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Se mere på \u003Ca href=\"/wiki/Wikipedia:Discord\" title=\"Wikipedia:Discord\"\u003EWikipedia:Discord\u003C/a\u003E\u003C/div\u003E\u003Cp\u003E\u003Ci\u003E\u003Csmall\u003E(\u003Ca href=\"/wiki/Hj%C3%A6lp:Sitenotice\" title=\"Hjælp:Sitenotice\"\u003ELæs her om sitenotice\u003C/a\u003E)\u003C/small\u003E\u003C/i\u003E\u003C/p\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></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="Websted"> <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="Indhold" 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">Indhold</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">flyt til sidebjælken</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skjul</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">Indledning</div> </a> </li> <li id="toc-Notation_og_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notation_og_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Notation og definition</span> </div> </a> <ul id="toc-Notation_og_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Egenskaber" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Egenskaber"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Egenskaber</span> </div> </a> <ul id="toc-Egenskaber-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metode" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Metode"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Metode</span> </div> </a> <ul id="toc-Metode-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_også" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_også"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Se også</span> </div> </a> <ul id="toc-Se_også-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Indhold" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Vis/skjul indholdsfortegnelsen" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Vis/skjul indholdsfortegnelsen</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Invers funktion</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Gå til en artikel på et andet sprog. Tilgængelig på 61 sprog" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-61" 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">61 sprog</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%98%E1%88%8B%E1%88%BD_%E1%8A%A0%E1%88%B5%E1%88%A8%E1%8A%AB%E1%89%A2" title="መላሽ አስረካቢ – amharisk" lang="am" hreflang="am" data-title="መላሽ አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="amharisk" class="interlanguage-link-target"><span>አማርኛ</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_%D8%B9%D9%83%D8%B3%D9%8A%D8%A9" title="دالة عكسية – arabisk" lang="ar" hreflang="ar" data-title="دالة عكسية" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/T%C9%99rs_funksiya" title="Tərs funksiya – aserbajdsjansk" lang="az" hreflang="az" data-title="Tərs funksiya" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B4%D0%B2%D0%B0%D1%80%D0%BE%D1%82%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F" title="Адваротная функцыя – belarusisk" lang="be" hreflang="be" data-title="Адваротная функцыя" data-language-autonym="Беларуская" data-language-local-name="belarusisk" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Обратна функция – bulgarsk" lang="bg" hreflang="bg" data-title="Обратна функция" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Inverzna_funkcija" title="Inverzna funkcija – bosnisk" lang="bs" hreflang="bs" data-title="Inverzna funkcija" data-language-autonym="Bosanski" data-language-local-name="bosnisk" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3_inversa" title="Funció inversa – catalansk" lang="ca" hreflang="ca" data-title="Funció inversa" data-language-autonym="Català" data-language-local-name="catalansk" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86%DB%8C_%DA%BE%DB%95%DA%B5%DA%AF%DB%95%DA%95%D8%A7%D9%88%DB%95" title="فانکشنی ھەڵگەڕاوە – sorani" lang="ckb" hreflang="ckb" data-title="فانکشنی ھەڵگەڕاوە" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Inverzn%C3%AD_zobrazen%C3%AD" title="Inverzní zobrazení – tjekkisk" lang="cs" hreflang="cs" data-title="Inverzní zobrazení" data-language-autonym="Čeština" data-language-local-name="tjekkisk" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D1%83%D1%82%C4%83%D0%BD%D0%BB%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8" title="Кутăнла функци – tjuvasjisk" lang="cv" hreflang="cv" data-title="Кутăнла функци" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjisk" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Umkehrfunktion" title="Umkehrfunktion – tysk" lang="de" hreflang="de" data-title="Umkehrfunktion" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CF%84%CE%AF%CF%83%CF%84%CF%81%CE%BF%CF%86%CE%B7_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Αντίστροφη συνάρτηση – græsk" lang="el" hreflang="el" data-title="Αντίστροφη συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="græsk" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Inverse_function" title="Inverse function – engelsk" lang="en" hreflang="en" data-title="Inverse function" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_inversa" title="Función inversa – spansk" lang="es" hreflang="es" data-title="Función inversa" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/P%C3%B6%C3%B6rdfunktsioon" title="Pöördfunktsioon – estisk" lang="et" hreflang="et" data-title="Pöördfunktsioon" data-language-autonym="Eesti" data-language-local-name="estisk" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Alderantzizko_funtzio" title="Alderantzizko funtzio – baskisk" lang="eu" hreflang="eu" data-title="Alderantzizko funtzio" data-language-autonym="Euskara" data-language-local-name="baskisk" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D9%88%D8%A7%D8%B1%D9%88%D9%86" title="تابع وارون – persisk" lang="fa" hreflang="fa" data-title="تابع وارون" data-language-autonym="فارسی" data-language-local-name="persisk" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/K%C3%A4%C3%A4nteisfunktio" title="Käänteisfunktio – finsk" lang="fi" hreflang="fi" data-title="Käänteisfunktio" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque – fransk" lang="fr" hreflang="fr" data-title="Bijection réciproque" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_inversa" title="Función inversa – galicisk" lang="gl" hreflang="gl" data-title="Función inversa" data-language-autonym="Galego" data-language-local-name="galicisk" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%94%D7%A4%D7%99%D7%9B%D7%94" title="פונקציה הפיכה – hebraisk" lang="he" hreflang="he" data-title="פונקציה הפיכה" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%B2%E0%A5%8B%E0%A4%AE_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="प्रतिलोम फलन – hindi" lang="hi" hreflang="hi" data-title="प्रतिलोम फलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Inverzna_funkcija" title="Inverzna funkcija – kroatisk" lang="hr" hreflang="hr" data-title="Inverzna funkcija" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Inverz_f%C3%BCggv%C3%A9ny" title="Inverz függvény – ungarsk" lang="hu" hreflang="hu" data-title="Inverz függvény" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%AF%D5%A1%D5%A4%D5%A1%D6%80%D5%B1_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1" title="Հակադարձ ֆունկցիա – armensk" lang="hy" hreflang="hy" data-title="Հակադարձ ֆունկցիա" data-language-autonym="Հայերեն" data-language-local-name="armensk" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Function_inverse" title="Function inverse – interlingua" lang="ia" hreflang="ia" data-title="Function inverse" 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_invers" title="Fungsi invers – indonesisk" lang="id" hreflang="id" data-title="Fungsi invers" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Simetra_elemento" title="Simetra elemento – ido" lang="io" hreflang="io" data-title="Simetra elemento" 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/Andhverfa" title="Andhverfa – islandsk" lang="is" hreflang="is" data-title="Andhverfa" data-language-autonym="Íslenska" data-language-local-name="islandsk" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_inversa" title="Funzione inversa – italiensk" lang="it" hreflang="it" data-title="Funzione inversa" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%86%E5%86%99%E5%83%8F" title="逆写像 – japansk" lang="ja" hreflang="ja" data-title="逆写像" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B5%D1%80%D1%96_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Кері функция – kasakhisk" lang="kk" hreflang="kk" data-title="Кері функция" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%8D%E0%B2%AF%E0%B2%B8%E0%B3%8D%E0%B2%A4_%E0%B2%AB%E0%B2%B2%E0%B2%A8" title="ವ್ಯಸ್ತ ಫಲನ – kannada" lang="kn" hreflang="kn" data-title="ವ್ಯಸ್ತ ಫಲನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%AD%ED%95%A8%EC%88%98" title="역함수 – koreansk" lang="ko" hreflang="ko" data-title="역함수" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio_inversa" title="Functio inversa – latin" lang="la" hreflang="la" data-title="Functio inversa" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Aplicaziun_recipruca" title="Aplicaziun recipruca – Lombard" lang="lmo" hreflang="lmo" data-title="Aplicaziun recipruca" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Инверзна функција – makedonsk" lang="mk" hreflang="mk" data-title="Инверзна функција" data-language-autonym="Македонски" data-language-local-name="makedonsk" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi_songsang" title="Fungsi songsang – malajisk" lang="ms" hreflang="ms" data-title="Fungsi songsang" data-language-autonym="Bahasa Melayu" data-language-local-name="malajisk" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%AA%E0%A4%B0%E0%A5%80%E0%A4%A4_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="विपरीत फलन – nepalesisk" lang="ne" hreflang="ne" data-title="विपरीत फलन" data-language-autonym="नेपाली" data-language-local-name="nepalesisk" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inverteerbaar" title="Inverteerbaar – nederlandsk" lang="nl" hreflang="nl" data-title="Inverteerbaar" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Invers_funksjon" title="Invers funksjon – nynorsk" lang="nn" hreflang="nn" data-title="Invers funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Invers_funksjon" title="Invers funksjon – bokmål" lang="nb" hreflang="nb" data-title="Invers funksjon" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_odwrotna" title="Funkcja odwrotna – polsk" lang="pl" hreflang="pl" data-title="Funkcja odwrotna" data-language-autonym="Polski" data-language-local-name="polsk" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_inversa" title="Função inversa – portugisisk" lang="pt" hreflang="pt" data-title="Função inversa" data-language-autonym="Português" data-language-local-name="portugisisk" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie_invers%C4%83" title="Funcție inversă – rumænsk" lang="ro" hreflang="ro" data-title="Funcție inversă" data-language-autonym="Română" data-language-local-name="rumænsk" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Обратная функция – russisk" lang="ru" hreflang="ru" data-title="Обратная функция" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Inverse_function" title="Inverse function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Inverse function" 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/Inverzn%C3%A9_zobrazenie_(funkcia)" title="Inverzné zobrazenie (funkcia) – slovakisk" lang="sk" hreflang="sk" data-title="Inverzné zobrazenie (funkcia)" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Inverzna_funkcija" title="Inverzna funkcija – slovensk" lang="sl" hreflang="sl" data-title="Inverzna funkcija" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni_i_anasjellt%C3%AB" title="Funksioni i anasjelltë – albansk" lang="sq" hreflang="sq" data-title="Funksioni i anasjelltë" data-language-autonym="Shqip" data-language-local-name="albansk" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Инверзна функција – serbisk" lang="sr" hreflang="sr" data-title="Инверзна функција" data-language-autonym="Српски / srpski" data-language-local-name="serbisk" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Invers_funktion" title="Invers funktion – svensk" lang="sv" hreflang="sv" data-title="Invers funktion" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%81%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="நேர்மாறுச் சார்பு – tamil" lang="ta" hreflang="ta" data-title="நேர்மாறுச் சார்பு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Inbersong_punsiyon" title="Inbersong punsiyon – tagalog" lang="tl" hreflang="tl" data-title="Inbersong punsiyon" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ters_fonksiyon" title="Ters fonksiyon – tyrkisk" lang="tr" hreflang="tr" data-title="Ters fonksiyon" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B1%D0%B5%D1%80%D0%BD%D0%B5%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Обернена функція – ukrainsk" lang="uk" hreflang="uk" data-title="Обернена функція" data-language-autonym="Українська" data-language-local-name="ukrainsk" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_ng%C6%B0%E1%BB%A3c" title="Hàm ngược – vietnamesisk" lang="vi" hreflang="vi" data-title="Hàm ngược" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8F%8D%E5%87%BD%E6%95%B0" title="反函数 – wu-kinesisk" lang="wuu" hreflang="wuu" data-title="反函数" data-language-autonym="吴语" data-language-local-name="wu-kinesisk" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%8D%E5%87%BD%E6%95%B8" title="反函數 – kinesisk" lang="zh" hreflang="zh" data-title="反函數" data-language-autonym="中文" data-language-local-name="kinesisk" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ge%CC%8Dk-h%C3%A2m-s%C3%B2%CD%98" title="Ge̍k-hâm-sò͘ – min-kinesisk" lang="nan" hreflang="nan" data-title="Ge̍k-hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min-kinesisk" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8F%8D%E5%87%BD%E6%95%B8" title="反函數 – kantonesisk" lang="yue" hreflang="yue" data-title="反函數" data-language-autonym="粵語" data-language-local-name="kantonesisk" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q191884#sitelinks-wikipedia" title="Redigér sproglinks" class="wbc-editpage">Redigér links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Navnerum"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Invers_funktion" title="Se indholdssiden [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a 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10%;border-collapse:collapse;border:1px solid #c0c090;background-color:#f8eaba;min-width:80%;box-sizing:border-box}.mw-parser-output .tmbox.mbox-small{min-width:0}.mw-parser-output .mediawiki .mbox-inside .tmbox{margin:2px 0;width:100%}.mw-parser-output .mbox-inside .tmbox.mbox-small{line-height:1.5em;font-size:100%}.mw-parser-output table.tmbox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output table.tmbox-delete{border:2px solid #b32424}.mw-parser-output table.tmbox-content{border:2px solid #f28500}.mw-parser-output table.tmbox-style{border:2px solid #fc3}.mw-parser-output table.tmbox-move{border:2px solid #9932cc}.mw-parser-output table.tmbox-protection,.mw-parser-output table.tmbox-notice{border:1px solid #c0c090}.mw-parser-output table.fmbox{clear:both;margin:0.2em 0;width:100%;border:1px solid #a2a9b1;background-color:#f8f9fa;box-sizing:border-box}.mw-parser-output table.fmbox-system{background-color:#f8f9fa}.mw-parser-output table.fmbox-warning{border:1px solid #bb7070;background-color:#ffdbdb}.mw-parser-output table.fmbox-editnotice{background-color:transparent}.mw-parser-output div.mw-warning-with-logexcerpt,.mw-parser-output div.mw-lag-warn-high,.mw-parser-output div.mw-cascadeprotectedwarning,.mw-parser-output div/**/#mw-protect-cascadeon,.mw-parser-output div.titleblacklist-warning,.mw-parser-output div.locked-warning{clear:both;margin:0.2em 0;border:1px solid #bb7070;background-color:#ffdbdb;padding:0.25em 0.9em;box-sizing:border-box}html body.mediawiki .mw-parser-output .mbox-small{clear:right;float:right;margin:4px 0 4px 1em;box-sizing:border-box;width:238px;font-size:88%;line-height:1.25em}html body.mediawiki .mw-parser-output .mbox-small-left{margin:4px 1em 4px 0;box-sizing:border-box;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .compact-ambox table .mbox-image,.mw-parser-output .compact-ambox table .mbox-imageright,.mw-parser-output .compact-ambox table .mbox-empty-cell{display:none}.mw-parser-output .compact-ambox table.ambox{border:none;border-collapse:collapse;background-color:transparent;margin:0 0 0 1.6em!important;padding:0!important;width:auto;display:block}body.mediawiki .mw-parser-output .compact-ambox table.mbox-small-left{font-size:100%;width:auto;margin:0}.mw-parser-output .compact-ambox table .mbox-text{padding:0;margin:0}.mw-parser-output .compact-ambox table .mbox-text-span{display:list-item;line-height:1.5em;list-style-type:square}.mw-parser-output .skin-vector .compact-ambox table .mbox-text-span{list-style-type:disc}.mw-parser-output .compact-ambox .hide-when-compact{display:none}.mw-parser-output .template-documentation{clear:both;margin:1em 0 0 0;border:1px solid #aaa;background-color:#ecfcf4;padding:1em}</style> <p>I <a href="/wiki/Matematik" title="Matematik">matematikken</a> er en <b>invers funktion</b> simpelthen en <a href="/wiki/Funktion_(matematik)" title="Funktion (matematik)">funktion</a> der gør det modsatte af en given funktion. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation_og_definition">Notation og definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invers_funktion&veaction=edit&section=1" title="Redigér afsnit: Notation og definition" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Invers_funktion&action=edit&section=1" title="Edit section's source code: Notation og definition"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ved hjælp af <a href="/wiki/Sammensat_funktion" title="Sammensat funktion">sammensatte funktioner</a> skal der for den inverse funktion gælde at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span> af funktionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> er lig x og <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> af funktionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span> er lig x: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(f(x))=x\,\land f(f^{-1}(x))=x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>∧</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(f(x))=x\,\land f(f^{-1}(x))=x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc31d241f02e48ed5787cff251cbade2d383e1ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.973ex; height:3.176ex;" alt="{\displaystyle f^{-1}(f(x))=x\,\land f(f^{-1}(x))=x\,}"></span></dd></dl> <p>De to funktioner ophæver således hinanden og kaldes derfor hinandens inverse. Grafisk spejles de to om linjen <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}"></span>. </p><p>Den hævede tekst "-1" i <i>f<sup>-1</sup></i> er ikke en <a href="/wiki/Eksponent" class="mw-redirect" title="Eksponent">eksponent</a>, men en forkortet skrivning af <i>f<sup>∘-1</sup></i>, hvor ∘ er <a href="/wiki/Sammensat_funktion" title="Sammensat funktion">sammensætningsoperatoren ∘</a>. Tilsvarende betyder <i>f</i> ²(x) normalt ikke kvadratet på <i>f</i>(x), men derimod <i>f</i> gjort to gange: <i>f<sup>∘2</sup></i>(x), <i>f</i>(<i>f</i>(x)). Undtagelser fra dette er der i <a href="/wiki/Trigonometri" title="Trigonometri">trigonometrien</a>, hvor sin²(x) sædvandligvis betyder kvadratet på sin(x). For at vise den inverse funktion bruger man derfor ofte forstavelsen arc. I <a href="/wiki/Infinitesimalregning" title="Infinitesimalregning">infinitesimalregningen</a> angiver n i <i>f</i> <sup>n</sup> typisk den n'te afledte af f. </p><p>Et eksempel på en funktion og dens inverse er <a href="/wiki/Eksponentialfunktion" class="mw-redirect" title="Eksponentialfunktion">eksponentialfunktionen</a> og <a href="/wiki/Logaritme" title="Logaritme">logaritmefunktionen</a>: </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 \log _{a}(a^{x})=x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}(a^{x})=x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83bfafa9fec6c05e7c0c9bfa049ec15bd47d367e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.1ex; height:2.843ex;" alt="{\displaystyle \log _{a}(a^{x})=x\,}"></span></dd></dl> <p>og </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 a^{\log _{a}(x)}=x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\log _{a}(x)}=x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca0349ebb764121149ed21552c9a64d0057c964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.468ex; height:2.843ex;" alt="{\displaystyle a^{\log _{a}(x)}=x\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Egenskaber">Egenskaber</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invers_funktion&veaction=edit&section=2" title="Redigér afsnit: Egenskaber" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Invers_funktion&action=edit&section=2" title="Edit section's source code: Egenskaber"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Når en invers funktion eksisterer er den unik</li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f^{-1}(x))'={\frac {1}{f'(f^{-1}(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f^{-1}(x))'={\frac {1}{f'(f^{-1}(x))}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348ef7bb841a22039dc33bb6d360f38cb4e3b283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.828ex; height:6.009ex;" alt="{\displaystyle (f^{-1}(x))'={\frac {1}{f'(f^{-1}(x))}}}"></span>, for alle <a href="/wiki/Reelle_tal" title="Reelle tal">reelle</a> x</li></ul> <div class="mw-heading mw-heading2"><h2 id="Metode">Metode</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invers_funktion&veaction=edit&section=3" title="Redigér afsnit: Metode" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Invers_funktion&action=edit&section=3" title="Edit section's source code: Metode"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Har man eksempelvis andengradspolynomiet: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)=x^{2}+x-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)=x^{2}+x-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742038ee5ec3e473ea05dddc48cafdac6033f3bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.327ex; height:3.176ex;" alt="{\displaystyle y=f(x)=x^{2}+x-2}"></span>, vil man kunne finde forskriften for den inverse funktion ved først at ombytte x og y og derefter isolere y: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y^{2}+y-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>y</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y^{2}+y-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f7e621cb26a88b2527709534377468023e7837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.642ex; height:3.009ex;" alt="{\displaystyle x=y^{2}+y-2}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x)=y={\frac {1}{2}}(12\pm {\sqrt {9+4x}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>12</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>9</mn> <mo>+</mo> <mn>4</mn> <mi>x</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(x)=y={\frac {1}{2}}(12\pm {\sqrt {9+4x}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/557fad46b937ae75e9e2f342d2cb3eeb0748f9ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.549ex; height:5.176ex;" alt="{\displaystyle f^{-1}(x)=y={\frac {1}{2}}(12\pm {\sqrt {9+4x}})}"></span> </p><p>Der er altså to løsninger, men kun den ene kan anvendes. Det skyldes at spejler man parablen i linjen y=x (for at få den inverse funktion) får man to værdier af y for hver x værdi og det er ikke en funktion. Man må vælge hvilken en af løsningerne man vil bruge. Altså hvilken gren af parablen man vil bruge. Parablen her bliver skåret i to dele af parablens toppunkt. </p><p>Tænker man sig som eksempel et trediegradspolynomium kan man få op til tre dele. Hver del modsvarer et stykke af funktionen. Funktionens graf deles op af eventuelle ekstrema. </p> <div class="mw-heading mw-heading2"><h2 id="Se_også"><span id="Se_ogs.C3.A5"></span>Se også</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Invers_funktion&veaction=edit&section=4" title="Redigér afsnit: Se også" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Invers_funktion&action=edit&section=4" title="Edit section's source code: Se også"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Multiplikativ_invers" class="mw-redirect" title="Multiplikativ invers">Multiplikativ invers</a></li> <li><a href="/w/index.php?title=Additiv_invers&action=edit&redlink=1" class="new" title="Additiv invers (ikke skrevet endnu)">Additiv invers</a></li></ul> <div class="navbox-styles"><style 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