiversity.org/wiki/" title="Collabora a Wikiversità"><img alt="Collabora a Wikiversità" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/18px-Wikiversity_logo_2017.svg.png" decoding="async" width="18" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/36px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> <a href="https://it.wikiversity.org/wiki/" class="extiw" title="v:">Wikiversità</a> contiene risorse sulla <b><a href="https://it.wikiversity.org/wiki/Funzioni" class="extiw" title="v:Funzioni">funzione</a></b></li> <li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sulla <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Functions_(mathematics)?uselang=it">funzione</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funzione_(matematica)&veaction=edit&section=24" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funzione_(matematica)&action=edit&section=24" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFTreccani.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/funzione"><span style="font-style:italic;">funzione</span></a>, su <span style="font-style:italic;">Treccani.it – Enciclopedie on line</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P3365" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_Italiana" class="citation libro" style="font-style:normal"> Leonida Tonelli e Salvatore Pincherle, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/funzione_(Enciclopedia-Italiana)/"><span style="font-style:italic;">FUNZIONE</span></a>, in <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Enciclopedia Italiana</a></span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 1932.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P4223" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFVocabolario_Treccani" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/vocabolario/funzione_res-f4951c70-dff0-11eb-94e0-00271042e8d9"><span style="font-style:italic;">Funzione</span></a>, su <span style="font-style:italic;"><a href="/wiki/Vocabolario_Treccani" title="Vocabolario Treccani">Vocabolario Treccani</a></span>, Thesaurus, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2018.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P5844" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSapere.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.sapere.it/enciclopedia/funzione+(matematica).html"><span style="font-style:italic;">funzione (matematica)</span></a>, su <span style="font-style:italic;">sapere.it</span>, <a href="/wiki/De_Agostini" title="De Agostini">De Agostini</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P6706" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_dei_ragazzi" class="citation libro" style="font-style:normal"> Luca Dell'Aglio, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/funzioni_(Enciclopedia-dei-ragazzi)/"><span style="font-style:italic;">Funzioni</span></a>, in <span style="font-style:italic;">Enciclopedia dei ragazzi</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2004-2006.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P9983" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/funzione_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">Funzione</span></a>, in <span style="font-style:italic;">Enciclopedia della Matematica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2013.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P9621" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFDizionario_di_Economia_e_Finanza" class="citation libro" style="font-style:normal"> Samantha Leorato, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/funzione-matematica_(Dizionario-di-Economia-e-Finanza)/"><span style="font-style:italic;">Funzione matematica</span></a>, in <span style="font-style:italic;">Dizionario di Economia e Finanza</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2012.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P9941" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/function-mathematics"><span style="font-style:italic;">function</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Function.html"><span style="font-style:italic;">Function</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSpringerEOM" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Function"><span style="font-style:italic;">Function</span></a>, su <span style="font-style:italic;"><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></span>, Springer e European Mathematical Society.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#P7554" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Analisi_matematica"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Analisi_matematica" title="Template:Analisi matematica"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Analisi_matematica&action=edit&redlink=1" class="new" title="Discussioni template:Analisi matematica (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Analisi_matematica&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Analisi_matematica" title="Analisi matematica">Analisi matematica</a></span></th></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Calcolo_infinitesimale" title="Calcolo infinitesimale">Calcolo infinitesimale</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Numero_reale" title="Numero reale">Numero reale</a><b> ·</b> <a href="/wiki/Infinitesimo" title="Infinitesimo">Infinitesimo</a><b> ·</b> <a href="/wiki/O-grande" title="O-grande">O-grande</a><b> ·</b> <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">Successione</a> (<a href="/wiki/Successione_di_funzioni" title="Successione di funzioni">di funzioni</a>)<b> ·</b> <a href="/wiki/Successione_di_Cauchy" title="Successione di Cauchy">Successione di Cauchy</a><b> ·</b> <a href="/wiki/Teorema_di_Bolzano-Weierstrass" title="Teorema di Bolzano-Weierstrass">Teorema di Bolzano-Weierstrass</a><b> ·</b> <a href="/wiki/Stima_asintotica" title="Stima asintotica">Stima asintotica</a><b> ·</b> <a href="/wiki/Limite_(matematica)" title="Limite (matematica)">Limite</a> (<a href="/wiki/Limite_di_una_funzione" title="Limite di una funzione">di una funzione</a><b> ·</b> <a href="/wiki/Limite_di_una_successione" title="Limite di una successione">di una successione</a><b> ·</b> <a href="/wiki/Forma_indeterminata" title="Forma indeterminata">Forma indeterminata</a>)<b> ·</b> <a href="/wiki/Teorema_del_confronto" title="Teorema del confronto">Teorema dei due carabinieri</a><b> ·</b> <a href="/wiki/Limite_notevole" title="Limite notevole">Limite notevole</a><b> ·</b> <a href="/wiki/Punto_di_accumulazione" title="Punto di accumulazione">Punto di accumulazione</a><b> ·</b> <a href="/wiki/Punto_isolato" title="Punto isolato">Punto isolato</a><b> ·</b> <a href="/wiki/Intorno" title="Intorno">Intorno</a><b> ·</b> <a href="/wiki/Serie_(matematica)" title="Serie (matematica)">Serie</a> (<a href="/wiki/Serie_di_funzioni" title="Serie di funzioni">di funzioni</a>)<b> ·</b> <a href="/wiki/Criteri_di_convergenza" title="Criteri di convergenza">Criteri di convergenza</a><b> ·</b> <a href="/wiki/Limite_di_funzioni_a_pi%C3%B9_variabili" title="Limite di funzioni a più variabili">Limite di funzioni a più variabili</a></td><td rowspan="7" class="navbox_image"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_kmplot.svg" class="mw-file-description" title="Analisi matematica"><img alt="Analisi matematica" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/100px-Nuvola_apps_kmplot.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/150px-Nuvola_apps_kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/200px-Nuvola_apps_kmplot.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>Analisi matematica</figcaption></figure></td></tr><tr><th colspan="1" class="navbox_group">Studio della continuità</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Funzione_continua" title="Funzione continua">Funzione continua</a><b> ·</b> <a href="/wiki/Punto_di_discontinuit%C3%A0" title="Punto di discontinuità">Punto di discontinuità</a><b> ·</b> <a href="/wiki/Continuit%C3%A0_uniforme" title="Continuità uniforme">Continuità uniforme</a><b> ·</b> <a href="/wiki/Funzione_lipschitziana" title="Funzione lipschitziana">Funzione lipschitziana</a><b> ·</b> <a href="/wiki/Teorema_di_Bolzano" title="Teorema di Bolzano">Teorema di Bolzano</a><b> ·</b> <a href="/wiki/Teorema_di_Weierstrass" title="Teorema di Weierstrass">Teorema di Weierstrass</a><b> ·</b> <a href="/wiki/Teorema_dei_valori_intermedi" title="Teorema dei valori intermedi">Teorema dei valori intermedi</a><b> ·</b> <a href="/wiki/Teorema_di_Heine-Cantor" title="Teorema di Heine-Cantor">Teorema di Heine-Cantor</a><b> ·</b> <a href="/wiki/Modulo_di_continuit%C3%A0" title="Modulo di continuità">Modulo di continuità</a><b> ·</b> <a href="/wiki/Funzione_semicontinua" class="mw-redirect" title="Funzione semicontinua">Funzione semicontinua</a><b> ·</b> <a href="/wiki/Continuit%C3%A0_separata" title="Continuità separata">Continuità separata</a><b> ·</b> <a href="/wiki/Teorema_di_approssimazione_di_Weierstrass" title="Teorema di approssimazione di Weierstrass">Teorema di approssimazione di Weierstrass</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Calcolo_differenziale" class="mw-redirect" title="Calcolo differenziale">Calcolo differenziale</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Derivata" title="Derivata">Derivata</a><b> ·</b> <a href="/wiki/Differenziale_(matematica)" title="Differenziale (matematica)">Differenziale</a><b> ·</b> <a href="/wiki/Regole_di_derivazione" title="Regole di derivazione">Regole di derivazione</a><b> ·</b> <a href="/wiki/Teorema_di_Fermat_sui_punti_stazionari" title="Teorema di Fermat sui punti stazionari">Teorema di Fermat</a><b> ·</b> <a href="/wiki/Teorema_di_Rolle" title="Teorema di Rolle">Teorema di Rolle</a><b> ·</b> <a href="/wiki/Teorema_di_Lagrange" title="Teorema di Lagrange">Teorema di Lagrange</a><b> ·</b> <a href="/wiki/Teorema_di_Cauchy_(analisi_matematica)" title="Teorema di Cauchy (analisi matematica)">Teorema di Cauchy</a><b> ·</b> <a href="/wiki/Teorema_di_Darboux" title="Teorema di Darboux">Teorema di Darboux</a><b> ·</b> <a href="/wiki/Teorema_di_Taylor" title="Teorema di Taylor">Teorema di Taylor</a><b> ·</b> <a href="/wiki/Serie_di_Taylor" title="Serie di Taylor">Serie di Taylor</a><b> ·</b> <a href="/wiki/Funzione_differenziabile" title="Funzione differenziabile">Funzione differenziabile</a><b> ·</b> <a href="/wiki/Gradiente_(funzione)" class="mw-redirect" title="Gradiente (funzione)">Gradiente</a><b> ·</b> <a href="/wiki/Matrice_jacobiana" title="Matrice jacobiana">Jacobiana</a><b> ·</b> <a href="/wiki/Matrice_hessiana" title="Matrice hessiana">Hessiana</a><b> ·</b> <a href="/wiki/Forma_differenziale" title="Forma differenziale">Forma differenziale</a><b> ·</b> <a href="/wiki/Generalizzazioni_della_derivata" title="Generalizzazioni della derivata">Generalizzazioni della derivata</a><b> ·</b> <a href="/wiki/Derivata_parziale" title="Derivata parziale">Derivata parziale</a><b> ·</b> <a href="/wiki/Derivata_mista" title="Derivata mista">Derivata mista</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Integrale" title="Integrale">Integrale</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Primitiva_(matematica)" title="Primitiva (matematica)">Primitiva</a><b> ·</b> <a href="/wiki/Integrale_di_Riemann" title="Integrale di Riemann">Integrale di Riemann</a><b> ·</b> <a href="/wiki/Integrale_improprio" title="Integrale improprio">Integrale improprio</a><b> ·</b> <a href="/wiki/Integrale_di_Lebesgue" title="Integrale di Lebesgue">Integrale di Lebesgue</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_del_calcolo_integrale" title="Teorema fondamentale del calcolo integrale">Teorema fondamentale</a><b> ·</b> <a href="/wiki/Metodi_di_integrazione" title="Metodi di integrazione">Metodi di integrazione</a><b> ·</b> <a href="/wiki/Categoria:Tavole_di_integrali" title="Categoria:Tavole di integrali">Tavole</a><b> ·</b> <a href="/wiki/Integrale_multiplo" title="Integrale multiplo">Integrale multiplo</a>, <a href="/wiki/Integrale_di_linea" title="Integrale di linea">di linea</a> (<a href="/wiki/Integrale_di_linea_di_prima_specie" title="Integrale di linea di prima specie">1ª specie</a><b> ·</b> <a href="/wiki/Integrale_di_linea_di_seconda_specie" title="Integrale di linea di seconda specie">2ª specie</a>) e <a href="/wiki/Integrale_di_superficie" title="Integrale di superficie">di superficie</a> (<a href="/wiki/Integrale_di_volume" title="Integrale di volume">di volume</a>)</td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Studio_di_funzione" title="Studio di funzione">Studio di funzione</a></th><td colspan="1" class="navbox_list navbox_odd"><a class="mw-selflink selflink">Funzione</a><b> ·</b> <a href="/wiki/Variabile_(matematica)" title="Variabile (matematica)">Variabile</a><b> ·</b> <a href="/wiki/Dominio_e_codominio" title="Dominio e codominio">Dominio e codominio</a><b> ·</b> <a href="/wiki/Funzioni_pari_e_dispari" title="Funzioni pari e dispari">Funzioni pari e dispari</a><b> ·</b> <a href="/wiki/Funzione_periodica" title="Funzione periodica">Funzione periodica</a><b> ·</b> <a href="/wiki/Funzione_monotona" title="Funzione monotona">Funzione monotona</a><b> ·</b> <a href="/wiki/Funzione_convessa" title="Funzione convessa">Funzione convessa</a><b> ·</b> <a href="/wiki/Massimo_e_minimo_di_una_funzione" title="Massimo e minimo di una funzione">Massimo e minimo di una funzione</a><b> ·</b> <a href="/wiki/Punto_angoloso" title="Punto angoloso">Punto angoloso</a><b> ·</b> <a href="/wiki/Cuspide_(matematica)" title="Cuspide (matematica)">Cuspide</a><b> ·</b> <a href="/wiki/Punto_di_flesso" title="Punto di flesso">Punto di flesso</a><b> ·</b> <a href="/wiki/Asintoto" title="Asintoto">Asintoto</a><b> ·</b> <a href="/wiki/Grafico_di_una_funzione" title="Grafico di una funzione">Grafico di una funzione</a><b> ·</b> <a href="/wiki/Funzione_iniettiva" title="Funzione iniettiva">Funzione iniettiva</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Disuguaglianza" title="Disuguaglianza">Disuguaglianze</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare">Disuguaglianza triangolare</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Cauchy-Schwarz" title="Disuguaglianza di Cauchy-Schwarz">Disuguaglianza di Cauchy-Schwarz</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Bernoulli" title="Disuguaglianza di Bernoulli">Bernoulli</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Jensen" title="Disuguaglianza di Jensen">Jensen</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_H%C3%B6lder" title="Disuguaglianza di Hölder">Hölder</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Young" title="Disuguaglianza di Young">Young</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Minkowski" title="Disuguaglianza di Minkowski">Minkowski</a></td></tr><tr><th colspan="1" class="navbox_group">Altro</th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Approssimazione_di_Stirling" title="Approssimazione di Stirling">Approssimazione di Stirling</a><b> ·</b> <a href="/wiki/Prodotto_di_Wallis" title="Prodotto di Wallis">Prodotto di Wallis</a><b> ·</b> <a href="/wiki/Funzione_Gamma" title="Funzione Gamma">Funzione Gamma</a><b> ·</b> <a href="/wiki/Teorema_delle_funzioni_implicite" title="Teorema delle funzioni implicite">Teorema delle funzioni implicite</a><b> ·</b> <a href="/wiki/Teorema_della_funzione_inversa" title="Teorema della funzione inversa">Teorema della funzione inversa</a><b> ·</b> <a href="/wiki/Condizione_di_H%C3%B6lder" title="Condizione di Hölder">Funzione hölderiana</a><b> ·</b> <a href="/wiki/Spazio_metrico" title="Spazio metrico">Spazio metrico</a><b> ·</b> <a href="/wiki/Spazio_normato" title="Spazio normato">Spazio normato</a><b> ·</b> <a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">Intervallo</a><b> ·</b> <a href="/wiki/Insieme_trascurabile" title="Insieme trascurabile">Insieme trascurabile</a><b> ·</b> <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">Insieme chiuso</a><b> ·</b> <a href="/wiki/Insieme_aperto" title="Insieme aperto">Insieme aperto</a><b> ·</b> <a href="/wiki/Palla_(matematica)" title="Palla (matematica)">Palla</a><b> ·</b> <a href="/wiki/Omeomorfismo" title="Omeomorfismo">Omeomorfismo</a><b> ·</b> <a href="/wiki/Omeomorfismo_locale" title="Omeomorfismo locale">Omeomorfismo locale</a><b> ·</b> <a href="/wiki/Diffeomorfismo" title="Diffeomorfismo">Diffeomorfismo</a><b> ·</b> <a href="/wiki/Diffeomorfismo_locale" title="Diffeomorfismo locale">Diffeomorfismo locale</a><b> ·</b> <a href="/wiki/Classe_C_di_una_funzione" title="Classe C di una funzione">Classe C di una funzione</a><b> ·</b> <a href="/wiki/Equazione_differenziale" title="Equazione differenziale">Equazione differenziale</a><b> ·</b> <a href="/wiki/Problema_di_Cauchy" title="Problema di Cauchy">Problema di Cauchy</a></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and 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href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4071510-3">4071510-3</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11946892t">cb11946892t</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11946892t">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007553160705171">987007553160705171</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="giapponese">JA</abbr></span>) <a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00564960">00564960</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output 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