Signum (wiskunde) - Wikipedia

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class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Eigenschappen</span> </div> </a> <ul id="toc-Eigenschappen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voorbeeld" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voorbeeld"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Voorbeeld</span> </div> </a> <button aria-controls="toc-Voorbeeld-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>Voorbeeld-subkopje inklappen</span> </button> <ul id="toc-Voorbeeld-sublist" class="vector-toc-list"> <li id="toc-Afleiding" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Afleiding"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Afleiding</span> </div> </a> <ul id="toc-Afleiding-sublist" 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vector-button-flush-left" title="Inhoudsopgave" > <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="Inhoudsopgave omschakelen" > <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">Inhoudsopgave omschakelen</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">Signum (wiskunde)</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="Ga naar een artikel in een andere taal. Beschikbaar in 43 talen" > <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-43" 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">43 talen</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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%A7%D9%84%D8%A5%D8%B4%D8%A7%D8%B1%D8%A9" title="دالة الإشارة – Arabisch" lang="ar" hreflang="ar" data-title="دالة الإشارة" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%B8%D0%B3%D0%BD%D1%83%D0%BC" title="Сигнум – Bulgaars" lang="bg" hreflang="bg" data-title="Сигнум" data-language-autonym="Български" data-language-local-name="Bulgaars" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9A%E0%A6%BF%E0%A6%B9%E0%A7%8D%E0%A6%A8_%E0%A6%85%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95" title="চিহ্ন অপেক্ষক – Bengaals" lang="bn" hreflang="bn" data-title="চিহ্ন অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bengaals" 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/Signum_funkcija" title="Signum funkcija – Bosnisch" lang="bs" hreflang="bs" data-title="Signum funkcija" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" 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_signe" title="Funció signe – Catalaans" lang="ca" hreflang="ca" data-title="Funció signe" data-language-autonym="Català" data-language-local-name="Catalaans" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Funkce_signum" title="Funkce signum – Tsjechisch" lang="cs" hreflang="cs" data-title="Funkce signum" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" 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/Sgn" title="Sgn – Tsjoevasjisch" lang="cv" hreflang="cv" data-title="Sgn" data-language-autonym="Чӑвашла" data-language-local-name="Tsjoevasjisch" 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/Vorzeichenfunktion" title="Vorzeichenfunktion – Duits" lang="de" hreflang="de" data-title="Vorzeichenfunktion" data-language-autonym="Deutsch" data-language-local-name="Duits" 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_%CF%80%CF%81%CE%BF%CF%83%CE%AE%CE%BC%CE%BF%CF%85" title="Συνάρτηση προσήμου – Grieks" lang="el" hreflang="el" data-title="Συνάρτηση προσήμου" data-language-autonym="Ελληνικά" data-language-local-name="Grieks" 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/Sign_function" title="Sign function – Engels" lang="en" hreflang="en" data-title="Sign function" data-language-autonym="English" data-language-local-name="Engels" 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/Signuma_funkcio" title="Signuma funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Signuma funkcio" 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_signo" title="Función signo – Spaans" lang="es" hreflang="es" data-title="Función signo" data-language-autonym="Español" data-language-local-name="Spaans" 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/Signumfunktsioon" title="Signumfunktsioon – Estisch" lang="et" hreflang="et" data-title="Signumfunktsioon" data-language-autonym="Eesti" data-language-local-name="Estisch" class="interlanguage-link-target"><span>Eesti</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_%D8%B9%D9%84%D8%A7%D9%85%D8%AA" title="تابع علامت – Perzisch" lang="fa" hreflang="fa" data-title="تابع علامت" data-language-autonym="فارسی" data-language-local-name="Perzisch" 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/Signum-funktio" title="Signum-funktio – Fins" lang="fi" hreflang="fi" data-title="Signum-funktio" data-language-autonym="Suomi" data-language-local-name="Fins" 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/Fonction_signe" title="Fonction signe – Frans" lang="fr" hreflang="fr" data-title="Fonction signe" data-language-autonym="Français" data-language-local-name="Frans" 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_signo" title="Función signo – Galicisch" lang="gl" hreflang="gl" data-title="Función signo" data-language-autonym="Galego" data-language-local-name="Galicisch" 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%99%D7%AA_%D7%94%D7%A1%D7%99%D7%9E%D7%9F" title="פונקציית הסימן – Hebreeuws" lang="he" hreflang="he" data-title="פונקציית הסימן" data-language-autonym="עברית" data-language-local-name="Hebreeuws" 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%9A%E0%A4%BF%E0%A4%B9%E0%A5%8D%E0%A4%A8_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="चिह्न फलन – Hindi" lang="hi" hreflang="hi" data-title="चिह्न फलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szignumf%C3%BCggv%C3%A9ny" title="Szignumfüggvény – Hongaars" lang="hu" hreflang="hu" data-title="Szignumfüggvény" data-language-autonym="Magyar" data-language-local-name="Hongaars" 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/Sgn" title="Sgn – Armeens" lang="hy" hreflang="hy" data-title="Sgn" data-language-autonym="Հայերեն" data-language-local-name="Armeens" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_tanda" title="Fungsi tanda – Indonesisch" lang="id" hreflang="id" data-title="Fungsi tanda" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Formerkisfall" title="Formerkisfall – IJslands" lang="is" hreflang="is" data-title="Formerkisfall" data-language-autonym="Íslenska" data-language-local-name="IJslands" 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_segno" title="Funzione segno – Italiaans" lang="it" hreflang="it" data-title="Funzione segno" data-language-autonym="Italiano" data-language-local-name="Italiaans" 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/%E7%AC%A6%E5%8F%B7%E9%96%A2%E6%95%B0" title="符号関数 – Japans" lang="ja" hreflang="ja" data-title="符号関数" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%ED%98%B8%ED%95%A8%EC%88%98" title="부호함수 – Koreaans" lang="ko" hreflang="ko" data-title="부호함수" data-language-autonym="한국어" data-language-local-name="Koreaans" 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/Signum_(functio)" title="Signum (functio) – Latijn" lang="la" hreflang="la" data-title="Signum (functio)" data-language-autonym="Latina" data-language-local-name="Latijn" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Signum_(matematikk)" title="Signum (matematikk) – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Signum (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - 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_signum" title="Funkcja signum – Pools" lang="pl" hreflang="pl" data-title="Funkcja signum" data-language-autonym="Polski" data-language-local-name="Pools" 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_sinal" title="Função sinal – Portugees" lang="pt" hreflang="pt" data-title="Função sinal" data-language-autonym="Português" data-language-local-name="Portugees" 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%9Bia_semn" title="Funcția semn – Roemeens" lang="ro" hreflang="ro" data-title="Funcția semn" data-language-autonym="Română" data-language-local-name="Roemeens" 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/Sgn" title="Sgn – Russisch" lang="ru" hreflang="ru" data-title="Sgn" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Signum_(matematika)" title="Signum (matematika) – Slowaaks" lang="sk" hreflang="sk" data-title="Signum (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slowaaks" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni_i_shenj%C3%ABs" title="Funksioni i shenjës – Albanees" lang="sq" hreflang="sq" data-title="Funksioni i shenjës" data-language-autonym="Shqip" data-language-local-name="Albanees" 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%A1%D0%B8%D0%B3%D0%BD%D1%83%D0%BC_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Сигнум функција – Servisch" lang="sr" hreflang="sr" data-title="Сигнум функција" data-language-autonym="Српски / srpski" data-language-local-name="Servisch" 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/Signumfunktionen" title="Signumfunktionen – Zweeds" lang="sv" hreflang="sv" data-title="Signumfunktionen" data-language-autonym="Svenska" data-language-local-name="Zweeds" 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%95%E0%AF%81%E0%AE%B1%E0%AE%BF%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="குறிச் சார்பு – Tamil" lang="ta" hreflang="ta" data-title="குறிச் 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%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87%E0%B8%AB%E0%B8%A1%E0%B8%B2%E0%B8%A2" title="ฟังก์ชันเครื่องหมาย – Thai" lang="th" hreflang="th" data-title="ฟังก์ชันเครื่องหมาย" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Sign_function" title="Sign function – Tagalog" lang="tl" hreflang="tl" data-title="Sign function" 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/%C4%B0%C5%9Faret_fonksiyonu" title="İşaret fonksiyonu – Turks" lang="tr" hreflang="tr" data-title="İşaret fonksiyonu" data-language-autonym="Türkçe" data-language-local-name="Turks" 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/Signum-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Signum-функція – Oekraïens" lang="uk" hreflang="uk" data-title="Signum-функція" data-language-autonym="Українська" data-language-local-name="Oekraïens" 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/Signum" title="Signum – Vietnamees" lang="vi" hreflang="vi" data-title="Signum" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamees" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%AC%A6%E5%8F%B7%E5%87%BD%E6%95%B0" title="符号函数 – Chinees" lang="zh" hreflang="zh" data-title="符号函数" data-language-autonym="中文" data-language-local-name="Chinees" class="interlanguage-link-target"><span>中文</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/Q236813#sitelinks-wikipedia" title="Taalkoppelingen bewerken" class="wbc-editpage">Koppelingen bewerken</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Signum_functie.PNG/390px-Signum_functie.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/6/65/Signum_functie.PNG 2x" data-file-width="441" data-file-height="255" /></a><figcaption>Grafiek van de signumfunctie (let op de <a href="/wiki/Nulpunt_(wiskunde)" title="Nulpunt (wiskunde)">nulwaarde</a> in de <a href="/wiki/Oorsprong_(wiskunde)" title="Oorsprong (wiskunde)">oorsprong</a>)</figcaption></figure> <p>Het <b>signum</b>, als functie vaak aangeduid als <b>sgn</b>, is een eenvoudige <a href="/wiki/Wiskunde" title="Wiskunde">wiskundige</a> <a href="/wiki/Functie_(wiskunde)" title="Functie (wiskunde)">functie</a> die, zoals de naam min of meer zegt<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>, het <a href="/wiki/Teken_(wiskunde)" title="Teken (wiskunde)">teken</a> van het <a href="/wiki/Argument_(wiskunde)" title="Argument (wiskunde)">argument</a> aangeeft. Een <a href="/wiki/Negatief_getal" title="Negatief getal">negatief getal</a> als argument heeft functiewaarde −1, een positief getal als argument heeft functiewaarde +1, argument 0 heeft functiewaarde 0. </p><p>Het gebruik van de functie signum maakt het in sommige gevallen mogelijk één uitdrukking te hanteren in plaats van de diverse gevallen te onderscheiden. In plaats van te schrijven: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}f_{\text{–}}(x)&{\mbox{als }}x<0\\f_{\text{0}}(x)&{\mbox{als }}x=0\\f_{\text{+}}(x)&{\mbox{als }}x>0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>–</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>0</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>+</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}f_{\text{–}}(x)&{\mbox{als }}x<0\\f_{\text{0}}(x)&{\mbox{als }}x=0\\f_{\text{+}}(x)&{\mbox{als }}x>0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8b67521e7a1d861f97601541ffe5541996b94b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:27.342ex; height:8.843ex;" alt="{\displaystyle f(x)={\begin{cases}f_{\text{–}}(x)&{\mbox{als }}x<0\\f_{\text{0}}(x)&{\mbox{als }}x=0\\f_{\text{+}}(x)&{\mbox{als }}x>0\end{cases}}}"></span></dd></dl> <p>kan volstaan worden met de uitdrukking: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\tfrac {1}{2}}(1-\operatorname {sgn}(x))|\operatorname {sgn}(x)|f_{\text{–}}(x)+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>–</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\tfrac {1}{2}}(1-\operatorname {sgn}(x))|\operatorname {sgn}(x)|f_{\text{–}}(x)+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7096ae8949ad278b3ddd76d39a5d8913b080faec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:36.828ex; height:3.509ex;" alt="{\displaystyle f(x)={\tfrac {1}{2}}(1-\operatorname {sgn}(x))|\operatorname {sgn}(x)|f_{\text{–}}(x)+}"></span> <dl><dd><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 +(1-|\operatorname {sgn}(x)|)f_{\text{0}}(x)+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>0</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +(1-|\operatorname {sgn}(x)|)f_{\text{0}}(x)+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/885af1f4243a58eb27443256ec1e3add0be2d894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.952ex; height:2.843ex;" alt="{\displaystyle +(1-|\operatorname {sgn}(x)|)f_{\text{0}}(x)+}"></span></dd> <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 +{\tfrac {1}{2}}(1+\operatorname {sgn}(x))|\operatorname {sgn}(x)|f_{\text{+}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>+</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +{\tfrac {1}{2}}(1+\operatorname {sgn}(x))|\operatorname {sgn}(x)|f_{\text{+}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b08408f4735c1bdd136ea36c5211c697e91fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.769ex; height:3.509ex;" alt="{\displaystyle +{\tfrac {1}{2}}(1+\operatorname {sgn}(x))|\operatorname {sgn}(x)|f_{\text{+}}(x)}"></span></dd></dl></dd></dl></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Eigenschappen">Eigenschappen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Signum_(wiskunde)&veaction=edit&section=1" title="Bewerk dit kopje: Eigenschappen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Signum_(wiskunde)&action=edit&section=1" title="De broncode bewerken van de sectie: Eigenschappen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elk <a href="/wiki/Re%C3%ABel_getal" title="Reëel getal">reëel getal</a> kan worden uitgedrukt als het <a href="/wiki/Vermenigvuldigen" title="Vermenigvuldigen">product</a> van de <a href="/wiki/Absolute_waarde" title="Absolute waarde">absolute waarde</a> en de signumfunctie ervan: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=|x|\cdot \operatorname {sgn}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=|x|\cdot \operatorname {sgn}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06ed751034799ed66ab69956ae339de925941111" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.241ex; height:2.843ex;" alt="{\displaystyle x=|x|\cdot \operatorname {sgn}(x)}"></span></dd></dl> <p>Voor <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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span> geldt: </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 \operatorname {sgn}(x)={\frac {x}{|x|}}={\frac {|x|}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x)={\frac {x}{|x|}}={\frac {|x|}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce80b460fc79e827ec1e06afeed0a3325963d98e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.626ex; height:6.509ex;" alt="{\displaystyle \operatorname {sgn}(x)={\frac {x}{|x|}}={\frac {|x|}{x}}}"></span></dd></dl> <p>en dan dus ook: </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 \operatorname {sgn}(x)={\frac {\sqrt {x^{2}}}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x)={\frac {\sqrt {x^{2}}}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7702d4d767afa265bf4c2db9b49f191c7f806d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.765ex; height:5.843ex;" alt="{\displaystyle \operatorname {sgn}(x)={\frac {\sqrt {x^{2}}}{x}}}"></span></dd></dl> <p>Voor <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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span> is de signumfunctie de <a href="/wiki/Afgeleide" title="Afgeleide">afgeleide</a> van de absolutewaardefunctie. </p><p>Voor alle reële waarden van <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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span> is de signumfunctie <a href="/wiki/Differentieerbaarheid" title="Differentieerbaarheid">differentieerbaar</a>, met afgeleide 0. </p> <div class="mw-heading mw-heading2"><h2 id="Voorbeeld">Voorbeeld</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Signum_(wiskunde)&veaction=edit&section=2" title="Bewerk dit kopje: Voorbeeld" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Signum_(wiskunde)&action=edit&section=2" title="De broncode bewerken van de sectie: Voorbeeld"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Bestand:Graph1007.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Graph1007.png/200px-Graph1007.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Graph1007.png/300px-Graph1007.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Graph1007.png/400px-Graph1007.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>Grafiek van de functie <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></figcaption></figure> <p>Een voorbeeld van het gebruik van het signum is de volgende functie (de <a href="/wiki/Grafiek_(wiskunde)" title="Grafiek (wiskunde)">grafiek</a> ervan staat in de hiernaast staande figuur): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=h_{1}(x)\cdot h_{2}(x)-2\operatorname {sgn}(x)-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=h_{1}(x)\cdot h_{2}(x)-2\operatorname {sgn}(x)-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2695f5a21b213d8184477580ee21d2e9f1cefe83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.163ex; height:2.843ex;" alt="{\displaystyle f(x)=h_{1}(x)\cdot h_{2}(x)-2\operatorname {sgn}(x)-2}"></span></dd></dl> <p>met: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}(x)=(-\operatorname {sgn}(x)+x-1)^{0{,}5\cdot \operatorname {sgn}(x)+1{,}5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}(x)=(-\operatorname {sgn}(x)+x-1)^{0{,}5\cdot \operatorname {sgn}(x)+1{,}5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f4ba5b27f356d00d24eed7f3b1745315d52714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.093ex; height:3.343ex;" alt="{\displaystyle h_{1}(x)=(-\operatorname {sgn}(x)+x-1)^{0{,}5\cdot \operatorname {sgn}(x)+1{,}5}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{2}(x)=2{,}5-1{,}5\cdot \operatorname {sgn}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{2}(x)=2{,}5-1{,}5\cdot \operatorname {sgn}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e70c313a356b9a93ca6376f29812315bd899c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.604ex; height:2.843ex;" alt="{\displaystyle h_{2}(x)=2{,}5-1{,}5\cdot \operatorname {sgn}(x)}"></span></dd></dl> <p>Subsitutie van <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> in de functie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8880a2e4243a2fe5157e574a0547ef3d5d373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}"></span> geeft: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}(0)=(-\operatorname {sgn}(0)+0-1)^{{\frac {1}{2}}\operatorname {sgn}(0)+{\frac {3}{2}}}=(-1)^{\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>0</mn> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}(0)=(-\operatorname {sgn}(0)+0-1)^{{\frac {1}{2}}\operatorname {sgn}(0)+{\frac {3}{2}}}=(-1)^{\frac {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de097813561c437df06e24137ac283a8019592e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.821ex; height:4.009ex;" alt="{\displaystyle h_{1}(0)=(-\operatorname {sgn}(0)+0-1)^{{\frac {1}{2}}\operatorname {sgn}(0)+{\frac {3}{2}}}=(-1)^{\frac {3}{2}}}"></span></dd></dl> <p>Deze laatste uitdrukking heeft <i>geen</i> reële waarde. De functie <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> bestaat daarmee niet voor <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>. </p><p>Deze functie kan ook beschreven worden met het voorschrift: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}4x&{\mbox{als }}x<0\\(x-2)^{2}-4&{\mbox{als }}x>0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}4x&{\mbox{als }}x<0\\(x-2)^{2}-4&{\mbox{als }}x>0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a32fe64cad95532bd4aaad58d02a034482ef44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.429ex; height:6.176ex;" alt="{\displaystyle f(x)={\begin{cases}4x&{\mbox{als }}x<0\\(x-2)^{2}-4&{\mbox{als }}x>0\end{cases}}}"></span></dd></dl> <p>Hierna volgt de afleiding daarvan. </p> <div class="mw-heading mw-heading3"><h3 id="Afleiding">Afleiding</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Signum_(wiskunde)&veaction=edit&section=3" title="Bewerk dit kopje: Afleiding" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Signum_(wiskunde)&action=edit&section=3" title="De broncode bewerken van de sectie: Afleiding"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Allereerst is: </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 -\operatorname {sgn}(x)+x-1=x-0\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mn>0</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\operatorname {sgn}(x)+x-1=x-0\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db420c07e12117ed063248dd251b8eee070e160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.632ex; height:2.843ex;" alt="{\displaystyle -\operatorname {sgn}(x)+x-1=x-0\quad }"></span> voor <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<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}"></span></li></ul> <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 -\operatorname {sgn}(x)+x-1=x-2\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\operatorname {sgn}(x)+x-1=x-2\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4836a122cb935f4502f49e259b749fe21d674479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.632ex; height:2.843ex;" alt="{\displaystyle -\operatorname {sgn}(x)+x-1=x-2\quad }"></span> voor <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span></dd></dl> <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 0{,}5\cdot \operatorname {sgn}(x)+1{,}5=1\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0{,}5\cdot \operatorname {sgn}(x)+1{,}5=1\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1aceb6a4de5dfbb5811fbf2100438d19e42e073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.557ex; height:2.843ex;" alt="{\displaystyle 0{,}5\cdot \operatorname {sgn}(x)+1{,}5=1\quad }"></span> voor <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<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}"></span></li></ul> <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 0{,}5\cdot \operatorname {sgn}(x)+1{,}5=2\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>=</mo> <mn>2</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0{,}5\cdot \operatorname {sgn}(x)+1{,}5=2\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc50f0e64b4d9865121aa3ea333a56a423f19b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.557ex; height:2.843ex;" alt="{\displaystyle 0{,}5\cdot \operatorname {sgn}(x)+1{,}5=2\quad }"></span> voor <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span></dd></dl> <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 2{,}5-1{,}5\cdot \operatorname {sgn}(x)=4\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{,}5-1{,}5\cdot \operatorname {sgn}(x)=4\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a78a7adbe9de31af512b12f637fd2e80a0b5c7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.557ex; height:2.843ex;" alt="{\displaystyle 2{,}5-1{,}5\cdot \operatorname {sgn}(x)=4\quad }"></span> voor <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<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}"></span></li></ul> <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 2{,}5-1{,}5\cdot \operatorname {sgn}(x)=1\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{,}5-1{,}5\cdot \operatorname {sgn}(x)=1\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8458741995806a02d02aefc7bc896c3005cb3358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.557ex; height:2.843ex;" alt="{\displaystyle 2{,}5-1{,}5\cdot \operatorname {sgn}(x)=1\quad }"></span> voor <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span></dd></dl> <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 2\cdot \operatorname {sgn}(x)=-2\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot \operatorname {sgn}(x)=-2\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4682db1820653182c6dbe579698879b2e2d98704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.743ex; height:2.843ex;" alt="{\displaystyle 2\cdot \operatorname {sgn}(x)=-2\quad }"></span> voor <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<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}"></span></li></ul> <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 2\cdot \operatorname {sgn}(x)=2\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot \operatorname {sgn}(x)=2\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597794b4d1a24608754dd6d04316c4ee98067a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.935ex; height:2.843ex;" alt="{\displaystyle 2\cdot \operatorname {sgn}(x)=2\quad }"></span> voor <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span></dd></dl> <p>Het functievoorschrift kan hiermee nu worden geïnterpreteerd als: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(x-{\mbox{(0 of 2)}})^{\mbox{(1 of 2)}}\cdot ({\mbox{4 of 1}})-{\mbox{(−2 of 2)}}-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>(0 of 2)</mtext> </mstyle> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>(1 of 2)</mtext> </mstyle> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>4 of 1</mtext> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>(−2 of 2)</mtext> </mstyle> </mrow> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(x-{\mbox{(0 of 2)}})^{\mbox{(1 of 2)}}\cdot ({\mbox{4 of 1}})-{\mbox{(−2 of 2)}}-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1db6bf91e7f32e6ce64467976b14e3ad48e20825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.737ex; height:3.843ex;" alt="{\displaystyle f(x)=(x-{\mbox{(0 of 2)}})^{\mbox{(1 of 2)}}\cdot ({\mbox{4 of 1}})-{\mbox{(−2 of 2)}}-2}"></span></li></ul> <p>Als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> negatief is, geldt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(x-0)^{1}\cdot 4+2-2=x\cdot 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(x-0)^{1}\cdot 4+2-2=x\cdot 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64fd72bbdb9e1d4c642007157e72ed4676ef16ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.829ex; height:3.176ex;" alt="{\displaystyle f(x)=(x-0)^{1}\cdot 4+2-2=x\cdot 4}"></span></dd></dl> <p>Is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> positief, dan is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(x-2)^{2}\cdot 1-2-2=(x-2)^{2}-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(x-2)^{2}\cdot 1-2-2=(x-2)^{2}-4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9d8bd15f4731d4bda5fb683f8a165f6f3eb9e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.857ex; height:3.176ex;" alt="{\displaystyle f(x)=(x-2)^{2}\cdot 1-2-2=(x-2)^{2}-4}"></span></dd></dl> <p>Dus is het voorschrift van <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> inderdaad te schrijven als: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\begin{cases}4x&{\mbox{als }}x<0\\(x-2)^{2}-4&{\mbox{als }}x>0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>als </mtext> </mstyle> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}4x&{\mbox{als }}x<0\\(x-2)^{2}-4&{\mbox{als }}x>0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a32fe64cad95532bd4aaad58d02a034482ef44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.429ex; height:6.176ex;" alt="{\displaystyle f(x)={\begin{cases}4x&{\mbox{als }}x<0\\(x-2)^{2}-4&{\mbox{als }}x>0\end{cases}}}"></span></dd></dl> <p>Verder is, en zie ook de grafiek van <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> hierboven: </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 \lim _{x\uparrow 0}f(x)=(1+0-1)^{1}\cdot (2{,}5+1{,}5)+2-2=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">↑</mo> <mn>0</mn> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\uparrow 0}f(x)=(1+0-1)^{1}\cdot (2{,}5+1{,}5)+2-2=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6130f20f8f41260a806e0bceb56f517e6dc920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.703ex; height:4.676ex;" alt="{\displaystyle \lim _{x\uparrow 0}f(x)=(1+0-1)^{1}\cdot (2{,}5+1{,}5)+2-2=0}"></span></dd> <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 \lim _{x\downarrow 0}f(x)=(-1+0-1)^{2}\cdot (2{,}5-1{,}5)-2-2=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\downarrow 0}f(x)=(-1+0-1)^{2}\cdot (2{,}5-1{,}5)-2-2=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86188231fc842c4d91c629754e75ef176e4d6596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.511ex; height:4.676ex;" alt="{\displaystyle \lim _{x\downarrow 0}f(x)=(-1+0-1)^{2}\cdot (2{,}5-1{,}5)-2-2=0}"></span></dd></dl> <p>Daarmee kan dan de functie <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> voor <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> continu gemaakt worden door te <i>definiëren</i>: </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(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d308c32c9894b88115262081194321ae7d9bbf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}"></span></dd></dl> <p>De continumakende waarde voor <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> is dus gelijk aan <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>. Hier is er dus sprake van <a href="/wiki/Discontinu%C3%AFteit#Ophefbare_discontinuïteit" title="Discontinuïteit">ophefbare discontinuïteit</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Toepassing">Toepassing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Signum_(wiskunde)&veaction=edit&section=4" title="Bewerk dit kopje: Toepassing" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Signum_(wiskunde)&action=edit&section=4" title="De broncode bewerken van de sectie: Toepassing"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Bestand:Signum(superellips).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Signum%28superellips%29.png/250px-Signum%28superellips%29.png" decoding="async" width="250" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Signum%28superellips%29.png/375px-Signum%28superellips%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Signum%28superellips%29.png/500px-Signum%28superellips%29.png 2x" data-file-width="757" data-file-height="510" /></a><figcaption>Kromme van Lamé met <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 k={\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af22181ad135347ede389a554e60901cfc877c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.968ex; height:3.509ex;" alt="{\displaystyle k={\tfrac {1}{2}}}"></span></figcaption></figure> <p>Een lid van de familie krommen van Lamé (de zogenoemde <a href="/wiki/Superellips" title="Superellips">superellipsen</a>) wordt in een <a href="/wiki/Cartesisch_co%C3%B6rdinatenstelsel" title="Cartesisch coördinatenstelsel">cartesisch coördinatenstelsel</a> gedefinieerd door: </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 \left|{\frac {x}{a}}\right|^{k}+\left|{\frac {y}{b}}\right|^{k}=1\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>b</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\frac {x}{a}}\right|^{k}+\left|{\frac {y}{b}}\right|^{k}=1\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3b73ffcc90ad0c41500611df39331d81fa4a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.346ex; height:5.676ex;" alt="{\displaystyle \left|{\frac {x}{a}}\right|^{k}+\left|{\frac {y}{b}}\right|^{k}=1\quad }"></span> met <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 k={\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af22181ad135347ede389a554e60901cfc877c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.968ex; height:3.509ex;" alt="{\displaystyle k={\tfrac {1}{2}}}"></span> en <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4689ff0639c824e92e7469fae91926382aa24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.522ex; height:2.509ex;" alt="{\displaystyle a,b>0}"></span></dd></dl> <p>Deze <a href="/wiki/Kromme" title="Kromme">kromme</a> is, evenals een ellips die met <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 k=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd301789e1f25a3da4be297ff637754ebee5f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=2}"></span> ook tot die familie behoort, symmetrisch in de <i>x</i>- en de <i>y</i>-as. Met <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=3,b=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=3,b=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c713b83b5632fb896e336de4afb22f612a54327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.783ex; height:2.509ex;" alt="{\displaystyle a=3,b=2}"></span> is de lengte van de grote as gelijk aan 6; die van de kleine as is 4. Wordt deze vergelijking geschreven als: </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 \left(\left|{\frac {x}{3}}\right|^{\frac {1}{4}}\right)^{\!2}+\left(\left|{\frac {y}{2}}\right|^{\frac {1}{4}}\right)^{\!2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>3</mn> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mn>2</mn> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\left|{\frac {x}{3}}\right|^{\frac {1}{4}}\right)^{\!2}+\left(\left|{\frac {y}{2}}\right|^{\frac {1}{4}}\right)^{\!2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5aa7535407817f44b1ef23ed4cc93dd16f62fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.021ex; height:8.009ex;" alt="{\displaystyle \left(\left|{\frac {x}{3}}\right|^{\frac {1}{4}}\right)^{\!2}+\left(\left|{\frac {y}{2}}\right|^{\frac {1}{4}}\right)^{\!2}=1}"></span></dd></dl> <p>dan ligt het, ten behoeve van een <a href="/wiki/Parametervergelijking" title="Parametervergelijking">parametervergelijking</a> met de parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in [0,2\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in [0,2\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dbc9ed8510c75442ce1d2e73f021258fc7e04c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.665ex; height:2.843ex;" alt="{\displaystyle t\in [0,2\pi ]}"></span>, voor de hand te stellen: </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{cases}\left|{\frac {x}{3}}\right|^{\frac {1}{4}}=\cos(t)\\\left|{\frac {y}{2}}\right|^{\frac {1}{4}}=\sin(t)\end{cases}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>3</mn> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mn>2</mn> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\left|{\frac {x}{3}}\right|^{\frac {1}{4}}=\cos(t)\\\left|{\frac {y}{2}}\right|^{\frac {1}{4}}=\sin(t)\end{cases}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86ac3509bdaf28457f2802230ef2a66ab36f1fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:18.804ex; height:9.509ex;" alt="{\displaystyle {\begin{cases}\left|{\frac {x}{3}}\right|^{\frac {1}{4}}=\cos(t)\\\left|{\frac {y}{2}}\right|^{\frac {1}{4}}=\sin(t)\end{cases}}\quad }"></span> of <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 \quad {\begin{cases}\left|x\right|=3\cos ^{4}(t)\\\left|y\right|=2\sin ^{4}(t)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>=</mo> <mn>3</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mi>y</mi> <mo>|</mo> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\begin{cases}\left|x\right|=3\cos ^{4}(t)\\\left|y\right|=2\sin ^{4}(t)\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9089c7a38ebbacfe34b10778bd2f9769f31f8974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.903ex; height:6.176ex;" alt="{\displaystyle \quad {\begin{cases}\left|x\right|=3\cos ^{4}(t)\\\left|y\right|=2\sin ^{4}(t)\end{cases}}}"></span></dd></dl> <p>Met de signumfunctie zijn beide laatste relaties dan te schrijven als: </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{cases}x=3\cos ^{4}(t)\cdot \operatorname {sgn}(\cos(t))\\y=2\sin ^{4}(t)\cdot \operatorname {sgn}(\sin(t))\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>=</mo> <mn>3</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x=3\cos ^{4}(t)\cdot \operatorname {sgn}(\cos(t))\\y=2\sin ^{4}(t)\cdot \operatorname {sgn}(\sin(t))\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4339f20556e94af87adddfcb99ab5b27a10731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.906ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}x=3\cos ^{4}(t)\cdot \operatorname {sgn}(\cos(t))\\y=2\sin ^{4}(t)\cdot \operatorname {sgn}(\sin(t))\end{cases}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Noten">Noten</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Signum_(wiskunde)&veaction=edit&section=5" title="Bewerk dit kopje: Noten" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Signum_(wiskunde)&action=edit&section=5" title="De broncode bewerken van de sectie: Noten"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"><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">Lat. <i>signum</i> (meervoud <i>signa</i>) komt van het werkwoord <i>signare</i> dat inkerven, markeren betekent. <i>Signum</i> betekent daarmee dus iets als 'dat wat gemarkeerd is'.</span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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