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gabay sa pagbabasa ng laktaw-laktaw sa Wikipedia"><span>Mga nilalaman</span></a></li><li id="n-Napiling-nilalaman" class="mw-list-item"><a href="/wiki/Wikipedia:Mga_napiling_artikulo"><span>Napiling nilalaman</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Natatangi:Alin_man" title="Magpalabas ng kahit-anong artikulo [x]" accesskey="x"><span>Alinmang artikulo</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Patungkol"><span>Patungkol sa Wikipedia</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Wikipedia:Kasalukuyang_pangyayari" title="Hanapin ang mga impormasyon tungkol sa mga kamakailang kaganapan"><span>Mga kaganapan</span></a></li> </ul> </div> </div> <div id="p-Pakikihalubilo" class="vector-menu mw-portlet mw-portlet-Pakikihalubilo" > <div class="vector-menu-heading"> Pakikihalubilo </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" 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[r]" accesskey="r"><span>Mga huling binago</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Wikipedia:Makipag-ugnayan_sa_amin" title="Paano makokontak ang Wikipedia"><span>Makipag-ugnayan</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Wikipedia:Tulong" title="Ang lugar para maghanap ng tulong sa Wikipedia."><span>Tulong</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Unang_Pahina" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="Ang Malayang Ensiklopedya" src="/static/images/mobile/copyright/wikipedia-tagline-tl.svg" width="119" height="13" style="width: 7.4375em; height: 0.8125em;"> </span> </a> 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type="search" name="search" placeholder="Maghanap sa Wikipedia" aria-label="Maghanap sa Wikipedia" autocapitalize="sentences" title="Maghanap sa Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Natatangi:Maghanap"> </div> <button class="cdx-button cdx-search-input__end-button">Hanapin</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Mga pansariling kagamitan"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Itsura"> <div 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class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_tl.wikipedia.org&uselang=tl" class=""><span>Donasyon</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Natatangi:Likhain_ang_kuwenta&returnto=Punsiyon+%28matematika%29" title="Mas maganda kung gagawa ka ng account at magla-login; gayunpaman, hindi ka pinipilit gawin ito" class=""><span>Gumawa ng account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Natatangi:Paglagda_ng_tagagamit&returnto=Punsiyon+%28matematika%29" title="Mas maganda kung magla-login ka; gayunpaman, hindi ka pinipilit gawin ito [o]" accesskey="o" class=""><span>Mag-login</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Marami pang pagpipilian" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Mga pansariling kagamitan" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Mga pansariling kagamitan</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_tl.wikipedia.org&uselang=tl"><span>Donasyon</span></a></li><li id="pt-contribute" class="mw-list-item"><a href="/wiki/Natatangi:Contribute"><span class="vector-icon mw-ui-icon-edit mw-ui-icon-wikimedia-edit"></span> <span>Mag-ambag</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Natatangi:Likhain_ang_kuwenta&returnto=Punsiyon+%28matematika%29" title="Mas maganda kung gagawa ka ng account at magla-login; gayunpaman, hindi ka pinipilit gawin ito"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Gumawa ng account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Natatangi:Paglagda_ng_tagagamit&returnto=Punsiyon+%28matematika%29" title="Mas maganda kung magla-login ka; gayunpaman, hindi ka pinipilit gawin ito [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Mag-login</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Mga pahina para sa naka-logout na mga patnugot o editor <a href="/wiki/Wikipedia:Maligayang_pagdating!" aria-label="Alamin pa ang tungkol sa pagpapatnugot"><span>alamin pa</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Natatangi:Usapan_ko" title="Usapang tungkol sa mga pagbabagong ginawa sa ip address na ito [n]" accesskey="n"><span>Usapan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sayt"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Mga nilalaman" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Mga nilalaman</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">ilipat sa gilid</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">itago</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Simula</div> </a> </li> <li id="toc-Depinisyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Depinisyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Depinisyon</span> </div> </a> <button aria-controls="toc-Depinisyon-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>Ipakita/Itago ang subseksyon na Depinisyon</span> </button> <ul id="toc-Depinisyon-sublist" class="vector-toc-list"> <li id="toc-Notasyon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notasyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Notasyon</span> </div> </a> <ul id="toc-Notasyon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mga_uri_ng_punsiyon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_uri_ng_punsiyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Mga uri ng punsiyon</span> </div> </a> <button aria-controls="toc-Mga_uri_ng_punsiyon-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>Ipakita/Itago ang subseksyon na Mga uri ng punsiyon</span> </button> <ul id="toc-Mga_uri_ng_punsiyon-sublist" class="vector-toc-list"> <li id="toc-Elementaryang_mga_punsiyon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementaryang_mga_punsiyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Elementaryang mga punsiyon</span> </div> </a> <ul id="toc-Elementaryang_mga_punsiyon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_punsiyong_alhebraiko" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mga_punsiyong_alhebraiko"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Mga punsiyong alhebraiko</span> </div> </a> <ul id="toc-Mga_punsiyong_alhebraiko-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mga_elementaryang_transendental_na_punsiyon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mga_elementaryang_transendental_na_punsiyon"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Mga elementaryang transendental na punsiyon</span> </div> </a> <ul id="toc-Mga_elementaryang_transendental_na_punsiyon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mga_sanggunian" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mga_sanggunian"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mga sanggunian</span> </div> </a> <ul id="toc-Mga_sanggunian-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="Mga nilalaman" 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="Itago/Ipakita ang talaan ng mga nilalaman" > <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">Itago/Ipakita ang talaan ng mga nilalaman</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">Punsiyon (matematika)</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="Pumunta sa isang artikulo na nakasulat sa ibang wika. Mayroon ito sa 120 (na) wika" > <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 wika</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) – Alemannic" lang="gsw" hreflang="gsw" data-title="Funktion (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" 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 – Amharic" lang="am" hreflang="am" data-title="አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" 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 – Arabic" lang="ar" hreflang="ar" data-title="دالة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9" title="دالة – Moroccan Arabic" lang="ary" hreflang="ary" data-title="دالة" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática – Asturian" lang="ast" hreflang="ast" data-title="Función matemática" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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) – Azerbaijani" lang="az" hreflang="az" data-title="Funksiya (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – Bashkir" lang="ba" hreflang="ba" data-title="Функция (математика)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Funkc%C4%97j%C4%97" title="Funkcėjė – Samogitian" lang="sgs" hreflang="sgs" data-title="Funkcėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Функцыя (матэматыка) – Belarusian" lang="be" hreflang="be" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Функция" data-language-autonym="Български" data-language-local-name="Bulgarian" 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) – Bangla" lang="bn" hreflang="bn" data-title="অপেক্ষক (গণিত)" data-language-autonym="বাংলা" data-language-local-name="Bangla" 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) – Bosnian" lang="bs" hreflang="bs" data-title="Funkcija (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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 – Catalan" lang="ca" hreflang="ca" data-title="Funció" data-language-autonym="Català" data-language-local-name="Catalan" 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) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فانکشن (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" 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) – Czech" lang="cs" hreflang="cs" data-title="Funkce (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" 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) – Chuvash" lang="cv" hreflang="cv" data-title="Функци (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" 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 – Welsh" lang="cy" hreflang="cy" data-title="Ffwythiant" data-language-autonym="Cymraeg" data-language-local-name="Welsh" 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) – Danish" lang="da" hreflang="da" data-title="Funktion (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" 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) – German" lang="de" hreflang="de" data-title="Funktion (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" 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 – Greek" lang="el" hreflang="el" data-title="Συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" 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) – English" lang="en" hreflang="en" data-title="Function (mathematics)" data-language-autonym="English" data-language-local-name="English" 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) – Spanish" lang="es" hreflang="es" data-title="Función (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" 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) – Estonian" lang="et" hreflang="et" data-title="Funktsioon (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Estonian" 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) – Basque" lang="eu" hreflang="eu" data-title="Funtzio (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" 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 – Persian" lang="fa" hreflang="fa" data-title="تابع" data-language-autonym="فارسی" data-language-local-name="Persian" 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 – Finnish" lang="fi" hreflang="fi" data-title="Funktio" data-language-autonym="Suomi" data-language-local-name="Finnish" 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) – Fijian" lang="fj" hreflang="fj" data-title="Cakacaka (fika)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" 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) – French" lang="fr" hreflang="fr" data-title="Fonction (mathématiques)" data-language-autonym="Français" data-language-local-name="French" 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 – Northern Frisian" lang="frr" hreflang="frr" data-title="Funksion" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" 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) – Irish" lang="ga" hreflang="ga" data-title="Feidhm (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" 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 – Galician" lang="gl" hreflang="gl" data-title="Función" data-language-autonym="Galego" data-language-local-name="Galician" 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 – Hebrew" lang="he" hreflang="he" data-title="פונקציה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="फलन – Hindi" lang="hi" hreflang="hi" data-title="फलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Function" title="Function – Fiji Hindi" lang="hif" hreflang="hif" data-title="Function" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – Croatian" lang="hr" hreflang="hr" data-title="Funkcija (matematika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" 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) – Hungarian" lang="hu" hreflang="hu" data-title="Függvény (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" 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) – Armenian" lang="hy" hreflang="hy" data-title="Ֆունկցիա (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" 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) – Indonesian" lang="id" hreflang="id" data-title="Fungsi (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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) – Icelandic" lang="is" hreflang="is" data-title="Fall (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_(matematica)" title="Funzione (matematica) – Italian" lang="it" hreflang="it" data-title="Funzione (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%96%A2%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="関数 (数学) – Japanese" lang="ja" hreflang="ja" data-title="関数 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Fongshan_(matimatix)" title="Fongshan (matimatix) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Fongshan (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/fancu" title="fancu – Lojban" lang="jbo" hreflang="jbo" data-title="fancu" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%90_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ფუნქცია (მათემატიკა) – Georgian" lang="ka" hreflang="ka" data-title="ფუნქცია (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="Georgian" 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) – Kabyle" lang="kab" hreflang="kab" data-title="Tasɣent (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" 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) – Kazakh" lang="kk" hreflang="kk" data-title="Функция (математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" 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 – Korean" lang="ko" hreflang="ko" data-title="함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio" title="Functio – Latin" lang="la" hreflang="la" data-title="Functio" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Funktioun_(Mathematik)" title="Funktioun (Mathematik) – Luxembourgish" lang="lb" hreflang="lb" data-title="Funktioun (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Fonzion_(matematega)" title="Fonzion (matematega) – Lombard" lang="lmo" hreflang="lmo" data-title="Fonzion (matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%95%E0%BA%B3%E0%BA%A5%E0%BA%B2_(%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94)" title="ຕຳລາ (ຄະນິດສາດ) – Lao" lang="lo" hreflang="lo" data-title="ຕຳລາ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – Lithuanian" lang="lt" hreflang="lt" data-title="Funkcija (matematika)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" 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 – Latvian" lang="lv" hreflang="lv" data-title="Funkcija" data-language-autonym="Latviešu" data-language-local-name="Latvian" 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) – Macedonian" lang="mk" hreflang="mk" data-title="Функција (математика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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) – Mongolian" lang="mn" hreflang="mn" data-title="Функц (математик)" data-language-autonym="Монгол" data-language-local-name="Mongolian" 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 – Malay" lang="ms" hreflang="ms" data-title="Fungsi" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" 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 – Burmese" lang="my" hreflang="my" data-title="ဖန်ရှင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" 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) – Low German" lang="nds" hreflang="nds" data-title="Afbillen (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" 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) – Dutch" lang="nl" hreflang="nl" data-title="Functie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" 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 – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><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) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Funksjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><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) – Occitan" lang="oc" hreflang="oc" data-title="Aplicacion (matematicas)" data-language-autonym="Occitan" data-language-local-name="Occitan" 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 – Polish" lang="pl" hreflang="pl" data-title="Funkcja" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Fonsion" title="Fonsion – Piedmontese" lang="pms" hreflang="pms" data-title="Fonsion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%81%D9%86%DA%A9%D8%B4%D9%86" title="فنکشن – Western Punjabi" lang="pnb" hreflang="pnb" data-title="فنکشن" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Função (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Função (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Funcție" data-language-autonym="Română" data-language-local-name="Romanian" 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) – Russian" lang="ru" hreflang="ru" data-title="Функция (математика)" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Yakut" lang="sah" hreflang="sah" data-title="Функция. Функция чэрчитэ, суолталарын түмсээнэ" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><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) – Sicilian" lang="scn" hreflang="scn" data-title="Funzioni (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" 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) – Scots" lang="sco" hreflang="sco" data-title="Function (mathematics)" data-language-autonym="Scots" data-language-local-name="Scots" 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-Croatian" lang="sh" hreflang="sh" data-title="Funkcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" 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) – Slovak" lang="sk" hreflang="sk" data-title="Zobrazenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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) – Slovenian" lang="sl" hreflang="sl" data-title="Funkcija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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 – Inari Sami" lang="smn" hreflang="smn" data-title="Funktio" data-language-autonym="Anarâškielâ" data-language-local-name="Inari Sami" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Murimo_(Masvomhu)" title="Murimo (Masvomhu) – Shona" lang="sn" hreflang="sn" data-title="Murimo (Masvomhu)" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Shaqada_(xisaabta)" title="Shaqada (xisaabta) – Somali" lang="so" hreflang="so" data-title="Shaqada (xisaabta)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni" title="Funksioni – Albanian" lang="sq" hreflang="sq" data-title="Funksioni" data-language-autonym="Shqip" data-language-local-name="Albanian" 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) – Serbian" lang="sr" hreflang="sr" data-title="Функција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" 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 – Swedish" lang="sv" hreflang="sv" data-title="Funktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Funkcyjo" title="Funkcyjo – Silesian" lang="szl" hreflang="szl" data-title="Funkcyjo" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="சார்பு – Tamil" lang="ta" hreflang="ta" data-title="சார்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ฟังก์ชัน (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="ฟังก์ชัน (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fonksiyon" title="Fonksiyon – Turkish" lang="tr" hreflang="tr" data-title="Fonksiyon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><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) – Tatar" lang="tt" hreflang="tt" data-title="Функция (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" 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 – Uyghur" lang="ug" hreflang="ug" data-title="فۇنكسىيە" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="Uyghur" 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) – Ukrainian" lang="uk" hreflang="uk" data-title="Функція (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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) – Uzbek" lang="uz" hreflang="uz" data-title="Funksiya (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Funkcii_(matematik)" title="Funkcii (matematik) – Veps" lang="vep" hreflang="vep" data-title="Funkcii (matematik)" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_s%E1%BB%91" title="Hàm số – Vietnamese" lang="vi" hreflang="vi" data-title="Hàm số" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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" data-title="פונקציע" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B5%9C%E2%B4%B0%E2%B5%99%E2%B5%96%E2%B5%8F%E2%B5%9C_(%E2%B5%9C%E2%B5%93%E2%B5%99%E2%B5%8F%E2%B4%B0%E2%B4%BD%E2%B5%9C)" title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ) – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ)" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%87%BD%E6%95%B0" title="函数 – Chinese" lang="zh" hreflang="zh" data-title="函数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%98%A0%E5%B0%84" title="映射 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="映射" data-language-autonym="文言" 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</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="tl" dir="ltr"><div role="note" class="hatnote navigation-not-searchable">Para sa ibang mga gamit, tingnan ang <a href="/wiki/Punsiyon" class="mw-disambig" title="Punsiyon">Punsiyon</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/Talaksan:Graph_of_example_function.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Graph_of_example_function.svg/250px-Graph_of_example_function.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Graph_of_example_function.svg/375px-Graph_of_example_function.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Graph_of_example_function.svg/500px-Graph_of_example_function.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption><a href="/wiki/Grapo_ng_punsiyon" class="mw-redirect" title="Grapo ng punsiyon">Grapo</a> ng isang punsiyon,<br /> <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{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mstyle displaystyle="false" scriptlevel="1" /> </mtd> </mtr> <mtr> <mtd /> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow> <mn>3</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849e95193ac47959d04871bd7819452dc57455b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.945ex; margin-bottom: -0.227ex; width:23.359ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}"></span><br /> Ang x sa kanan ang bariabulong input na kumakatawan sa isang halaga ipinasok at ang nasa kaliwa na f(x) ang output na inilabas na halaga o resulta ng nakwentang halaga ng punsiyon sa ibinigay na argumentong x.</figcaption></figure> <p>Ang <b>punsiyon</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> (<a href="/wiki/Wikang_Ingles" title="Wikang Ingles">Ingles</a>: <i>function</i>) o <b>bunin</b> <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> sa <a href="/wiki/Matematika" title="Matematika">matematika</a>, ay nag-uugnay ng isang bilang o input (binibigay na bilang) na argumento ng punsiyon, sa halaga ng kalalabasang halaga o <i>output</i>. Ang punsiyon ay nagtatakda lamang ng isang kinalalabasan na output sa isang ipinasok na halaga o <i>input</i>. Ang halaga ng isang punsiyong <i>f</i> na may argumentong <i>x</i> ay tinutukoy ng <i>f</i>(<i>x</i>) at binabasang "<i>f</i> ng <i>x</i>". Ang isang halimbawa ang <i>f</i>(<i>x</i>) = 2<i>x</i> kung saan ang punsiyong <i>f</i> ay nag-uugnay ng input na <i>x</i> sa bilang na dalawang beses na kasing laki nito. Kung ang <i>x</i> = 5, kung gayon ang <i>f</i>(<i>x</i>) = 10. </p><p>Ang mga argumento ng punsiyon at halaga ay maaaring mga <a href="/wiki/Real_na_bilang" class="mw-redirect" title="Real na bilang">real na bilang</a> o mga <a href="/w/index.php?title=Elemento_(matematika)&action=edit&redlink=1" class="new" title="Elemento (matematika) (hindi pa naisusulat)">elemento</a> ng anumang ibinigay na <a href="/wiki/Pangkat_(matematika)" title="Pangkat (matematika)">pangkat</a>. Halimbawa, ang isang punsiyon ay maaaring mag-ugnay ng letrang A sa bilang na 1, ang letrang B sa bilang na 2 at iba pa. Ang pangkat ng lahat ng mga input para sa isang punsiyon ay tinatawag na <a href="/wiki/Sakop_(matematika)" title="Sakop (matematika)">sakop</a> (Ingles: <i>domain</i>) at ang pangkat ng lahat ng mga output ang <a href="/wiki/Saklaw_(matematika)" title="Saklaw (matematika)">saklaw</a> (Ingles: <i>range</i>) o imahe (Ingles: <i>image) nito.</i> </p><p>Maraming mga paraang upang ilarawan o ikatawan ang isang punsiyon gaya ng isang <a href="/wiki/Pormula" title="Pormula">pormula</a> o <a href="/wiki/Algoritmo" title="Algoritmo">algoritmo</a> na kumukwenta sa output para sa isang ibinigay na input na isang larawan o <a href="/wiki/Grapo" class="mw-redirect mw-disambig" title="Grapo">grapo</a> ng input-ouput na <a href="/w/index.php?title=Isinaayos_na_mga_pares&action=edit&redlink=1" class="new" title="Isinaayos na mga pares (hindi pa naisusulat)">isinaayos na mga pares</a> ng punsiyon o isang <a href="/w/index.php?title=Tabla_(matematika)&action=edit&redlink=1" class="new" title="Tabla (matematika) (hindi pa naisusulat)">tabla</a> ng mga output sa mga napiling input. Ang isang punsiyon ay maaari ring ilarawan sa pamamagitan ng mga ugnayan nito sa ibang mga punsiyon, halimbawa bilang <a href="/wiki/Inbersong_punsiyon" title="Inbersong punsiyon">inbersong punsiyon</a> o solusyon ng isang <a href="/wiki/Diperensiyal_na_ekwasyon" class="mw-redirect" title="Diperensiyal na ekwasyon">diperensiyal na ekwasyon</a>. Sa analohiya sa <a href="/wiki/Aritmetika" title="Aritmetika">aritmetika</a>, posibleng maglarawan ng punsiyong <a href="/wiki/Adisyon" class="mw-redirect" title="Adisyon">adisyon</a>, <a href="/wiki/Multiplikasyon" class="mw-redirect" title="Multiplikasyon">multiplikasyon</a> at ibang mga <a href="/w/index.php?title=Matematikal_na_operasyon&action=edit&redlink=1" class="new" title="Matematikal na operasyon (hindi pa naisusulat)">matematikal na operasyon</a> upang lumikha ng mga bagong punsiyon. Ang isang mahalagang operasyon na inilalarawa sa mga punsiyon ang <a href="/w/index.php?title=Komposisyon_ng_mga_punsiyon&action=edit&redlink=1" class="new" title="Komposisyon ng mga punsiyon (hindi pa naisusulat)">komposisyon ng mga punsiyon</a> kung saan ang output mula sa isang punsiyon ay nagiging input ng iba pang punsiyon. </p><p>Ang mga pormal na depinisyon ng mga punsiyon ay nagbibigay ng pangkat ng mga input (sakop o domain), pangkat ng pinares na input at output at isang pangkat na tinatawag na codomain kung saan ang mga output ay itinakdang mahuhulog. Ang mga kalipunan ng mga punsiyon na may parehong domain at codomain ay tinatawag na mga <a href="/w/index.php?title=Espasyong_punsiyon&action=edit&redlink=1" class="new" title="Espasyong punsiyon (hindi pa naisusulat)">espasyong punsiyon</a>, na ang mga katangian ay pinag-aaral sa mga matematikal na disiplina gaya ng <a href="/w/index.php?title=Real_na_analisis&action=edit&redlink=1" class="new" title="Real na analisis (hindi pa naisusulat)">real na analisis</a> at <a href="/w/index.php?title=Kompleks_na_analisis&action=edit&redlink=1" class="new" title="Kompleks na analisis (hindi pa naisusulat)">kompleks na analisis</a>. Ang mga punsiyon at ang iba't ibang mga <a href="/wiki/Analogo" class="mw-redirect" title="Analogo">analogo</a> nito o paglalahat gaya ng mga <a href="/w/index.php?title=Punktor&action=edit&redlink=1" class="new" title="Punktor (hindi pa naisusulat)">punktor</a> (Ingles: <i>functor</i>) ng <a href="/w/index.php?title=Teoriyang_kategorya&action=edit&redlink=1" class="new" title="Teoriyang kategorya (hindi pa naisusulat)">teoriyang kategorya</a> ang mga "sental na bagay ng imbestigasyon" sa karamihan ng mga larangan sa <a href="/wiki/Matematika" title="Matematika">modernong matematika</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Depinisyon">Depinisyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Punsiyon_(matematika)&veaction=edit&section=1" title="Baguhin seksiyon: Depinisyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Punsiyon_(matematika)&action=edit&section=1" title="Edit section's source code: Depinisyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1738803/mw-parser-output/.tmulti">.mw-parser-output .tmulti .thumbinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{text-align:left;background-color:transparent}.mw-parser-output .tmulti .thumbcaption-center{text-align:center;background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .thumbcaption{text-align:center}}</style><div class="thumb tmulti tright"><div class="thumbinner" style="width:204px;max-width:204px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Talaksan:Injection_keine_Injektion_2a.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Injection_keine_Injektion_2a.svg/200px-Injection_keine_Injektion_2a.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Injection_keine_Injektion_2a.svg/300px-Injection_keine_Injektion_2a.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Injection_keine_Injektion_2a.svg/400px-Injection_keine_Injektion_2a.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div><div class="thumbcaption">Ang diagramang ito ay kumakatawan sa isang punsiyon na may sakop/domain na <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 \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span>, codomain na <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,B,C,D\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A,B,C,D\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7166a22b1ae5dd3e429b71c09560a89e917c7da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.624ex; height:2.843ex;" alt="{\displaystyle \{A,B,C,D\}}"></span> at pangkat ng isinaayos na mga pares na <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 \{(1,D),(2,C),(3,C)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,D),(2,C),(3,C)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2cb6b0bcc1d86bf9ac946083478c13ccf2049c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.867ex; height:2.843ex;" alt="{\displaystyle \{(1,D),(2,C),(3,C)\}}"></span>. Ang imahe ay <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,D\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{C,D\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d7072a1a4163ba50559f58f329c8daff5ba0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.049ex; height:2.843ex;" alt="{\displaystyle \{C,D\}}"></span>.</div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Talaksan:Injection_keine_Injektion_1.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Injection_keine_Injektion_1.svg/200px-Injection_keine_Injektion_1.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Injection_keine_Injektion_1.svg/300px-Injection_keine_Injektion_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Injection_keine_Injektion_1.svg/400px-Injection_keine_Injektion_1.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div><div class="thumbcaption">Ito ay <b>hindi</b> kumakatawan sa isang punsiyon dahil ang 2 ang unang elemento sa higit sa isang isinaayos na pares, sa partikular ang (2,B) at (2,C) ay parehong mga elemento ng pangkat ng isinaayos na mga pares.</div></div></div></div></div> <p>Ang isang tiyak na depinisyon ng isang punsiyon ay isang isinaayos na triple ng mga <a href="/wiki/Pangkat_(matematika)" title="Pangkat (matematika)">pangkat</a> na isinulat na (<i>X</i>, <i>Y</i>, <i>F</i>) kung saan ang <i>X</i> ang sakop o domain, ang <i>Y</i> ang codomain, at ang <i>F</i> ang pangkat ng mga isinaayos na pares (<i>a</i>, <i>b</i>).<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> Sa bawat mga isinaayos na pares, ang unang elementong <i>a</i> ay mula sa sakop/domain, ang ikalawang elementong <i>b</i> ay mula sa codomain at ang isang kinakailangang kondisyon ay ang bawat elemento sa sakop/domain ang unang elemento sa eksaktong isang isinaayos na pares. Ang pangkat ng lahat ng <i>b</i> ay tinatawag na <a href="/wiki/Imahe_(matematika)" title="Imahe (matematika)">imahe</a> ng punsiyon at hindi kinakailangang maging buo ng codomain. Ang karamihan sa mga may-akda ay gumagamit ng salitang "range" upang ipakahulugang imahe samantalang ang iba ay gumagamit ng codomain upang pakahulugang range. </p><p>Ang notasyong <i>f</i>:<i>X</i>→<i>Y</i> ay nagpapakitang ang <i>f</i> ay isang punsiyn na may sako/domain na <i>X</i> at codomain na <i>Y</i>, at ang punsiyong <i>f</i> ay sinasabing <i>mapa</i> o <i>associate/kaugnay</i> na mga elemento <i>Y</i>. </p><p>Imbis na pag-usapan ang isang <i>punsiyonh</i> at tukuyin ang domain at codoamin sa depinisyon nito, normal na mas konbinyenteng pag-usapan ang mga punksiyon na may spesipikong domain at codomai na mga punsiyon ng isang ibinigay na uri at pagkatapos ay ilalarawan ng pangkat ng mga isinaayos na pares na <i>F</i>. Halimbawa, ang isang punsiyon mula sa mga <a href="/wiki/Real_na_bilang" class="mw-redirect" title="Real na bilang">real</a> tungol sa real ay bibigyan ng pangkat ng mga pares na (<i>x</i>, 2<i>x</i>) kung saan ang <i>x</i> ay isang <a href="/wiki/Real_na_bilang" class="mw-redirect" title="Real na bilang">real</a>. </p><p>Kung ang domain at codomain ay parehong mga pangkat ng mga <a href="/wiki/Real_na_bilang" class="mw-redirect" title="Real na bilang">real na bilang</a> gaya ng karaniwang kaso, sinasabi nating ang <i>f</i> ay isang may halagang <a href="/wiki/Real_na_bilang" class="mw-redirect" title="Real na bilang">real</a> na punsiyon ng isang real na <a href="/wiki/Bariabulo" class="mw-redirect" title="Bariabulo">bariabulo</a> at ang pag-aaral ng mga gayong punsiyon ay tinatawag na mga <a href="/w/index.php?title=Real_na_bariabulo&action=edit&redlink=1" class="new" title="Real na bariabulo (hindi pa naisusulat)">real na bariabulo</a>. Kung ang domain at codomain ay parehong pangkat ng mga <a href="/wiki/Kompleks_na_bilang" class="mw-redirect" title="Kompleks na bilang">kompleks na bilang</a>, kung gayon, ating sasabihing ang <i>f</i> ay isang may halagang kompleks na punsiyon ng isang kompleks na bariabulo. Ang pag-aaral ng mga punsiyong ito ay tinatawag na mga <a href="/w/index.php?title=Kompleks_na_bariabulo&action=edit&redlink=1" class="new" title="Kompleks na bariabulo (hindi pa naisusulat)">kompleks na bariabulo</a>. Sa karamihan ng mga sitwasyon, ang domain at codomain ay nauunawan mula sa konteksto at ang tanging ugnayan sa pagitan ng mga input at output ang ibinigay, ngunit kung ang <i>f</i>(<i>x</i>) = √<i>x</i>, kung gayon sa mga real na bariabulo, ang sakop/domain ay limitado sa hindi-negatibong mga bilang samantalang sa mga kompleks na bariabul, ang domain ang lahat ng mga kompleks na bilang. </p><p>Lalo na sa <a href="/wiki/Teoriya_ng_pangkat" class="mw-redirect" title="Teoriya ng pangkat">teoriya ng pangkat</a>, ang isang punsiyong <i>f</i> ay kalimitang inilalarawan bilang pangkat ng mga isinaayos na pares. Ang domain ay simpleng ang pangkat ng elemento na lumalabas bilang unang elemento ng pares at walang hayagang codomain na hiwalay mula sa imahe. </p><p>Ang isang punsiyon ay maaari ring tawaging <b><a href="/w/index.php?title=Mapa_(matematika)&action=edit&redlink=1" class="new" title="Mapa (matematika) (hindi pa naisusulat)">mapa</a></b> o <b>pagmamapa/mapping</b>. Gayunpaman, ang ilang mga may-akda ay gumagamit ng mga terminong "punsiyon" at "mapa" upang tukuyin ang iba't ibang uri ng mga punsiyon. Ang ibang mga spesipikong uri ng mga punsiyon ay kinabibilangan ng <b><a href="/w/index.php?title=Punsiyonal_(matematika)&action=edit&redlink=1" class="new" title="Punsiyonal (matematika) (hindi pa naisusulat)">punsiyonal</a></b>(Ingles: <b>functional</b>) at mga <b><a href="/w/index.php?title=Operador_(matematika)&action=edit&redlink=1" class="new" title="Operador (matematika) (hindi pa naisusulat)">operador</a>(Ingles: </b>operator<b>).</b> </p><p>Ang isang spesipikong input sa isang punsiyon ay tinatawag na <b>argumento</b> ng punsiyon. Sa bawat argumentong halagang <i>x</i>, ang tumutugong walang katulad(unique) na <i>y</i> sa codomain ay tinatawag na <b>halaga</b> sa <i>x</i>, <b>output</b> ng ƒ para sa argumentong <i>x</i> o ang <b><a href="/wiki/Imahe_(matematika)" title="Imahe (matematika)">imahe</a></b> ng <i>x</i> <b>sa ilalim</b> ng ƒ. Ang imahe ng <i>x</i> ay maaaring isulat bilang ƒ(<i>x</i>) o bilang <i>y</i>. </p><p>Ang <a href="/wiki/Grapo" class="mw-redirect mw-disambig" title="Grapo">grapo</a> ng punsiyon ang pangkat nito ng mga isinaayos na pares na <i>F</i>. Ito ay isang <a href="/wiki/Abstraksiyon" class="mw-redirect" title="Abstraksiyon">abstraksiyon</a> ng ideya ng grapo bilang larawan na nagpapakita ng punsiyon na binalangkas sa isang pares ng mga <a href="/w/index.php?title=Aksis_na_koordinado&action=edit&redlink=1" class="new" title="Aksis na koordinado (hindi pa naisusulat)">aksis na koordinado</a>. Halimbawa, ang (3, 9), ang punto sa taas ng 3 sa horisontal na aksis sa kanan at 9 sa bertikal na aksis ay nakahimlay sa grapo ng <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=x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1108c4c9ee8ac7de90b77f9bd27415b13b6bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.638ex; height:3.009ex;" alt="{\displaystyle y=x^{2}}"></span>. </p><p>Ang sakop/domain na <i>X</i> ay maaaring <a href="/wiki/Void" class="mw-redirect" title="Void">void</a> ngunit kung ang <i>X = ∅</i>, kung gayon ang <i>F = ∅</i>. Ang codomain na <i>Y</i> ay maaari ring void, ngunit kung ang <i>Y = ∅</i>, kung gayon ang <i>X = ∅</i> at ang <i>F = ∅</i>. Ang mga gayong <a href="/w/index.php?title=Walang_lamang_punsiyon&action=edit&redlink=1" class="new" title="Walang lamang punsiyon (hindi pa naisusulat)">walang lamang punsiyon</a> ay hindi karaniwan ngunit ang teoriya ay nagsisiguro ng eksistensiya ng mga ito. </p><p>Ang pangkat ng lahat ng mga punsiyong <i>f</i>:<i>X</i>→<i>Y</i> ay minsang tinutukoy ng i<i>Y</i><sup><i>X</i></sup>. Kung ang <i>X</i> ay <a href="/wiki/Inpinidad" class="mw-redirect" title="Inpinidad">inpinido</a>(infinite) at mayroon higit sa isang elemnto sa <i>Y</i>, kung gayon mayroon <a href="/w/index.php?title=Hindi_mabibilang_na_pangkat&action=edit&redlink=1" class="new" title="Hindi mabibilang na pangkat (hindi pa naisusulat)">hindi mabibilang na maraming</a> mga punsiyon mula sa <i>X</i> tungo sa <i>Y</i> bagaman ang taning mabibilang marami sa mga ito ang maaaring ihayag ng isang pormula o algoritmo. </p><p>Sa mga ilang bahagi ng <a href="/wiki/Matematika" title="Matematika">matematika</a>, kabilang ang <a href="/w/index.php?title=Teoriya_ng_rekursiyon&action=edit&redlink=1" class="new" title="Teoriya ng rekursiyon (hindi pa naisusulat)">teoriya ng rekursiyon</a> at <a href="/w/index.php?title=Punsiyonal_na_analisis&action=edit&redlink=1" class="new" title="Punsiyonal na analisis (hindi pa naisusulat)">punsiyonal na analisis</a>, konbinyenteng pag-aralan ang mga <a href="/w/index.php?title=Parsiyal_na_punsiyon&action=edit&redlink=1" class="new" title="Parsiyal na punsiyon (hindi pa naisusulat)">parsiyal na punsiyon</a> kung saan ang ilang mga halaga ng domain ay walang kaugnayan/association sa <a href="/wiki/Grapo" class="mw-redirect mw-disambig" title="Grapo">grapo</a>, i.e. isang-halagang mga ugnayan. Halimbawa, ang punsiyon <i>f</i> upang ang <i>f</i>(<i>x</i>) = 1/<i>x</i> ay hindi naglalarawan ng isang halaga para sa <i>x</i> = 0, at kaya ay isa lamang parsiyal na punsiyon mula sa real na linya tungo sa real na linya. Ang terminong <i>kabuuang punsiyon</i> ay maaaring gamitin upang bigyang diin ang katotohanang ang bawat elemento ng domain ay lumilitaw bilang unang elemento ng isinaayos ng pares sa grapo. Sa ibang mga bahgai ng matematika, ang hindi-isang-halagang mga ugnayan ay katulad ding isinanib sa mga punsiyon. Ito ay tinatawag na mga <a href="/w/index.php?title=Maraming_halagang_punsiyon&action=edit&redlink=1" class="new" title="Maraming halagang punsiyon (hindi pa naisusulat)">maraming halagang punsiyon</a> na may tumutugong terminong <a href="/w/index.php?title=Isang-halagang_punsiyon&action=edit&redlink=1" class="new" title="Isang-halagang punsiyon (hindi pa naisusulat)">isang-halagang punsiyon</a> para sa mga ordinaryong punsiyon. </p><p>Maraming mga operasyon sa <a href="/wiki/Teoriya_ng_pangkat" class="mw-redirect" title="Teoriya ng pangkat">teoriya ng pangkat</a> gaya ng <a href="/wiki/Kapangyarihang_pangkat" title="Kapangyarihang pangkat">kapangyarihang pangkat</a> ay may <a href="/w/index.php?title=Klase_(teoriya_ng_pangkat)&action=edit&redlink=1" class="new" title="Klase (teoriya ng pangkat) (hindi pa naisusulat)">klase</a> ng lahat ng mga pangkat bilang domain ng mga ito at kaya bagaman ang mga ito ay inpormal na inilalarawan bilang mga punsiyon, ang mga ito ay hindi nagkakasya sa set-teoretikal na depinisyon ibinalangkas sa itaas dahil ang klase ay hindi kinakailangang isang pangkat. </p><p>Ang punsiyon ay isang espesyal na kaso ng mas pangkalahatang matematikal na konseptong <a href="/w/index.php?title=Relasyon_(matematika)&action=edit&redlink=1" class="new" title="Relasyon (matematika) (hindi pa naisusulat)">relasyon</a>(ugnayan) kung saan ang restriksiyon ang bawat elemento ng domain ay lumilitaw na unang elemento sa isa at tanging isang isinaayos na pares lamang inalis. Sa ibang salita, ang elemento ng domain ay maaaring hindi unang elemento ng anumang isiniaayos na pares o maaaring unang elemento ng dalawa o higit pang isinaayos na pares. Ang relasyon ay "may isang halaga" kapag kung ang elemento ng domain ang unang elemento, ito ay hindi unang elemento ng anumang ibang isinaayos na pares. Ang relasyon ay isang "kaliwang-kabuuan" o simpleng "kabuuan" kung ang bawat elemento ng domain ang unang elemento ng isang isinaayos na pares. Kaya ang punsiyon ay isang buon, may isang halagang relasyon. </p> <div class="mw-heading mw-heading3"><h3 id="Notasyon">Notasyon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Punsiyon_(matematika)&veaction=edit&section=2" title="Baguhin seksiyon: Notasyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Punsiyon_(matematika)&action=edit&section=2" title="Edit section's source code: Notasyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ang pormal na deskripsiyon ng isang punsiyon ay karaniwang sumasangkot sa pangalan ng punsiyon, sakop/domain nito, codomain nito at isang patakarang ng tugunan(correspondence). Kaya kalimitan nating nakikita ang isang dalawang-bahaging notasyon na ang halimbawa ang </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{aligned}f\colon \mathbb {N} &\to \mathbb {R} \\n&\mapsto {\frac {n}{\pi }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>π</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f\colon \mathbb {N} &\to \mathbb {R} \\n&\mapsto {\frac {n}{\pi }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc88100c8beee542479e34e89f4ff35f4dfdaa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.048ex; margin-bottom: -0.29ex; width:10.587ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}f\colon \mathbb {N} &\to \mathbb {R} \\n&\mapsto {\frac {n}{\pi }}\end{aligned}}}"></span></dd></dl> <p>kung saan ang unang bahagi ay binabasang: </p> <ul><li>Ang "ƒ ay isang punsiyon mula sa <b>N</b> tungo sa <b>R</b>" (kalimitang isinulat na "Let (hayaan) ang ƒ: <i>X</i> → <i>Y</i>" upang pakahulugang "Hayaang ang ƒ ay maging isang punsiyon mula sa <i>X</i> tungo sa <i>Y</i>"), o</li> <li>"ƒ ay isang punsiyon sa <b>N<i> sa </i></b><i>R'</i>", o</li> <li>"ƒ ay isang may halagang-<b>R</b>-na punsiyon ng isang may halagang-<b>N</b> na bariabulo",</li></ul> <p>at ang ikalawang bahagi ay binabasang: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mspace width="thinmathspace" /> </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/205e33e6845813cc72ca346b896a7945f90ca373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,}"></span> ay namamapa sa <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 {\frac {n}{\pi }}.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>π</mi> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{\pi }}.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fecc72af4ba9d4e8ad9d2e3db29d654dc6670a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:3.265ex; height:4.676ex;" alt="{\displaystyle {\frac {n}{\pi }}.\,\!}"></span></li></ul> <p>Dito, ang punsiyong pinangalanang "ƒ" ay may mga <a href="/wiki/Natural_na_bilang" class="mw-redirect" title="Natural na bilang">natural na bilang</a> bilang sakop/domain, <a href="/wiki/Real_na_bilang" class="mw-redirect" title="Real na bilang">real na bilang</a> bilang codoamin at nagmamapa ng <i>n</i> sa sarili nito na hinati ng π. Sa hindi pormal na paglalarawan, ang mahabang anyong ito ay maaaring paiklin na </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(n)={\frac {n}{\pi }},\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>π</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\frac {n}{\pi }},\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0464dcef0207a7a9f643806b11a53cd251bf89e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:10.846ex; height:4.676ex;" alt="{\displaystyle f(n)={\frac {n}{\pi }},\,\!}"></span></dd></dl> <p>kung saan ang <i>f</i>(<i>n</i>) ay binabasa bilang "f bilang punsiyon ng n" o "f ng n". Mayroon ilang pagkawala ng impormasyon: hindi na natin hayagang ibinibigay ang domain na <b>N</b> at codomain na <b>R</b>. </p><p>Karaniwang inaalis ang <a href="/w/index.php?title=Parentheses&action=edit&redlink=1" class="new" title="Parentheses (hindi pa naisusulat)">parentheses</a> sa palibot ng argumento kapag walang walang maliit na tsansa ng pagkalito kaya sin <i>x</i>. Ito ay tinatawag na <a href="/w/index.php?title=Notasyong_prefix&action=edit&redlink=1" class="new" title="Notasyong prefix (hindi pa naisusulat)">notasyong prefix</a>. Ang pagsusulat ng punsiyon pagkatapos ng agumento gaya ng sa <i>x</i> ƒ, ay tinatawag na <a href="/w/index.php?title=Notasyong_postfix&action=edit&redlink=1" class="new" title="Notasyong postfix (hindi pa naisusulat)">notasyong postfix</a>. Halimbawa, ang <a href="/wiki/Paktoryal" title="Paktoryal">paktoryal</a> na punsiyon ay kinagagawiang isulat na <i>n</i>! bagaman ang paglalahat nito na <a href="/w/index.php?title=Punsiyong_gama&action=edit&redlink=1" class="new" title="Punsiyong gama (hindi pa naisusulat)">punsiyong gama</a> ay isinusulat na Γ(<i>n</i>). Ang mga parenteheses ay ginagamit par rin upang lutasin ang mga kalituhan at tumukoy ng pangunguna(precedence) bagaman sa ilang mga pormal na pagtatakda, ang konsistenteng paggamit ng notasyong prefix o postfix ay nag-aalis ng pangangailangan para sa anumang parentheses. </p><p>Upang ilarawan ang isang punsiyon, ang notasyong tuldok(dot notation) ay minsang ginagmait upang bigyang diin ang punsiyonal na kalikasan ng ekspresyon ng hindi nagtatakda ng espesyal na simbol sa <a href="/w/index.php?title=Baribaulo&action=edit&redlink=1" class="new" title="Baribaulo (hindi pa naisusulat)">baribaulo</a>. Halimbawa, ang <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 \scriptstyle a(\cdot )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>a</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle a(\cdot )^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6744e555eface082db79d767aaa02b20cad314b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.438ex; height:2.509ex;" alt="{\displaystyle \scriptstyle a(\cdot )^{2}}"></span> ay tumatayo para sa punsiyong <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 \textstyle x\mapsto ax^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle x\mapsto ax^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f691dd4d0aea1e9524631e3cf2bbf03f6651add" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.557ex; height:2.509ex;" alt="{\displaystyle \textstyle x\mapsto ax^{2}}"></span>, ang <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 \scriptstyle \int _{a}^{\,\cdot }f(u)du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mo>⋅</mo> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>u</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfcf988419b3bc72c63a4ca3409347509d1ee9a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.304ex; height:2.343ex;" alt="{\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du}"></span> ay tumatayo para sa punsiyong <a href="/wiki/Integral" class="mw-redirect" title="Integral">integral</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 \scriptstyle x\mapsto \int _{a}^{x}f(u)du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>x</mi> <mo stretchy="false">↦</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>u</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c0ade0c28f08bbd558b545185656528b45ea4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.892ex; height:2.509ex;" alt="{\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du}"></span> at iba pa. </p> <div class="mw-heading mw-heading2"><h2 id="Mga_uri_ng_punsiyon">Mga uri ng punsiyon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Punsiyon_(matematika)&veaction=edit&section=3" title="Baguhin seksiyon: Mga uri ng punsiyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Punsiyon_(matematika)&action=edit&section=3" title="Edit section's source code: Mga uri ng punsiyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Elementaryang_mga_punsiyon">Elementaryang mga punsiyon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Punsiyon_(matematika)&veaction=edit&section=4" title="Baguhin seksiyon: Elementaryang mga punsiyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Punsiyon_(matematika)&action=edit&section=4" title="Edit section's source code: Elementaryang mga punsiyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Elementaryang_mga_punsiyon&action=edit&redlink=1" class="new" title="Elementaryang mga punsiyon (hindi pa naisusulat)">Elementaryang mga punsiyon</a> ay mga punsiyon na nilikha para sa mga pundamental na operasyon ng matematika gaya ng <a href="/wiki/Adisyon" class="mw-redirect" title="Adisyon">adisyon</a>, <a href="/w/index.php?title=Substraksiyon&action=edit&redlink=1" class="new" title="Substraksiyon (hindi pa naisusulat)">substraksiyon</a>, <a href="/wiki/Multiplikasyon" class="mw-redirect" title="Multiplikasyon">multiplikasyon</a> at <a href="/wiki/Dibisyon" class="mw-redirect" title="Dibisyon">dibisyon</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Mga_punsiyong_alhebraiko">Mga punsiyong alhebraiko</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Punsiyon_(matematika)&veaction=edit&section=5" title="Baguhin seksiyon: Mga punsiyong alhebraiko" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Punsiyon_(matematika)&action=edit&section=5" title="Edit section's source code: Mga punsiyong alhebraiko"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Polinomial" class="mw-redirect" title="Polinomial">Polinomial</a> <ul><li><a href="/wiki/Linyar_na_punsiyon" class="mw-redirect" title="Linyar na punsiyon">Linyar na punsiyon</a>: Unang digri na polinomial, ang grap ay diretsong linya graph is a.</li> <li><a href="/wiki/Kwadratiko_na_punsiyon" class="mw-redirect" title="Kwadratiko na punsiyon">Kwadratiko na punsiyon</a>: Ikalawang digri na polinomial, ang grap ay parabola.</li> <li><a href="/wiki/Kubiko_na_punsiyon" title="Kubiko na punsiyon">Kubiko na punsiyon</a>: Ikatlong digri na polinomial.</li> <li><a href="/wiki/Kwartiko_na_punsiyon" title="Kwartiko na punsiyon">Kwartiko na punsiyon</a>: Ikaapat na digri na polinomial</li> <li><a href="/wiki/Kwintiko_na_punsiyon" title="Kwintiko na punsiyon">Kwintiko na punsiyon</a>: Ikalimang digri na polinomial.</li> <li><a href="/wiki/Sekstiko_na_punsiyon" title="Sekstiko na punsiyon">Sekstiko na punsiyon</a>: Ikaanim na digri na polynomial.</li></ul></li> <li><a href="/wiki/Rasyonal_na_punsiyon" title="Rasyonal na punsiyon">Rasyonal na punsiyon</a>: <a href="/wiki/Rasyo" title="Rasyo">rasyo</a> ng dalawang polinomial.</li> <li><a href="/wiki/Ika_n_na_ugat" class="mw-redirect" title="Ika n na ugat">ika n na ugat</a> <ul><li><a href="/wiki/Ugat_ng_Kwadrado" class="mw-redirect" title="Ugat ng Kwadrado">Ugat ng Kwadrado</a></li> <li><a href="/wiki/Ugat_ng_Kubiko" class="mw-redirect" title="Ugat ng Kubiko">Ugat ng Kubiko</a></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Mga_elementaryang_transendental_na_punsiyon">Mga elementaryang transendental na punsiyon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Punsiyon_(matematika)&veaction=edit&section=6" title="Baguhin seksiyon: Mga elementaryang transendental na punsiyon" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Punsiyon_(matematika)&action=edit&section=6" title="Edit section's source code: Mga elementaryang transendental na punsiyon"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Eksponensiyal_na_punsiyon" title="Eksponensiyal na punsiyon">Eksponensiyal na punsiyon</a></li> <li><a href="/wiki/Hyperboliko_na_punsiyon" class="mw-redirect" title="Hyperboliko na punsiyon">Hyperboliko na punsiyon</a></li> <li><a href="/wiki/Logaritmo" title="Logaritmo">Logaritmo</a> <ul><li><a href="/w/index.php?title=Natural_na_logaritmo&action=edit&redlink=1" class="new" title="Natural na logaritmo (hindi pa naisusulat)">Natural na logaritmo</a></li> <li><a href="/w/index.php?title=Karaniwang_logaritmo&action=edit&redlink=1" class="new" title="Karaniwang logaritmo (hindi pa naisusulat)">Karaniwang logaritmo</a></li> <li><a href="/w/index.php?title=Binaryong_logaritmo&action=edit&redlink=1" class="new" title="Binaryong logaritmo (hindi pa naisusulat)">Binaryong logaritmo</a></li> <li><a href="/w/index.php?title=Walang_hanggang_logaritmo&action=edit&redlink=1" class="new" title="Walang hanggang logaritmo (hindi pa naisusulat)">Walang hanggang logaritmo</a></li></ul></li> <li><a href="/w/index.php?title=Kapangyarihang_punsiyon&action=edit&redlink=1" class="new" title="Kapangyarihang punsiyon (hindi pa naisusulat)">Kapangyarihang punsiyon</a></li> <li>Peryodikong mga punsiyon <ul><li><a href="/wiki/Trigonometriya" title="Trigonometriya">Trigonometrikong punsiyon</a>: <a href="/wiki/Sine" class="mw-disambig" title="Sine">sine</a>, <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>, <a href="/wiki/Tangent" title="Tangent">tangent</a>, at iba pa.</li> <li><a href="/w/index.php?title=Sawtooth_na_alon&action=edit&redlink=1" class="new" title="Sawtooth na alon (hindi pa naisusulat)">Sawtooth na alon</a></li> <li><a href="/w/index.php?title=Kwadradong_alon&action=edit&redlink=1" class="new" title="Kwadradong alon (hindi pa naisusulat)">Kwadradong alon</a></li> <li><a href="/w/index.php?title=Trianggulong_alon&action=edit&redlink=1" class="new" title="Trianggulong alon (hindi pa naisusulat)">Trianggulong alon</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Mga_sanggunian">Mga sanggunian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Punsiyon_(matematika)&veaction=edit&section=7" title="Baguhin seksiyon: Mga sanggunian" class="mw-editsection-visualeditor"><span>baguhin</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Punsiyon_(matematika)&action=edit&section=7" title="Edit section's source code: Mga sanggunian"><span>baguhin ang wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2121818">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.lingvozone.com/main.jsp?action=translation&do=dictionary&language_id_from=23&language_id_to=38&word=function">Salin ng <i>function</i></a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r2132731">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFMarissa_R._Enriquez2012" class="citation book cs1">Marissa R. Enriquez (2012). <i>English-Tagalog Tagalog-English Dictionary</i>. Amos Books, Inc. p. 424. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Natatangi:Mga_pinagmulang_aklat/971-0324-24-1" title="Natatangi:Mga pinagmulang aklat/971-0324-24-1"><bdi>971-0324-24-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=English-Tagalog+Tagalog-English+Dictionary&rft.pages=424&rft.pub=Amos+Books%2C+Inc.&rft.date=2012&rft.isbn=971-0324-24-1&rft.au=Marissa+R.+Enriquez&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APunsiyon+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_book" title="Padron:Cite book">cite book</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2132731"><cite id="CITEREFEthan_D_Bloch2011" class="citation book cs1">Ethan D Bloch (2011). <a rel="nofollow" class="external text" href="https://archive.org/details/proofsfundamenta0002bloc"><i>Proofs and Fundamentals: A First Course in Abstract Mathematics</i></a>. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/proofsfundamenta0002bloc/page/131">131</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Natatangi:Mga_pinagmulang_aklat/978-1-4419-7126-5" title="Natatangi:Mga pinagmulang aklat/978-1-4419-7126-5"><bdi>978-1-4419-7126-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+and+Fundamentals%3A+A+First+Course+in+Abstract+Mathematics&rft.pages=131&rft.pub=Springer&rft.date=2011&rft.isbn=978-1-4419-7126-5&rft.au=Ethan+D+Bloch&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fproofsfundamenta0002bloc&rfr_id=info%3Asid%2Ftl.wikipedia.org%3APunsiyon+%28matematika%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Padron:Cite_book" title="Padron:Cite book">cite book</a>}}</code>: CS1 maint: date auto-translated (<a href="/wiki/Kategorya:CS1_maint:_date_auto-translated" title="Kategorya:CS1 maint: date auto-translated">link</a>)</span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r2023235">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Alhebra" title="Alhebra">Alhebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alhebrang_basal" title="Alhebrang basal">Basal</a></li> <li><a href="/wiki/Alhebrang_elementaryo" title="Alhebrang elementaryo">Elementaryo</a></li> <li><a href="/w/index.php?title=Alhebrang_homolohika&action=edit&redlink=1" class="new" title="Alhebrang homolohika (hindi pa naisusulat)">Homolohika</a></li> <li><a href="/w/index.php?title=Alhebrang_komutatibo&action=edit&redlink=1" class="new" title="Alhebrang komutatibo (hindi pa naisusulat)">Komutatibo</a></li> <li><a href="/wiki/Alhebrang_linyar" title="Alhebrang linyar">Linyar</a> <ul><li><a href="/wiki/Alhebrang_multilinyar" title="Alhebrang multilinyar">multi</a></li></ul></li> <li><a href="/w/index.php?title=Alhebrang_pangkalahatan&action=edit&redlink=1" class="new" title="Alhebrang 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title="Teorya ng ayos (hindi pa naisusulat)">ayos</a></li> <li><a href="/wiki/Teorya_ng_grap" title="Teorya ng grap">grap</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Heometriya" title="Heometriya">Heometriya</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Heometriyang_alhebraiko" title="Heometriyang alhebraiko">Alhebraiko</a></li> <li><a href="/w/index.php?title=Heometriyang_aritmetiko&action=edit&redlink=1" class="new" title="Heometriyang aritmetiko (hindi pa naisusulat)">Aritmetiko</a></li> <li><a href="/wiki/Heometriyang_deribatibo" title="Heometriyang deribatibo">Deribatibo</a></li> <li><a href="/wiki/Heometriyang_diskreto" class="mw-redirect" title="Heometriyang diskreto">Diskreto</a></li> <li><a href="/w/index.php?title=Heometriyang_Euclides&action=edit&redlink=1" class="new" title="Heometriyang Euclides (hindi pa naisusulat)">Euclides</a></li> <li><a href="/w/index.php?title=Heometriyang_natatapos&action=edit&redlink=1" class="new" title="Heometriyang natatapos (hindi pa naisusulat)">Natatapos</a></li> <li><a href="/wiki/Heometriyang_pasuri" title="Heometriyang pasuri">Pasuri</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Teorya_ng_bilang" title="Teorya ng bilang">Teorya ng bilang</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=Alhebraikong_teorya_ng_bilang&action=edit&redlink=1" class="new" title="Alhebraikong teorya ng bilang (hindi pa naisusulat)">Alhebraiko</a></li> <li><a href="/wiki/Aritmetika" title="Aritmetika">Aritmetika</a></li> <li><a href="/w/index.php?title=Heometriyang_diopantika&action=edit&redlink=1" class="new" title="Heometriyang diopantika (hindi pa naisusulat)">Heometriyang diopantika</a></li> <li><a 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