{"id":58,"date":"2023-09-17T11:10:24","date_gmt":"2023-09-17T11:10:24","guid":{"rendered":"https:\/\/mathority.org\/nl\/samenstelling-van-functies-samengestelde-functie\/"},"modified":"2023-09-17T11:10:24","modified_gmt":"2023-09-17T11:10:24","slug":"samenstelling-van-functies-samengestelde-functie","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/samenstelling-van-functies-samengestelde-functie\/","title":{"rendered":"Samenstelling van functies (samengestelde functie)"},"content":{"rendered":"<p>In dit artikel leggen we uit wat de samengestelde functie (of samenstelling van functies) is. Bovendien kunt u verschillende voorbeelden zien van samengestelde functies en hoe het domein van dit soort functies wordt berekend. Tenslotte vind je de eigenschappen van functiesamenstelling en een aantal stap-voor-stap oefeningen om te oefenen. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-composicion-de-funciones\"><\/span> Wat is functiesamenstelling?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Functiesamenstelling bestaat uit het <strong>achtereenvolgens<\/strong> evalueren van dezelfde waarde van de onafhankelijke variabele (x) in twee of meer functies.<\/strong> Als u bijvoorbeeld de functies (gof)(x) samenstelt, ontstaat de samengestelde functie g[f(x)]. <\/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\/composition-des-fonctions.webp\" alt=\"samenstelling van functies\" class=\"wp-image-637\" width=\"319\" height=\"115\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> De uitdrukking van de samengestelde 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-5e3eba5c152156d18705b3afd68f487e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\circ f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\"><\/p>\n<p> we lezen \u201cf samengesteld met g\u201d of \u201cf gevolgd door g\u201d.<\/p>\n<p> Houd er rekening mee dat de volgorde belangrijk is bij het samenstellen van functies; de functie rechts van het compositiesymbool wordt als eerste toegepast<\/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-876a58c2d2bc4ab866c0af7a5cdbf4b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan de functie links van het compositiesymbool <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6c9247c09d7827160e8316d83ce2b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"26\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-composicion-de-funciones\"><\/span> Voorbeeld van functiesamenstelling<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Laten we, gegeven de definitie van een samengestelde functie, een voorbeeld bekijken van hoe de samenstelling van twee functies kan worden berekend.<\/p>\n<ul>\n<li> Gegeven de volgende twee verschillende 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-a9c8e90e6e8bb1e0837f424caaf5d1a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3x+1 \\qquad g(x)=\\cfrac{x+4}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"243\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Bereken de samengestelde 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-624e8051bd11c44546f1270c91d069ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<p> en evalueer het daarin<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf60bc9fcf312a246a055c15ee98033c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=3.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> De samenstelling van functies<\/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-624e8051bd11c44546f1270c91d069ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<p> Dit betekent dat we de volgende samengestelde functie moeten uitvoeren:<\/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-cb23add9f37be01b1ebf5923aef845ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(x) = g\\Big(f(x)\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"162\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om het op te lossen, vervangen we<\/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> door zijn algebra\u00efsche 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-179039983ee415873b7367ac255c1023_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big(f(x)\\Big)= g\\Big(3x+1\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"166\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p> En nu nemen we de functie van<em> <\/em><\/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-8d124ddf8086d5d941fcb1de7c075c0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=\\cfrac{x+4}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"100\" style=\"vertical-align: -12px;\"><\/p>\n<p>en we plaatsen de uitdrukking<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4b24384ffd361d326faea85da151496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\"><\/p>\n<p> waar er een is<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-845f2902b8bebf60c3c7372a7fbe4d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"19\" style=\"vertical-align: 0px;\"><\/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-9407b8a097013423f71626368a65940b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big(3x+1\\Big)=\\cfrac{(3x+1)+4}{2}=\\cfrac{3x+5}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"275\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Op deze manier hebben we al de functie <em>f<\/em> berekend die bestaat uit <em>g<\/em> :<\/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-b46e81ab3496bd4daad7560eabda1caa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(x)=\\cfrac{3x+5}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"152\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Tenslotte, om de samengestelde functie in te evalueren<\/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-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> Bereken eenvoudig het beeld van de functie in de genoemde waarde: <\/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-b1a5472ce52fe01b6809224912a25539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(3)=\\cfrac{3\\cdot 3+5}{2}=\\cfrac{14}{2}=7\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"237\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"dominio-de-la-funcion-compuesta\"><\/span> Samengesteld functiedomein<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Wanneer we bewerkingen op functies uitvoeren, is het domein van de resulterende functie normaal gesproken het snijpunt van de domeinen van de oorspronkelijke functies. Aan deze eigenschap wordt echter niet voldaan door de functiesamenstelling.<\/p>\n<p> Het <strong>domein van functiesamenstelling<\/strong><\/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-a1e321d1fe5674142070eefd17519b77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g\\circ f)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<p> is equivalent aan de verzameling van alle waarden van x in het domein van 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-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> zoals<\/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> behoort tot het domein van 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-dce98a79808d5694e9c7a9f4545e45f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" 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-cf27df12b66bbfdc7516fe47410cbf5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom}(g\\circ f)=\\{x\\in\\text{Dom}(f)\\ | \\ f(x)\\in \\text{Dom}(g)\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"354\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Om het domein van een samengestelde functie te berekenen, moet u daarom eerst het domein van elke functie afzonderlijk vinden, en vervolgens het domein van de functie die het resultaat is van de bewerking. Het domein van de samenstelling van de functies zal dus bestaan uit alle waarden die aan de vorige wiskundige voorwaarde voldoen.<\/p>\n<p> \ud83d\udc49 Onthoud: als je een probleem tegenkomt waarvan je niet weet hoe je het moet oplossen, kun je het ons vragen in de reacties hieronder! <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-de-la-composicion-de-funciones\"><\/span> Eigenschappen van functiesamenstelling<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Samengestelde functies hebben de volgende kenmerken:<\/p>\n<ul>\n<li> De samenstelling van functies heeft de associatieve eigenschap, daarom is de volgende vergelijking altijd waar:<\/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-bdfe1df3ab8d42f8d2af7f0d33628dba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\circ (g\\circ h)=(f\\circ g)\\circ h\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"178\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Over het algemeen is de functiesamenstelling niet commutatief, dus de volgorde van de bewerking bepaalt het resultaat:<\/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-a641ded39b05f710e2cbc89f7c309275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\circ g\\neq g\\circ f\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<ul>\n<li> Het neutrale element van de samenstelling van functies komt overeen met de identiteitsfunctie\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da6458e64d1478ad56231146ce88db89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<p> Elke functie die is samengesteld met de identiteitsfunctie resulteert dus in de functie zelf:<\/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-161782350ae4c16617cbe0795e0fa35a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\circ id = id \\circ f = f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"143\" 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-d5ce8a7d2dbca796654cb483808271e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"id = x\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Het berekenen van de inverse van de samenstelling van twee functies komt neer op het eerst vinden van de inverse van elke functie en vervolgens het bepalen van de samengestelde 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-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> De inverse functie fungeert ook als een symmetrisch element van de samengestelde functie, aangezien de samenstelling van een functie met zijn inverse equivalent is aan de identiteitsfunctie:<\/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-b801f112fdb719fad2b8c605ad831f0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ f^{-1})^{-1}=(f^{-1}\\circ f)=id=x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"254\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> De afgeleide van de samenstelling van twee functies wordt berekend met behulp van de kettingregel:<\/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-edf8bcb549ad025afce59e50c1cfbb94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl(g\\circ f\\bigr)'(x)=g'\\Bigl(f(x)\\Bigr)\\cdot f'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"223\" style=\"vertical-align: -11px;\"><\/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 de kettingregel?<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-composicion-de-funciones\"><\/span> Opgeloste oefeningen over de samenstelling van functies<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Gegeven de volgende twee functies:<\/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-8d22ce1010c1d1f2baa34bd5b96e4ca8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x-2 \\qquad g(x)= 5x + 4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Bereken de samenstellingen van de functies <em>f<\/em> samengesteld met <em>g<\/em> en <em>g<\/em> samengesteld met <em>f<\/em> . <\/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-a1e321d1fe5674142070eefd17519b77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g\\circ f)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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-8e664a72ceacc815125057558ab04c6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ g)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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\"> De samenstelling van functies<\/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-624e8051bd11c44546f1270c91d069ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<p> betekent om de volgende samengestelde functie te berekenen:<\/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-cb23add9f37be01b1ebf5923aef845ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(x) = g\\Big(f(x)\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"162\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus om het op te lossen, vervangen we<\/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> vanwege zijn uitdrukking:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc434b7ab7aea14a249be3943671119b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" 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-13cb5a87becd27ff70f8fb5fb3b13239_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big(f(x)\\Big)= g\\Big(x-2\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"157\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> EN<\/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-bf53f7369c2b0dfc1c60397d3dab8cd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big(x-2\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"68\" style=\"vertical-align: -11px;\"><\/p>\n<p> Dit betekent dat in de uitdrukking van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a63a5d2293c895c25f8684ae82cefbf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x) =5x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<p> je moet de variabele vervangen<\/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<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f860e374f2a5f611e892f304ee0cc2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-2:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\"><\/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-575d7c9f8dde8d78c1dd910ba1557ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big(x-2\\Big) = 5(x-2) +4= 5x-10+4 = 5x-6\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"374\" style=\"vertical-align: -11px;\"><\/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-df045f4ebc5cdfdcbd99a12307235058_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\left(g \\circ f\\right)(x) = 5x-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Aan de andere kant, om de functie <em>g<\/em> te vinden die uit <em>f<\/em> bestaat, moet je dezelfde procedure uitvoeren, maar dan in de omgekeerde volgorde:<\/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-2e9be8f1bab4b7607441c7c1b7e3ad0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\left(f \\circ g\\right)(x)&amp;= f\\Big(g(x)\\Big)\\\\[2ex]&amp;=f\\Big(5x+4\\Big)\\\\[2ex]&amp;=(5x+4)-2\\\\[2ex]&amp;=\\bm{5x+2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"193\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Deze oefening demonstreert ook de eigenschap dat samengestelde functies niet commutatief zijn, omdat het resultaat afhangt van de volgorde waarin de functies worden toegepast.<\/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> Gegeven de volgende twee functies:<\/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-3e7ff3894ffbafb8d56296b5b3d0a080_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x) =x^2-3 \\qquad g(x)=\\frac{2x+3}{x+4}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"251\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Berekent de samenstelling van functies <em>f<\/em> samengesteld uit <em>g<\/em> . <\/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-a1e321d1fe5674142070eefd17519b77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g\\circ f)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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\"> De functie <em>f<\/em> samengesteld uit <em>g<\/em> betekent het oplossen van de volgende samengestelde 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-cb23add9f37be01b1ebf5923aef845ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(x) = g\\Big(f(x)\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"162\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We vervangen daarom de functie f(x) door zijn 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-28b41bd35ba991a996cf3a890aaeadf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big(f(x)\\Big)= g\\Big(x^2-3 \\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"165\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En nu 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-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<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce3abcbe3e7405ea80705e3c169ae20d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: 0px;\"><\/p>\n<p> in de uitdrukking van de functie g(x):<\/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-bf0bb412f9088845027b57d83a91073d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}g\\Big(x^2-3\\Big)&amp;=\\cfrac{2(x^2-3)+3}{(x^2-3)+4}\\\\[2ex]&amp;=\\cfrac{2x^2-6+3}{x^2+1}\\\\[2ex]&amp;=\\cfrac{2x^2-3}{x^2+1}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"175\" width=\"206\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kort gezegd is het resultaat van de functiesamenstelling: <\/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-6d625dc7f6b8e7af0988037119d473f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\left(g \\circ f\\right)(x) =} \\cfrac{\\bm{2x^2-3}}{\\bm{x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"155\" style=\"vertical-align: -14px;\"><\/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> Gegeven de volgende twee kwadratische functies:<\/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-a594c87e42327318112e78714904aa81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x) =x^2 \\qquad g(x)=g(x)= x^2-4x+8\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"314\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Bepaal het resultaat van de volgende samenstelling van functies: <\/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-b7186993d7a14fceff2669b6d3f10585_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g\\circ f)(2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" 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-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-706bfe49f02e183102e665a522eb803c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<p> bestaat uit het vinden van de volgende samengestelde 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-217743a7af2baefbc83e1c377f3f804f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(2) = g\\Big(f(2)\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"160\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus om de samengestelde functie op te lossen, berekenen we 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-7bd87dc0308123b84fbe51bb86e0b009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2) :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-c3421a45cc1c0ad35b18520e81ddc031_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"75\" 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-671b30c79ed994c1923982d4a09d2eae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=2^2=4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Daarom, als <\/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-69585295dd3a781afd9ebde9750865bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=4 :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-f64414212eaa56a80be0b0ac9bd1be23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(g \\circ f\\right)(2) = g\\Big(f(2)\\Big) = g\\big(4\\big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"218\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om de waarde van de samengestelde functie te vinden, hoeft u dus alleen maar te berekenen <\/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-e05e67fe2bbccdbd12d182052a93df62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(4) :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" 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-9fc21e0e44e696ed36cd6e9a535cff09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\left(g \\circ f\\right)(2)&amp;=g\\Big(f(2)\\Big)\\\\[2ex]&amp;= g\\big(4\\big)\\\\[2ex]&amp;=4^2-4\\cdot 4+8 \\\\[2ex]&amp;= 16 - 16 + 8\\\\[2ex]&amp;= 8\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"198\" width=\"198\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Samenvattend is het resultaat van het functiecompositieprobleem: <\/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-8d3e091b61117ac43ec8d17d4aa1d5b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\left(g \\circ f\\right)(2) =8}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/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> Gegeven de volgende twee functies:<\/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-0dabd181964a078fe5348b19efc168d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{2x-2}{-x+7}\\qquad g(x)= x^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"256\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Zoek het resultaat van <em>g<\/em> samengesteld met <em>f<\/em> bij x=2: <\/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-47390afb10cf5e3de6b0ff87edeff73b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ g)(2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" 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\"> In dit geval moeten we de volgende samengestelde functie berekenen:<\/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-e6de0edda1d8a4151fc88faa013a012e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(f \\circ g\\right)(2) = f\\Big(g(2)\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"160\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus eerst vinden we <\/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-809314c764c52d861c5b9671f5139d5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(2) :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" 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-c4f1e7070d6f986eac4f1483f9a30198_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=x^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"104\" 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-94e98ec0f41ba6635ee79cf96968e7ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(2)=2^2-1=4-1 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"198\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En dus, zoals <\/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-82da28e3a6fb3ea74c8e8ef8d022b202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(2)=3 :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" 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-bfe82e73b68c3b618cb0795bb6530af7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(f \\circ g\\right)(2) = f\\Big(g(2)\\Big) = f\\big(3\\big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"219\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om de samengestelde functie op te lossen, moeten we dus berekenen <\/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-b6d1361868e53506ddeca640a71aa321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3) :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-9ec461a36a477af7bc703fd48bc3d4c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\left(f \\circ g\\right)(2)&amp;=f\\Big(g(2)\\Big)\\\\[2ex]&amp;= f\\big(3\\big)\\\\[2ex]&amp;=\\cfrac{2\\cdot 3-2}{-3+7}\\\\[2ex]&amp;=\\cfrac{6-2}{-3+7}\\\\[2ex]&amp;=\\cfrac{4}{4}\\\\[2ex]&amp;=1\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"305\" width=\"162\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Concluderend is het resultaat van het uitoefenen van de samengestelde functies: <\/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-8860734e1b816965b945b8ddf117912d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\left(f \\circ g\\right)(2)=1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 5<\/h3>\n<p> Gegeven de volgende drie functies:<\/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-cde9a25a0c078e5e4491aef0682310d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = x+1 \\qquad g(x)= 3x - 5\\qquad h(x) = \\sqrt{x} - 3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"389\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Bereken de volgende samenstelling van de 3 functies: <\/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-2ff91e1d3fdea1de8b629e4c17cee6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(h \\circ g \\circ f )(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" 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\"> 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-cf14fee9bc19b3aaefabcac820afb69b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(h \\circ g \\circ f\\right)(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" style=\"vertical-align: -5px;\"><\/p>\n<p> Dit betekent dat we de volgende samengestelde functie moeten berekenen:<\/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-b6525a7b5ac0a86cc710225df5ebdf7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(h \\circ g \\circ f\\right)(x) = h \\bigg( g\\Big(f(x)\\Big) \\bigg)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"225\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus bepalen we 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-0efee4b8fc162ee4ee30c15f1dac01c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big( f(x)\\Big):\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"74\" style=\"vertical-align: -11px;\"><\/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-a0ab9465f2aebf9fb7c81298eab5b8ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}g\\Big( f(x)\\Big)&amp;=g\\Big(x+1 \\Big)\\\\[2ex]&amp;= 3(x+1)-5\\\\[2ex]&amp;= 3x+3-5\\\\[2ex]&amp;= 3x-2\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"154\" width=\"181\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En nu gaan we berekenen<\/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-0512ac0197a64c0dce8e3a14b97efd1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h \\bigg( g\\Big(f(x)\\Big) \\bigg)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"98\" style=\"vertical-align: -17px;\"><\/p>\n<p> . Om dit te doen, vervangen we de gevonden uitdrukking from<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fde695528fc22ce481c7750cd392927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\\Big(f(x)\\Big)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"62\" style=\"vertical-align: -11px;\"><\/p>\n<p> waar een verschijnt<\/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> in 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-240be4d8f4d59436eb46fefb7d2f7633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h(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-89fbb7189eaf2dd3c622466c7313806d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}h \\bigg( g\\Big(f(x)\\Big) \\bigg)&amp;= h \\bigg(3x-2\\bigg)\\\\[2ex]&amp;= \\sqrt{3x-2} - 3\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"83\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We kunnen de samengestelde functie niet verder vereenvoudigen. De samenstelling van de drie functies resulteert dus in een irrationele 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-d3d99aa6d620db9304fa7121387dfd95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\left(h \\circ g \\circ f\\right)(x) =\\sqrt{3x-2} - 3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"222\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In dit artikel leggen we uit wat de samengestelde functie (of samenstelling van functies) is. Bovendien kunt u verschillende voorbeelden zien van samengestelde functies en hoe het domein van dit soort functies wordt berekend. Tenslotte vind je de eigenschappen van functiesamenstelling en een aantal stap-voor-stap oefeningen om te oefenen. Wat is functiesamenstelling? Functiesamenstelling bestaat uit &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/samenstelling-van-functies-samengestelde-functie\/\"> <span class=\"screen-reader-text\">Samenstelling van functies (samengestelde 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-58","post","type-post","status-publish","format-standard","hentry","category-functie-representatie"],"yoast_head":"<!-- This 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