{"id":59,"date":"2023-09-17T11:09:46","date_gmt":"2023-09-17T11:09:46","guid":{"rendered":"https:\/\/mathority.org\/nl\/omgekeerde-of-wederkerige-functie\/"},"modified":"2023-09-17T11:09:46","modified_gmt":"2023-09-17T11:09:46","slug":"omgekeerde-of-wederkerige-functie","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/omgekeerde-of-wederkerige-functie\/","title":{"rendered":"Inverse (of wederzijdse) functie"},"content":{"rendered":"<p>In dit artikel leggen we uit wat een inverse (of reciproque) functie is en hoe je de inverse van een functie kunt berekenen. Je ontdekt ook hoe je gemakkelijk kunt weten of een functie een inverse heeft of niet, en wat de eigenschappen van dit soort functies zijn. Tenslotte kun je oefenen met stapsgewijze oefeningen op inverse functies. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-funcion-inversa\"><\/span> Wat is de inverse functie?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>De inverse functie, ook wel de reciproque functie genoemd, is de functie waarvan het domein het bereik is van een andere functie (de oorspronkelijke functie) en waarvan het bereik het domein is van de oorspronkelijke functie. De inverse functie van de functie <em>f<\/em> wordt uitgedrukt met het symbool <em>f <sup>-1<\/sup><\/em> .<\/strong><\/p>\n<p> Daarom is de inverse functie van <em>f(x)<\/em> de functie die aan de volgende voorwaarde voldoet: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/resultat-multiplication-de-polynomes.jpg\" alt=\"omgekeerde of wederkerige functie\" class=\"wp-image-685\" width=\"208\" height=\"210\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Goud<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58f82ae2e066a1c3d7f8abc65f1a5caf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\"><\/p>\n<p> is de inverse functie van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-84385e96f3e4a161ac693910846e9e0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f.\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Het concept van een inverse functie kan ook worden gedefinieerd met behulp van functiesamenstelling, aangezien elke functie die is samengesteld met zijn inverse functie gelijk is aan de identiteitsfunctie:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8209c8ece304984d292d18d3f5968fb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ f^{-1})(x)=(f^{-1}\\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"244\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zie:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\">wat is functiesamenstelling?<\/a><\/span><\/p>\n<p> Dus als aan de vorige vergelijking is voldaan, betekent dit dat<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58f82ae2e066a1c3d7f8abc65f1a5caf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\"><\/p>\n<p> is de inverse functie (of wederkerige functie) van <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-84385e96f3e4a161ac693910846e9e0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f.\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-funcion-inversa\"><\/span> Voorbeeld van inverse functie<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Laten we, gegeven de definitie van inverse functie, een voorbeeld oplossen om de betekenis ervan beter te begrijpen.<\/p>\n<ul>\n<li> Bepaal of de volgende functies inverse van elkaar zijn:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7dbebbad962bbec034d4203171f49422_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x+1\\qquad g(x)=\\cfrac{x-1}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"243\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Als de twee functies het omgekeerde van elkaar zijn, wordt aan de volgende twee voorwaarden voldaan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-facf310c96df4f7102c40db7ba7bbc5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x \\qquad \\qquad  (g \\circ f)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Laten we dus controleren of aan beide vergelijkingen is voldaan. Wij controleren eerst<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d73ab8f95080e2c6ec60813b7c9edb66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36f7d5f4d815a58f808114ae471c31dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\displaystyle\\left(f \\circ g\\right)(x)&amp; = f\\Big(g(x)\\Big)\\\\[2ex]&amp;= f\\left( \\frac{x-1}{2} \\right)\\\\[2ex]&amp; = 2\\left( \\frac{x-1}{2} \\right)+1\\\\[2ex]&amp; =x-1+1\\\\[2ex]&amp;=\\bm{x} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"240\" width=\"212\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> \u27a4 Als je de berekening die we zojuist hebben gemaakt niet begrijpt, ga dan naar de bovenstaande link voor <em>Wat is de samenstelling van functies?<\/em> , leggen we uit hoe je dit soort bewerkingen met functies kunt oplossen.<\/p>\n<p> Zodat<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76ab7a77ded5af1f65f660a89117dd63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> ja, het is gelukt. \u2705<\/p>\n<p> Laten we nu de gelijkheid controleren<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42d46171e7088d03e62c95620856d280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x) = x :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-83423fe7deab472f1a7e9575ec7af8c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\left(g \\circ f\\right)(x)&amp;= g\\Big(f(x)\\Big)\\\\[2ex]&amp;= g\\Big(2x +1 \\Big)\\\\[2ex]&amp;=\\cfrac{(2x+1)-1}{2}\\\\[2ex]&amp;= \\cfrac{2x}{2}\\\\[2ex]&amp;=\\bm{x} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"249\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> En de invertibiliteitsvoorwaarde<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f9fbc925d54530fcfa7ee7678e65b952_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> het is ook gelukt. \u2705<\/p>\n<p> Concluderend: aangezien beide vergelijkingen gelden, <strong>zijn de twee functies inverse van elkaar.<\/strong><\/p>\n<p> Hieronder ziet u beide functies grafisch weergegeven. Merk op dat <strong>de grafieken van twee inverse functies symmetrisch zijn<\/strong> rond de bissectrice van het eerste en derde kwadrant: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonction-inverse.webp\" alt=\"omgekeerde functie\" class=\"wp-image-673\" width=\"486\" height=\"402\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-saber-si-una-funcion-tiene-inversa\"><\/span> Hoe weet je of een functie een inverse heeft?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Een functie heeft een inverse functie als het een injectieve functie is<\/strong> , dat wil zeggen als elke waarde in zijn hele domein overeenkomt met een enkele waarde in zijn interval. <\/p>\n<div class=\"wp-block-columns are-vertically-aligned-center is-layout-flex wp-container-135\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Exponenti\u00eble functie met inverse functie<\/span> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-etape-par-etape-des-fonctions-exponentielles.webp\" alt=\"\" class=\"wp-image-251\" width=\"254\" height=\"250\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Kwadratische functie zonder inverse functie<\/span> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-d-une-fonction-quadratique-ou-parabole.webp\" alt=\"\" class=\"wp-image-137\" width=\"222\" height=\"250\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<p> De linker exponenti\u00eble functie heeft bijvoorbeeld een inverse functie omdat elke <em>x<\/em> overeenkomt met een enkele waarde van <em>f(x)<\/em> . Aan de andere kant heeft de rechter kwadratische functie geen inverse functie, omdat deze verschillende waarden van <em>x<\/em> heeft waarvan de afbeeldingen gelijk zijn (bijvoorbeeld <em>f(1)=f(3)=2)<\/em> .<\/p>\n<p> Op dezelfde manier bestaat een bijectieve functie uit een functie die zowel injectief als surjectief is, daarom heeft elke bijectieve functie ook een inverse functie.<\/p>\n<p> Aan de andere kant moet je er rekening mee houden dat de inverse functie niet hetzelfde is als de <em>multiplicatieve inverse van een functie<\/em> , maar eerder twee verschillende concepten. Om de multiplicatieve inverse van een functie te vinden, berekent u eenvoudigweg 1 correspondentie met de genoemde functie.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4bf5aff2d6025747e58082ac697b06d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f^{-1}(x) \\neq \\cfrac{1}{f(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"113\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> In de volgende sectie zullen we zien hoe we de inverse functie kunnen bepalen. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-la-funcion-inversa\"><\/span> Hoe de inverse functie te vinden<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Om de inverse van een functie te berekenen, moeten de volgende stappen worden uitgevoerd: <\/p>\n<div style=\"padding-top: 30px; padding-bottom: 10px; padding-right: 30px; padding-left: 15px; border: 2px dashed #FF9B28; border-radius:30px; margin-bottom:25px\">\n<ol style=\"color:#FF8A05; font-weight: bold;border:\">\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Vervang <em>f(x)<\/em> door <em>y<\/em> .<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Verander alle <em>x<\/em> in <em>y<\/em> en omgekeerd.<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Wis de <em>y-<\/em> variabele.<\/span><\/li>\n<li> <span style=\"color:#101010;font-weight: normal;\">Vervang de variabele <em>y<\/em> door <em>f <sup>-1<\/sup> (x)<\/em> . De inverse functie is de uitdrukking gevonden voor <em>f <sup>-1<\/sup> (x)<\/em> .<\/span><\/li>\n<\/ol>\n<\/div>\n<p> Om precies te zien hoe de inverse functie wordt berekend, bepalen we als voorbeeld de inverse van de volgende functie:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7bd000cffbbb2432fd2c53b683d15c5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) =4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Allereerst moeten we vervangen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Voor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-824b16de72dde879834460d93bd88610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Nu veranderen we alles<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> van de functie door<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , en vice versa:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a9c604955a9823146b45d8d7fcf955d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y= 4x+5 \\quad \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\quad x= 4y+5\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"317\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Vervolgens wissen we de variabele <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a67252387800a0b4cf9d5de3f47992ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=4y+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5c3b6dd8da6675c67d8d97143338bc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-5=4y\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3b7fce291e0a2e8783d43d7bf3091dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-5}{4}=y\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"73\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-94afb1cafad9006eb47ba582cf925cb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{x-5}{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> En tenslotte de inverse functie van<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> is de algebra\u00efsche uitdrukking die we hebben verkregen door te isoleren <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c678d462718983cd6440a900898d488b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{f^{-1}(x) = } \\cfrac{\\bm{x-5}}{\\bm{4}}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"115\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-funcion-inversa\"><\/span> Opgeloste oefeningen van de inverse functie<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Hieronder hebben we verschillende stap-voor-stap oefeningen over de inverse functie voorbereid, zodat u kunt oefenen.<\/p>\n<p> \ud83d\udc49 Onthoud dat als je niet begrijpt hoe je een oefening moet oplossen of wilt dat wij een probleem voor je oplossen, je ons in de reacties kunt schrijven!<\/p>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Controleer of de volgende twee functies invers (of wederkerig) zijn of niet: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b15be151088ebf4c65087b5f4e00938_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3x-7\\qquad g(x)=\\cfrac{x+7}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"243\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Om de twee functies inverse van elkaar te laten zijn, moet het volgende waar zijn:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f64196e1a5f56f51ec1b1164eceb60b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x)=x \\qquad \\qquad (g \\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het is daarom noodzakelijk om te controleren of aan deze twee voorwaarden is voldaan. Wij controleren eerst <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11b7c4f21f3fb296e8a27a20cf8d1c3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x)=x :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbbdf5bb2c308a1973c5f2d99e822ad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\displaystyle\\left(f \\circ g\\right)(x)&amp;= f\\Big(g(x)\\Big)\\\\[2ex]&amp;= f\\left( \\frac{x+7}{3} \\right)\\\\[2ex]&amp;= 3 \\left(\\frac{x +7}{3} \\right) - 7  \\\\[2ex] &amp; =x + 7 - 7 \\\\[2ex]&amp;= \\bm{x}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"240\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nog,<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76ab7a77ded5af1f65f660a89117dd63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> ja, het is gelukt. \u2705<\/p>\n<p class=\"has-text-align-left\"> Laten we nu de andere samenstelling van functies bekijken <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7b058926aba155b6b765d5724492a92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x)=x :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac5819584c8358b812c8214a902e4f08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\left(g \\circ f\\right)(x)&amp;= g\\Big(f(x)\\Big)\\\\[2ex]&amp;= g\\left(3x-7\\right)\\\\[2ex]&amp;=\\cfrac{(3x-7)+7}{3}\\\\[2ex]&amp;=\\cfrac{3x}{3}\\\\[2ex]&amp;=\\bm{x}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"239\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Waardoor<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f9fbc925d54530fcfa7ee7678e65b952_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> het is ook gelukt. \u2705<\/p>\n<p class=\"has-text-align-left\"> Hoe komt dat<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3c53166029da197c6e17e19d3809563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> En<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db26718e178acd21a2f1f266b9a3c3b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> , de twee functies zijn het omgekeerde van elkaar.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 2<\/h3>\n<p> Bereken de inverse (of reciproque functie) van de volgende polynoomfunctie van de eerste graad: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e230d2b35a28b2bdfc77f06634de138b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-2x-3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Het eerste dat u moet doen om de functie om te keren, is door de term te vervangen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Voor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a4a9dab43d26aecc4bb0ce18ea508c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"}y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b754ae8eb949f8f8d85ba36ebabb52d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-2x-3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu veranderen we de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> door<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , en vice versa:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-653d447dc0dcd501486c99f8cb1d40f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-2x-3 \\ \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\ x =-2y-3\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"322\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En dan laten wij los <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42c5d877f3a0a6fd971e65ade72afdb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x =-2y-3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23c71b5b09a6e2cef29e449dcd1efa16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x +3=-2y\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9611a1a60c491236193f13a0542011bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x +3}{-2}=y\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"73\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b026add174bb871e220df0191afdbd76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = \\cfrac{x +3}{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het is ons al gelukt om vrij te geven<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> . Daarom is de inverse functie van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Oosten: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2197f512ad2a7360c966f64b78e75d53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{f^{-1}(x) = -} \\cfrac{\\bm{x+3}}{\\bm{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"133\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 3<\/h3>\n<p> Inverteer de volgende kwadratische polynoomfunctie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8200d353eb8e21c4c04f3601fc802489_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Om de inverse functie te vinden, zullen we de procedure volgen die we hierboven hebben gezien. Dus we zullen bellen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> naar de functie<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d47514334dca13875d8344a469e1c7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=x^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ten tweede passen we de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> voor de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , en vice versa:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-29de23ff2053fc75656c8ff4d7fa41a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=x^2-1 \\ \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\ x =y^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"291\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte isoleren we de variabele <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e69105ebcca11f4a74f3f3657e3e6440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x =y^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5728627e7e38f4515904dee980b00694_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+1=y^2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1cccfbd1b731987173cb8242913f3015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x+1}=y\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a8de5ae78dde4cbb3d501a56dec92e6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\pm \\sqrt{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In dit geval heeft de verkregen functie echter twee afbeeldingen voor elk element van zijn domein (het positieve beeld en het negatieve beeld). <strong>Er is dus geen inverse functie<\/strong> van de probleemfunctie.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 4<\/h3>\n<p> Bepaal de inverse functie (of reciproque functie) van de volgende rationale functie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a7ccc4e4bbef9c4997233bfc9271bf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x-1}{2x+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"110\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst vervangen wij<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Voor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-15a4196ec4380923616fae6f09302cb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{x-1}{2x+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"85\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu veranderen we de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> teller en noemer<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , en vice versa:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf93d67efcb45f42b278c4bbdd134482_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{x-1}{2x+3} \\ \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\ x=\\cfrac{y-1}{2y+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"298\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En dan laten wij los <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5501ae45c059ce0992ae626ae90dccd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{y-1}{2y+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"85\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De uitdrukking<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3535e891270ab3ae4ed4aad6a498984_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y +3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -4px;\"><\/p>\n<p> deelt de hele rechterkant van de vergelijking, dus we kunnen deze vermenigvuldigen door de hele linkerkant van de vergelijking te vermenigvuldigen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a7e58aaef340adec944945b7feb0f8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\\cdot (2y+3)=y-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b837ac50ef3677cc95e0a2d439a76d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2yx+3x=y-1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"132\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wij stellen alle voorwaarden aan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> aan de ene kant van de vergelijking, en de andere termen aan de andere kant:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-525563a5b84daff8946bfe896e5d30d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2yx-y=-3x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"145\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om op te helderen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42796b3d2f8b77c564fefa868de8d64a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"}y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , halen we de gemeenschappelijke factor uit de linkerkant van de vergelijking:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d21c3a81a922f2d84a75a4568b97dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y(2x-1)=-3x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En als postbode<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cbbee5c6fcee058c32f9f067554f4679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -5px;\"><\/p>\n<p> is om de hele linkerkant van de vergelijking te vermenigvuldigen, we kunnen dit doen door de hele rechterkant te delen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5ac1a61f0a3683aa8cde1c6ad21a1cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-3x-1}{2x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"107\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het is ons al gelukt om vrij te geven<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> . Dus de inverse functie van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Oosten: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40fd07fe81e86ffb37c7b857f43a140f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{f^{-1}(x)=} \\cfrac{\\bm{-3x-1}}{\\bm{2x-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"138\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-de-la-funcion-inversa\"><\/span> Eigenschappen van de inverse functie<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> De inverse functie heeft de volgende kenmerken:<\/p>\n<ul>\n<li> De inverse functie is uniek, dat wil zeggen: als een functie inverteerbaar is, is er voor deze functie slechts \u00e9\u00e9n inverse functie.<\/li>\n<\/ul>\n<ul>\n<li> Het domein van de inverse functie is het bereik (of bereik) van de oorspronkelijke functie.<\/li>\n<\/ul>\n<ul>\n<li> Op dezelfde manier is het pad van de inverse functie equivalent aan het domein van de oorspronkelijke functie.<\/li>\n<\/ul>\n<ul>\n<li> Elke functie die uit zijn inverse functie bestaat, geeft de identiteitsfunctie (x).<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8209c8ece304984d292d18d3f5968fb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ f^{-1})(x)=(f^{-1}\\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"244\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> De grafiek van een functie en de grafiek van zijn inverse functie zijn symmetrisch rond de bissectrice van het eerste en derde kwadrant.<\/li>\n<\/ul>\n<ul>\n<li> De inverse van de inverse functie is gelijk aan de oorspronkelijke functie:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-53db614ba7e0fdd5be384f53e05c6d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(f^{-1}\\right)^{-1}=f\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"95\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Het omkeren van een samengestelde functie komt overeen met het afzonderlijk berekenen van de inverse van elke functie en het vervolgens samenstellen van de inverse functies.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e5f9cc891e8322cf737caf8fa76a92f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ g)^{-1}=f^{-1}\\circ g^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Een functie is tegelijkertijd continu met zijn inverse functie, of met andere woorden: als een functie continu is, zal zijn inverse functie dat ook zijn.<\/li>\n<\/ul>\n<ul>\n<li> Als een functie differentieerbaar is en de afgeleide op geen enkel moment verdwijnt\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7e8b3cc99b3a65d07a0bfbdd9680062_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<p> , zal de inverse functie ervan ook differentieerbaar zijn.<\/li>\n<\/ul>\n<p> Bovendien kan de afgeleide van de inverse functie worden berekend door de <strong>inverse functiestelling<\/strong> toe te passen, waarvan de formule is:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1aa57f06b0b96e0900dd47e0bf4ad66c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(f^{-1}\\right)'(y)=\\cfrac{1}{f'(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"139\" style=\"vertical-align: -17px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In dit artikel leggen we uit wat een inverse (of reciproque) functie is en hoe je de inverse van een functie kunt berekenen. Je ontdekt ook hoe je gemakkelijk kunt weten of een functie een inverse heeft of niet, en wat de eigenschappen van dit soort functies zijn. Tenslotte kun je oefenen met stapsgewijze oefeningen &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/omgekeerde-of-wederkerige-functie\/\"> <span class=\"screen-reader-text\">Inverse (of wederzijdse) functie<\/span> Lees meer &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[49],"tags":[],"class_list":["post-59","post","type-post","status-publish","format-standard","hentry","category-functie-representatie"],"yoast_head":"<!-- This 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