{"id":1298,"date":"2025-07-24T04:07:12","date_gmt":"2025-07-24T04:07:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1298"},"modified":"2026-03-18T03:02:03","modified_gmt":"2026-03-18T03:02:03","slug":"inverse-functions-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-functions-3\/","title":{"raw":"Inverse Functions: Fresh Take","rendered":"Inverse Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Verify inverse functions using composition.<\/li>\r\n \t<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.<\/li>\r\n \t<li>Find or evaluate the inverse of a function.<\/li>\r\n \t<li>Use the graph of a one-to-one function to graph its inverse function on the same axes.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Inverse Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Inverse Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">An inverse function [latex]f^{-1}(x)[\/latex] \"undoes\" what the original function [latex]f(x)[\/latex] does<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Input and output are swapped: if [latex]f(a) = b[\/latex], then [latex]f^{-1}(b) = a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f^{-1}(x)[\/latex] represents the inverse of [latex]f(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The \"-1\" is not an exponent; [latex]f^{-1}(x) \\neq \\frac{1}{f(x)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Representation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The graph of [latex]f^{-1}(x)[\/latex] is a reflection of [latex]f(x)[\/latex] over the line [latex]y = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex](a, b)[\/latex] is on [latex]f(x)[\/latex], then [latex](b, a)[\/latex] is on [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conditions for Inverse Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The original function must be one-to-one (injective)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each output corresponds to exactly one input<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Passes the horizontal line test<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verifying Inverse Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These identities should hold for all [latex]x[\/latex] in the domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given that [latex]{h}^{-1}\\left(6\\right)=2[\/latex], what are the corresponding input and output values of the original function [latex]h?[\/latex][reveal-answer q=\"664604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"664604\"][latex]h(2)=6[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-geafgffh-TSztRfzmk0M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/TSztRfzmk0M?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-geafgffh-TSztRfzmk0M\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844443&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-geafgffh-TSztRfzmk0M&amp;vembed=0&amp;video_id=TSztRfzmk0M&amp;video_target=tpm-plugin-geafgffh-TSztRfzmk0M\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+an+Inverse+Function+From+a+Table_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find an Inverse Function From a Table\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)={x}^{3}-4[\/latex] and [latex]g\\left(x\\right)=\\sqrt[3]{x+4}[\/latex], is [latex]g={f}^{-1}?[\/latex][reveal-answer q=\"226656\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"226656\"]Yes[\/hidden-answer]<\/section><section class=\"textbox interact\" aria-label=\"Interact\"><iframe src=\"https:\/\/lumenlearning.h5p.com\/content\/1290756028512507138\/embed\" width=\"1088\" height=\"637\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/section><section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)={\\left(x - 1\\right)}^{3}\\text{and}g\\left(x\\right)=\\sqrt[3]{x}+1[\/latex], is [latex]g={f}^{-1}?[\/latex][reveal-answer q=\"384680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"384680\"]Yes[\/hidden-answer]<\/section>\r\n<h2>Determine the Domain and Range of an Inverse Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Relationship between function and inverse domains\/ranges:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Range of [latex]f(x)[\/latex] = Domain of [latex]f^{-1}(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain of [latex]f(x)[\/latex] = Range of [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One-to-one functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Only one-to-one functions have inverse functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pass the horizontal line test<\/li>\r\n \t<li>Identifying one-to-one functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Graphical method: Horizontal line test<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Algebraic method: [latex]f(a) = f(b)[\/latex] implies [latex]a = b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Restricting domains:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Non-one-to-one functions can be made invertible by restricting their domains<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The restricted domain becomes the range of the inverse function<\/li>\r\n \t<li>Domain restriction:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose a portion of the function that is one-to-one<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ensures the inverse is a valid function<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Uniqueness of inverse functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A function has only one inverse on a given domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>How to Find Domain and Range of Inverse Functions<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For one-to-one functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Write the range of [latex]f(x)[\/latex] as the domain of [latex]f^{-1}(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the domain of [latex]f(x)[\/latex] as the range of [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For restricted functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the restriction that makes the function one-to-one<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The restricted domain of [latex]f(x)[\/latex] becomes the range of [latex]f^{-1}(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The range of the restricted [latex]f(x)[\/latex] becomes the domain of [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">The domain of the function [latex]f[\/latex] is [latex]\\left(1,\\infty \\right)[\/latex] and the range of the function [latex]f[\/latex] is [latex]\\left(\\mathrm{-\\infty },-2\\right)[\/latex]. Find the domain and range of the inverse function.[reveal-answer q=\"146498\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"146498\"]The domain of the function [latex]{f}^{-1}[\/latex] is [latex]\\left(-\\infty \\text{,}-2\\right)[\/latex] and the range of the function [latex]{f}^{-1}[\/latex] is [latex]\\left(1,\\infty \\right)[\/latex].[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeaafchb-rsJ14O5-KDw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/rsJ14O5-KDw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aeaafchb-rsJ14O5-KDw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844444&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aeaafchb-rsJ14O5-KDw&amp;vembed=0&amp;video_id=rsJ14O5-KDw&amp;video_target=tpm-plugin-aeaafchb-rsJ14O5-KDw\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Restrict+the+Domain+to+Make+a+Function+1+to+1%2C+Then+Find+the+Inverse_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Finding and Evaluating Inverse Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Inverting tabular functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Interchange domain and range\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain of [latex]f(x)[\/latex] = Range of [latex]f^(-1)(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range of [latex]f(x)[\/latex] = Domain of [latex]f^(-1)(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Swap inputs and outputs<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluating inverses from graphs:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use vertical axis for inverse input\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical extent of [latex]f(x)[\/latex] = Domain of [latex]f^(-1)(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use horizontal axis for inverse output\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\" style=\"list-style-type: none;\">\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal extent of [latex]f(x)[\/latex] = Range of [latex]f^(-1)(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Finding inverse functions from formulas:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex] in terms of [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Replace [latex]y[\/latex] with [latex]f^(-1)(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Methods for Finding and Evaluating Inverse Functions<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For tabular functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Swap the input and output columns<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Read the new table for the inverse function values<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For graphical functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use the [latex]y[\/latex]-axis of the original function as the input for the inverse<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Read the corresponding [latex]x[\/latex]-value as the output of the inverse<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For formula-based functions:\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Verify the function is one-to-one<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Replace [latex]f(x)[\/latex] with [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interchange [latex]x[\/latex] and [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rename the resulting expression as [latex]f^(-1)(x)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Using the table below,\u00a0find and interpret\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\text{ }f\\left(60\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\text{ }{f}^{-1}\\left(60\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<table summary=\"Two rows and five columns. The first row is labeled\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]t\\text{ (minutes)}[\/latex]<\/td>\r\n<td>30<\/td>\r\n<td>50<\/td>\r\n<td>60<\/td>\r\n<td>70<\/td>\r\n<td>90<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex]<\/td>\r\n<td>20<\/td>\r\n<td>40<\/td>\r\n<td>50<\/td>\r\n<td>60<\/td>\r\n<td>70<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"401025\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"401025\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f\\left(60\\right)=50[\/latex]. In [latex]60[\/latex] minutes, [latex]50[\/latex] miles are traveled.<\/li>\r\n \t<li>[latex]{f}^{-1}\\left(60\\right)=70[\/latex]. To travel [latex]60[\/latex] miles, it will take [latex]70[\/latex] minutes.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Using the graph below,\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>find [latex]{g}^{-1}\\left(1\\right)[\/latex]<\/li>\r\n \t<li>estimate [latex]{g}^{-1}\\left(4\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18205520\/CNX_Precalc_Figure_01_07_0062.jpg\" alt=\"Graph of g(x).\" width=\"487\" height=\"254\" \/> Graph of g(x)[\/caption]\r\n\r\n[reveal-answer q=\"350455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"350455\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]3[\/latex]<\/li>\r\n \t<li>b. [latex]5.6[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve for [latex]x[\/latex] in terms of [latex]y[\/latex] given [latex]y=\\frac{1}{3}\\left(x - 5\\right)[\/latex][reveal-answer q=\"875458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875458\"][latex]x=3y+5[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]19175[\/ohm2_question]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adcgfdeh-q6y0ToEhT1E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/q6y0ToEhT1E?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-adcgfdeh-q6y0ToEhT1E\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844445&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-adcgfdeh-q6y0ToEhT1E&amp;vembed=0&amp;video_id=q6y0ToEhT1E&amp;video_target=tpm-plugin-adcgfdeh-q6y0ToEhT1E\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Lesson+-+Inverse+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLesson: Inverse Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hebaaahb-FIF8SdZkJc8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FIF8SdZkJc8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hebaaahb-FIF8SdZkJc8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844446&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hebaaahb-FIF8SdZkJc8&amp;vembed=0&amp;video_id=FIF8SdZkJc8&amp;video_target=tpm-plugin-hebaaahb-FIF8SdZkJc8\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Function+and+Inverse+Function+Values+Using+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Function and Inverse Function Values Using a Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Finding Inverse Functions and Their Graphs<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Graphical relationship between a function and its inverse:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Reflection over the line [latex]y = x[\/latex] (identity line)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain and range relationships:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain of [latex]f(x)[\/latex] becomes range of [latex]f^{-1}(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range of [latex]f(x)[\/latex] becomes domain of [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Restricting domains for invertibility:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Some functions need domain restrictions to be one-to-one<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Functions that are their own inverses:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Special cases where [latex]f(x) = f^{-1}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Concepts<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Reflection principle:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Graph of [latex]f^{-1}(x)[\/latex] is the reflection of [latex]f(x)[\/latex] over [latex]y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One-to-one functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Necessary for a function to have an inverse<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pass the horizontal line test<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity line:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The line [latex]y = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Acts as the \"mirror\" for reflecting graphs<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Restricted domains:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Make non-one-to-one functions invertible<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^2[\/latex] restricted to [latex][0, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Techniques for Graphing Inverse Functions<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify key points on the original function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercepts become [latex]x[\/latex]-intercepts for the inverse<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts become [latex]y[\/latex]-intercepts for the inverse<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflect these key points over [latex]y = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider domain and range:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptotes become horizontal asymptotes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal asymptotes become vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sketch the inverse function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Connect the reflected points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ensure the graph passes the vertical line test<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify inverse relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check if composition yields the identity function<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Verify inverse functions using composition.<\/li>\n<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.<\/li>\n<li>Find or evaluate the inverse of a function.<\/li>\n<li>Use the graph of a one-to-one function to graph its inverse function on the same axes.<\/li>\n<\/ul>\n<\/section>\n<h2>Inverse Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Inverse Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">An inverse function [latex]f^{-1}(x)[\/latex] &#8220;undoes&#8221; what the original function [latex]f(x)[\/latex] does<\/li>\n<li class=\"whitespace-normal break-words\">Input and output are swapped: if [latex]f(a) = b[\/latex], then [latex]f^{-1}(b) = a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f^{-1}(x)[\/latex] represents the inverse of [latex]f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The &#8220;-1&#8221; is not an exponent; [latex]f^{-1}(x) \\neq \\frac{1}{f(x)}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Representation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The graph of [latex]f^{-1}(x)[\/latex] is a reflection of [latex]f(x)[\/latex] over the line [latex]y = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex](a, b)[\/latex] is on [latex]f(x)[\/latex], then [latex](b, a)[\/latex] is on [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conditions for Inverse Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The original function must be one-to-one (injective)<\/li>\n<li class=\"whitespace-normal break-words\">Each output corresponds to exactly one input<\/li>\n<li class=\"whitespace-normal break-words\">Passes the horizontal line test<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verifying Inverse Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">These identities should hold for all [latex]x[\/latex] in the domain<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given that [latex]{h}^{-1}\\left(6\\right)=2[\/latex], what are the corresponding input and output values of the original function [latex]h?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q664604\">Show Solution<\/button><\/p>\n<div id=\"q664604\" class=\"hidden-answer\" style=\"display: none\">[latex]h(2)=6[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-geafgffh-TSztRfzmk0M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/TSztRfzmk0M?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-geafgffh-TSztRfzmk0M\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844443&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-geafgffh-TSztRfzmk0M&amp;vembed=0&amp;video_id=TSztRfzmk0M&amp;video_target=tpm-plugin-geafgffh-TSztRfzmk0M\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+an+Inverse+Function+From+a+Table_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find an Inverse Function From a Table\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)={x}^{3}-4[\/latex] and [latex]g\\left(x\\right)=\\sqrt[3]{x+4}[\/latex], is [latex]g={f}^{-1}?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q226656\">Show Solution<\/button><\/p>\n<div id=\"q226656\" class=\"hidden-answer\" style=\"display: none\">Yes<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\"><iframe loading=\"lazy\" src=\"https:\/\/lumenlearning.h5p.com\/content\/1290756028512507138\/embed\" width=\"1088\" height=\"637\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)={\\left(x - 1\\right)}^{3}\\text{and}g\\left(x\\right)=\\sqrt[3]{x}+1[\/latex], is [latex]g={f}^{-1}?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q384680\">Show Solution<\/button><\/p>\n<div id=\"q384680\" class=\"hidden-answer\" style=\"display: none\">Yes<\/div>\n<\/div>\n<\/section>\n<h2>Determine the Domain and Range of an Inverse Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Relationship between function and inverse domains\/ranges:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Range of [latex]f(x)[\/latex] = Domain of [latex]f^{-1}(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Domain of [latex]f(x)[\/latex] = Range of [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">One-to-one functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Only one-to-one functions have inverse functions<\/li>\n<li class=\"whitespace-normal break-words\">Pass the horizontal line test<\/li>\n<li>Identifying one-to-one functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Graphical method: Horizontal line test<\/li>\n<li class=\"whitespace-normal break-words\">Algebraic method: [latex]f(a) = f(b)[\/latex] implies [latex]a = b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Restricting domains:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Non-one-to-one functions can be made invertible by restricting their domains<\/li>\n<li class=\"whitespace-normal break-words\">The restricted domain becomes the range of the inverse function<\/li>\n<li>Domain restriction:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose a portion of the function that is one-to-one<\/li>\n<li class=\"whitespace-normal break-words\">Ensures the inverse is a valid function<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Uniqueness of inverse functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A function has only one inverse on a given domain<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>How to Find Domain and Range of Inverse Functions<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For one-to-one functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Write the range of [latex]f(x)[\/latex] as the domain of [latex]f^{-1}(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write the domain of [latex]f(x)[\/latex] as the range of [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For restricted functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the restriction that makes the function one-to-one<\/li>\n<li class=\"whitespace-normal break-words\">The restricted domain of [latex]f(x)[\/latex] becomes the range of [latex]f^{-1}(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The range of the restricted [latex]f(x)[\/latex] becomes the domain of [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">The domain of the function [latex]f[\/latex] is [latex]\\left(1,\\infty \\right)[\/latex] and the range of the function [latex]f[\/latex] is [latex]\\left(\\mathrm{-\\infty },-2\\right)[\/latex]. Find the domain and range of the inverse function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q146498\">Show Solution<\/button><\/p>\n<div id=\"q146498\" class=\"hidden-answer\" style=\"display: none\">The domain of the function [latex]{f}^{-1}[\/latex] is [latex]\\left(-\\infty \\text{,}-2\\right)[\/latex] and the range of the function [latex]{f}^{-1}[\/latex] is [latex]\\left(1,\\infty \\right)[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeaafchb-rsJ14O5-KDw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/rsJ14O5-KDw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aeaafchb-rsJ14O5-KDw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844444&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aeaafchb-rsJ14O5-KDw&amp;vembed=0&amp;video_id=rsJ14O5-KDw&amp;video_target=tpm-plugin-aeaafchb-rsJ14O5-KDw\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Restrict+the+Domain+to+Make+a+Function+1+to+1%2C+Then+Find+the+Inverse_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Finding and Evaluating Inverse Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Inverting tabular functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Interchange domain and range\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain of [latex]f(x)[\/latex] = Range of [latex]f^(-1)(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range of [latex]f(x)[\/latex] = Domain of [latex]f^(-1)(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Swap inputs and outputs<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluating inverses from graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use vertical axis for inverse input\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical extent of [latex]f(x)[\/latex] = Domain of [latex]f^(-1)(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Use horizontal axis for inverse output\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\" style=\"list-style-type: none;\">\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal extent of [latex]f(x)[\/latex] = Range of [latex]f^(-1)(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Finding inverse functions from formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex] in terms of [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Replace [latex]y[\/latex] with [latex]f^(-1)(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Methods for Finding and Evaluating Inverse Functions<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For tabular functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Swap the input and output columns<\/li>\n<li class=\"whitespace-normal break-words\">Read the new table for the inverse function values<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For graphical functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use the [latex]y[\/latex]-axis of the original function as the input for the inverse<\/li>\n<li class=\"whitespace-normal break-words\">Read the corresponding [latex]x[\/latex]-value as the output of the inverse<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For formula-based functions:\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Verify the function is one-to-one<\/li>\n<li class=\"whitespace-normal break-words\">Replace [latex]f(x)[\/latex] with [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Interchange [latex]x[\/latex] and [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Rename the resulting expression as [latex]f^(-1)(x)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Using the table below,\u00a0find and interpret<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\text{ }f\\left(60\\right)[\/latex]<\/li>\n<li>[latex]\\text{ }{f}^{-1}\\left(60\\right)[\/latex]<\/li>\n<\/ol>\n<table summary=\"Two rows and five columns. The first row is labeled\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]t\\text{ (minutes)}[\/latex]<\/td>\n<td>30<\/td>\n<td>50<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<td>90<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex]<\/td>\n<td>20<\/td>\n<td>40<\/td>\n<td>50<\/td>\n<td>60<\/td>\n<td>70<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q401025\">Show Solution<\/button><\/p>\n<div id=\"q401025\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]f\\left(60\\right)=50[\/latex]. In [latex]60[\/latex] minutes, [latex]50[\/latex] miles are traveled.<\/li>\n<li>[latex]{f}^{-1}\\left(60\\right)=70[\/latex]. To travel [latex]60[\/latex] miles, it will take [latex]70[\/latex] minutes.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Using the graph below,<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>find [latex]{g}^{-1}\\left(1\\right)[\/latex]<\/li>\n<li>estimate [latex]{g}^{-1}\\left(4\\right)[\/latex]<\/li>\n<\/ol>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18205520\/CNX_Precalc_Figure_01_07_0062.jpg\" alt=\"Graph of g(x).\" width=\"487\" height=\"254\" \/><figcaption class=\"wp-caption-text\">Graph of g(x)<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q350455\">Show Solution<\/button><\/p>\n<div id=\"q350455\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]3[\/latex]<\/li>\n<li>b. [latex]5.6[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve for [latex]x[\/latex] in terms of [latex]y[\/latex] given [latex]y=\\frac{1}{3}\\left(x - 5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q875458\">Show Solution<\/button><\/p>\n<div id=\"q875458\" class=\"hidden-answer\" style=\"display: none\">[latex]x=3y+5[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm19175\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=19175&theme=lumen&iframe_resize_id=ohm19175&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adcgfdeh-q6y0ToEhT1E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/q6y0ToEhT1E?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-adcgfdeh-q6y0ToEhT1E\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844445&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-adcgfdeh-q6y0ToEhT1E&amp;vembed=0&amp;video_id=q6y0ToEhT1E&amp;video_target=tpm-plugin-adcgfdeh-q6y0ToEhT1E\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Lesson+-+Inverse+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLesson: Inverse Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hebaaahb-FIF8SdZkJc8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FIF8SdZkJc8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hebaaahb-FIF8SdZkJc8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844446&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hebaaahb-FIF8SdZkJc8&amp;vembed=0&amp;video_id=FIF8SdZkJc8&amp;video_target=tpm-plugin-hebaaahb-FIF8SdZkJc8\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Function+and+Inverse+Function+Values+Using+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Function and Inverse Function Values Using a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Finding Inverse Functions and Their Graphs<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Graphical relationship between a function and its inverse:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reflection over the line [latex]y = x[\/latex] (identity line)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain and range relationships:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain of [latex]f(x)[\/latex] becomes range of [latex]f^{-1}(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range of [latex]f(x)[\/latex] becomes domain of [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Restricting domains for invertibility:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Some functions need domain restrictions to be one-to-one<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Functions that are their own inverses:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Special cases where [latex]f(x) = f^{-1}(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Concepts<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reflection principle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Graph of [latex]f^{-1}(x)[\/latex] is the reflection of [latex]f(x)[\/latex] over [latex]y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">One-to-one functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Necessary for a function to have an inverse<\/li>\n<li class=\"whitespace-normal break-words\">Pass the horizontal line test<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identity line:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The line [latex]y = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Acts as the &#8220;mirror&#8221; for reflecting graphs<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Restricted domains:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Make non-one-to-one functions invertible<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^2[\/latex] restricted to [latex][0, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Techniques for Graphing Inverse Functions<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify key points on the original function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercepts become [latex]x[\/latex]-intercepts for the inverse<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts become [latex]y[\/latex]-intercepts for the inverse<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reflect these key points over [latex]y = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Consider domain and range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical asymptotes become horizontal asymptotes<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal asymptotes become vertical asymptotes<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Sketch the inverse function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Connect the reflected points<\/li>\n<li class=\"whitespace-normal break-words\">Ensure the graph passes the vertical line test<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verify inverse relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check if composition yields the identity function<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":67,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Find an Inverse Function From a Table\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/TSztRfzmk0M\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/rsJ14O5-KDw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Lesson: Inverse 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