{"id":19,"date":"2023-09-17T11:09:46","date_gmt":"2023-09-17T11:09:46","guid":{"rendered":"https:\/\/mathority.org\/it\/funzione-inversa-o-reciproca\/"},"modified":"2023-09-17T11:09:46","modified_gmt":"2023-09-17T11:09:46","slug":"funzione-inversa-o-reciproca","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/funzione-inversa-o-reciproca\/","title":{"rendered":"Funzione inversa (o reciproca)."},"content":{"rendered":"<p>In questo articolo spieghiamo cos&#8217;\u00e8 una funzione inversa (o reciproca) e come calcolare l&#8217;inversa di una funzione. Scoprirai anche come sapere facilmente se una funzione ha un inverso o meno e le propriet\u00e0 di questo tipo di funzioni. Infine, puoi esercitarti con esercizi passo passo sulle funzioni inverse. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-funcion-inversa\"><\/span> Qual \u00e8 la funzione inversa?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>La funzione inversa, detta anche funzione reciproca, \u00e8 la funzione il cui dominio \u00e8 l&#8217;intervallo di un&#8217;altra funzione (la funzione originaria) e il cui intervallo \u00e8 il dominio della funzione originaria. La funzione inversa della funzione <em>f<\/em> si esprime con il simbolo <em>f <sup>-1<\/sup><\/em> .<\/strong><\/p>\n<p> Pertanto, la funzione inversa di <em>f(x)<\/em> \u00e8 la funzione che soddisfa la seguente condizione: <\/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=\"funzione inversa o reciproca\" class=\"wp-image-685\" width=\"208\" height=\"210\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Oro<\/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> \u00e8 la funzione inversa di<\/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> Il concetto di funzione inversa pu\u00f2 essere definito anche utilizzando la composizione di funzioni, poich\u00e9 qualsiasi funzione composta con la sua funzione inversa \u00e8 uguale alla funzione identit\u00e0:<\/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>Vedi:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/it\/composizione-di-funzioni-funzione-composita\/\">cos&#8217;\u00e8 la composizione della funzione?<\/a><\/span><\/p>\n<p> Quindi se l&#8217;equazione precedente \u00e8 soddisfatta, significa che<\/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> \u00e8 la funzione inversa (o funzione reciproca) di <\/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> Esempio di funzione inversa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Data la definizione di funzione inversa, risolviamo un esempio per comprenderne meglio il significato.<\/p>\n<ul>\n<li> Determina se le seguenti funzioni sono tra loro inverse:<\/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> Se le due funzioni sono tra loro inverse, saranno soddisfatte le seguenti 2 condizioni:<\/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> Quindi controlliamo se entrambe le equazioni sono soddisfatte. Per prima cosa controlliamo<\/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 Se non capisci il calcolo che abbiamo appena fatto, devi andare al link sopra <em>Qual \u00e8 la composizione delle funzioni?<\/em> , spieghiamo come risolvere questo tipo di operazioni con le funzioni.<\/p>\n<p> Affinch\u00e9<\/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> s\u00ec, \u00e8 compiuto. \u2705<\/p>\n<p> Ora controlliamo l&#8217;uguaglianza<\/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> E la condizione di invertibilit\u00e0<\/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> \u00e8 anche compiuto. \u2705<\/p>\n<p> In conclusione, poich\u00e9 valgono entrambe le equazioni, <strong>le due funzioni sono l&#8217;una l&#8217;inversa dell&#8217;altra.<\/strong><\/p>\n<p> Di seguito puoi vedere entrambe le funzioni rappresentate graficamente. Si noti che <strong>i grafici di due funzioni inverse sono simmetrici<\/strong> rispetto alla bisettrice del primo e del terzo quadrante: <\/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=\"funzione inversa\" 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> Come sapere se una funzione ha un inverso<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Una funzione ha una funzione inversa se \u00e8 una funzione iniettiva<\/strong> , cio\u00e8 se ogni valore nel suo intero dominio corrisponde a un singolo valore nel suo intervallo. <\/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;\">Funzione esponenziale con funzione inversa<\/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;\">Funzione quadratica senza funzione inversa<\/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> Ad esempio, la funzione esponenziale sinistra ha una funzione inversa perch\u00e9 ogni <em>x<\/em> corrisponde a un singolo valore di <em>f(x)<\/em> . D&#8217;altra parte, la funzione quadratica destra non ha una funzione inversa poich\u00e9 ha pi\u00f9 valori di <em>x<\/em> le cui immagini sono uguali (ad esempio <em>f(1)=f(3)=2)<\/em> .<\/p>\n<p> Allo stesso modo, una funzione biiettiva consiste in una funzione che \u00e8 sia iniettiva che suriettiva, quindi qualsiasi funzione biiettiva ha anche una funzione inversa.<\/p>\n<p> D&#8217;altra parte, dovresti tenere presente che la funzione inversa non \u00e8 la stessa cosa <em>dell&#8217;inverso moltiplicativo di una funzione<\/em> , ma piuttosto due concetti diversi. Per trovare l&#8217;inverso moltiplicativo di una funzione, calcola semplicemente 1 corrispondenza di detta funzione.<\/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> Nella prossima sezione vedremo come determinare la funzione inversa. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-la-funcion-inversa\"><\/span> Come trovare la funzione inversa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Per calcolare l&#8217;inversa di una funzione \u00e8 necessario eseguire i seguenti passaggi: <\/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;\">Sostituisci <em>f(x)<\/em> con <em>y<\/em> .<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Cambia tutto <em>x<\/em> in <em>y<\/em> e viceversa.<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Cancella la variabile <em>y<\/em> .<\/span><\/li>\n<li> <span style=\"color:#101010;font-weight: normal;\">Sostituisci la variabile <em>y<\/em> con <em>f <sup>-1<\/sup> (x)<\/em> . La funzione inversa \u00e8 l&#8217;espressione trovata per <em>f <sup>-1<\/sup> (x)<\/em> .<\/span><\/li>\n<\/ol>\n<\/div>\n<p> Affinch\u00e9 tu possa vedere esattamente come viene calcolata la funzione inversa, determiniamo come esempio l&#8217;inversa della seguente funzione:<\/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> Prima di tutto dobbiamo sostituire<\/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> Per<\/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> Ora cambiamo tutto<\/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> della funzione di<\/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> , e viceversa:<\/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> Quindi cancelliamo la variabile <\/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> E infine, la funzione inversa di<\/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> \u00e8 l&#8217;espressione algebrica che abbiamo ottenuto isolando <\/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> Esercizi risolti della funzione inversa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Di seguito abbiamo preparato diversi esercizi passo passo sulla funzione inversa in modo che tu possa esercitarti.<\/p>\n<p> \ud83d\udc49 Ricorda che se non capisci come risolvere un esercizio o vuoi che ti risolviamo un problema, puoi scrivercelo nei commenti!<\/p>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Controlla se le seguenti due funzioni sono inverse (o reciproche) o no: <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Affinch\u00e9 le due funzioni siano tra loro inverse, deve essere vero quanto segue:<\/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\"> \u00c8 quindi necessario verificare se le due condizioni sono soddisfatte. Per prima cosa controlliamo <\/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\"> Ancora,<\/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> s\u00ec, \u00e8 compiuto. \u2705<\/p>\n<p class=\"has-text-align-left\"> Ora controlliamo l&#8217;altra composizione delle funzioni <\/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\"> Per cui<\/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> \u00e8 anche compiuto. \u2705<\/p>\n<p class=\"has-text-align-left\"> Come \u00e8 possibile?<\/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> E<\/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> , le due funzioni sono una l&#8217;inversa dell&#8217;altra.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 2<\/h3>\n<p> Calcola l&#8217;inversa (o funzione reciproca) della seguente funzione polinomiale di primo grado: <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La prima cosa da fare per invertire la funzione \u00e8 sostituire il termine<\/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> Per<\/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\"> Ora cambiamo il<\/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> di<\/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> , e viceversa:<\/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\"> E poi rilasciamo <\/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\"> Siamo gi\u00e0 riusciti a rilasciare<\/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> . Pertanto, la funzione inversa di<\/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> Est: <\/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\">Esercizio 3<\/h3>\n<p> Invertire la seguente funzione polinomiale quadratica: <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per trovare la funzione inversa seguiremo il procedimento che abbiamo visto sopra. Quindi chiameremo<\/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> alla funzione<\/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\"> In secondo luogo, modifichiamo il file<\/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> per il<\/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> , e viceversa:<\/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\"> E infine isoliamo la variabile <\/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 questo caso per\u00f2 la funzione ottenuta ha due immagini per ogni elemento del suo dominio (l&#8217;immagine positiva e l&#8217;immagine negativa). Pertanto <strong>non esiste una funzione inversa<\/strong> della funzione problematica.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 4<\/h3>\n<p> Determinare la funzione inversa (o funzione reciproca) della seguente funzione razionale: <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Innanzitutto, sostituiamo<\/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> Per<\/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\"> Ora cambiamo il<\/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> numeratore e denominatore<\/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> , e viceversa:<\/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\"> E poi rilasciamo <\/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\"> L&#8217;espressione<\/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> divide l&#8217;intero lato destro dell&#8217;equazione, quindi possiamo moltiplicarlo moltiplicando l&#8217;intero lato sinistro dell&#8217;equazione: <\/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\"> Mettiamo tutti i termini con<\/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> da un lato dell&#8217;equazione e gli altri termini dall&#8217;altro:<\/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\"> Per chiarire<\/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> , estraiamo il fattore comune dal lato sinistro dell&#8217;equazione:<\/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\"> E come postino<\/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> \u00e8 moltiplicare l&#8217;intero lato sinistro dell&#8217;equazione, possiamo farlo dividendo l&#8217;intero lato destro:<\/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\"> Siamo gi\u00e0 riusciti a rilasciare<\/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> . Quindi la funzione inversa di<\/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> Est: <\/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> Propriet\u00e0 della funzione inversa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> La funzione inversa ha le seguenti caratteristiche:<\/p>\n<ul>\n<li> La funzione inversa \u00e8 unica, cio\u00e8 se una funzione \u00e8 invertibile, per questa funzione esiste una sola funzione inversa.<\/li>\n<\/ul>\n<ul>\n<li> Il dominio della funzione inversa \u00e8 l&#8217;intervallo (o intervallo) della funzione originale.<\/li>\n<\/ul>\n<ul>\n<li> Allo stesso modo, il percorso della funzione inversa \u00e8 equivalente al dominio della funzione originale.<\/li>\n<\/ul>\n<ul>\n<li> Qualsiasi funzione composta dalla sua funzione inversa d\u00e0 la funzione identit\u00e0 (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> Il grafico di una funzione e il grafico della sua funzione inversa sono simmetrici rispetto alla bisettrice del primo e del terzo quadrante.<\/li>\n<\/ul>\n<ul>\n<li> L&#8217;inverso della funzione inversa \u00e8 uguale alla funzione originale:<\/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> Invertire una funzione composta equivale a calcolare l&#8217;inversa di ciascuna funzione separatamente e quindi comporre le funzioni inverse.<\/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> Una funzione \u00e8 contemporaneamente continua con la sua funzione inversa, o in altre parole, se una funzione \u00e8 continua, lo sar\u00e0 anche la sua funzione inversa.<\/li>\n<\/ul>\n<ul>\n<li> Se una funzione \u00e8 differenziabile e la derivata non scompare in nessun momento\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> , anche la sua funzione inversa sar\u00e0 differenziabile.<\/li>\n<\/ul>\n<p> Inoltre, la derivata della funzione inversa pu\u00f2 essere calcolata applicando il <strong>teorema della funzione inversa<\/strong> , la cui formula \u00e8:<\/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 questo articolo spieghiamo cos&#8217;\u00e8 una funzione inversa (o reciproca) e come calcolare l&#8217;inversa di una funzione. Scoprirai anche come sapere facilmente se una funzione ha un inverso o meno e le propriet\u00e0 di questo tipo di funzioni. Infine, puoi esercitarti con esercizi passo passo sulle funzioni inverse. Qual \u00e8 la funzione inversa? La funzione &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/funzione-inversa-o-reciproca\/\"> <span class=\"screen-reader-text\">Funzione inversa (o reciproca).<\/span> Leggi altro &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":[17],"tags":[],"class_list":["post-19","post","type-post","status-publish","format-standard","hentry","category-rappresentazione-delle-funzioni"],"yoast_head":"<!-- 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