{"id":7950,"date":"2026-02-15T20:21:01","date_gmt":"2026-02-15T14:51:01","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7950"},"modified":"2026-02-15T20:21:09","modified_gmt":"2026-02-15T14:51:09","slug":"how-to-find-the-inverse-of-a-function","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/how-to-find-the-inverse-of-a-function\/","title":{"rendered":"How to find the inverse of a function"},"content":{"rendered":"\n<p>Here in this post , we will see how to find the inverse of a function<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Inverse Function<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A function can have inverse only if it is bijective<\/li>\n\n\n\n<li>Note that not all functions have inverses.  <\/li>\n\n\n\n<li>Sometime , the function may not be bijective from R -&gt; R, you can restrict the domain  and range of the function to make it one-to-one and onto  and then find the inverse of the restricted function.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">How to find the inverse of a function<\/h2>\n\n\n\n<p><strong>Method I<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>let f(x) be the function f(x)=4x -2<\/li>\n\n\n\n<li>Consider an arbitrary element y then by definition y=4x-2<\/li>\n\n\n\n<li>Find out the value of x in terms of y<br>$y=4x-2$<br>$y+2=4x$<br>$x= \\frac {y+2}{4}$<\/li>\n\n\n\n<li>Now the invertible function can be defined as<br>$g(x) = \\frac {x+2}{4}$<br><\/li>\n<\/ul>\n\n\n\n<p><strong>Method II<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>let f(x) be the function f(x)=4x -2<\/li>\n\n\n\n<li>Consider an arbitrary element y then by definition y=4x-2<\/li>\n\n\n\n<li>Swap the position of x and y<br>$y=4x-2$<br>$x=4y -2$<br>$y= \\frac {x+2}{4}$<\/li>\n\n\n\n<li>Now the invertible function can be defined as<br>$g(x) = \\frac {x+2}{4}$<\/li>\n<\/ul>\n\n\n\n<p><strong>Method III<\/strong><\/p>\n\n\n\n<p>Let f(x) be the function and $f^{-1}(x) $ be the inverse<\/p>\n\n\n\n<p>We can use below equation<\/p>\n\n\n\n<p>$f(f^{-1}(x)) =x$<\/p>\n\n\n\n<p>Example<\/p>\n\n\n\n<p>f(x)=4x -2<\/p>\n\n\n\n<p>$f(f^{-1}(x)) =x$<br>or<br>$4 f^{-1}(x) -2 =x$<br>$f^{-1}(x) = \\frac {x+2}{4}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved Examples<\/h2>\n\n\n\n<p><strong>Example 1<\/strong><\/p>\n\n\n\n<p>Let f: R -&gt; R given as $f(x) =x^3 + 1$, find the inverse $f^{-1}(x)$<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>let $y=x^3 + 1$<br>or $ x^3 =1 -y$<br>or $x = (1-y)^{1\/3}$<br>So inverse function is<\/p>\n\n\n\n<p>$f^{-1}(x)= (1-x)^{1\/3}$<\/p>\n\n\n\n<p><strong>Example<\/strong> 2<\/p>\n\n\n\n<p>Let  f: R -&gt;R given as  $f(x) = 4 \u2013 (x\u20137)^3$. Find the inverse $f^{-1}(x)$<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>Let $y= 4 \u2013 (x\u20137)^3$<br>$(x-7)^3=4-y$<br>$x= (4-y)^{1\/3}  + 7$<br>So inverse function is<\/p>\n\n\n\n<p>$f^{-1}(x)= (4-x)^{1\/3}  + 7$<\/p>\n\n\n\n<p><strong>Example<\/strong> 3<\/p>\n\n\n\n<p>if f: $[2,\\infty)$ -> ($-\\infty,4] $ given as f(x)=x(4-x). Find the inverse $f^{-1}(x)$<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>Let $y= x(4-x)$<br>$x^2 -4x + y=0$<br>$x= \\frac {4 \\pm \\sqrt {16-4y}}{2}$<br>$x= 2 \\pm \\sqrt {4-y}$<br>as $x \\geq 2$<br>$x= 2 + \\sqrt {4-y}$ <br>So inverse function is<\/p>\n\n\n\n<p>$f^{-1}(x)= 2 + \\sqrt {4-x}$<\/p>\n\n\n\n<p>I hope you like this post on How to find the inverse of a function<\/p>\n\n\n\n<p><strong>Related Articles<\/strong><\/p>\n\n\n\n<p><a href=\"https:\/\/physicscatalyst.com\/article\/inverse-trigonometric-function-formula\/\">Inverse Trigonometric Function Formulas<\/a><br><a href=\"https:\/\/physicscatalyst.com\/article\/domain-and-range-of-inverse-trigonometric-functions\/\">Domain and Range Of Inverse Trigonometric Functions<\/a><br><a href=\"https:\/\/physicscatalyst.com\/article\/many-one-functions-definition-examples\/\">Many one Functions<\/a><br><br><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here in this post , we will see how to find the inverse of a function Inverse Function How to find the inverse of a function Method I Method II Method III Let f(x) be the function and $f^{-1}(x) $ be the inverse We can use below equation $f(f^{-1}(x)) =x$ Example f(x)=4x -2 $f(f^{-1}(x)) =x$or$4 [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","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":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-7950","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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