{"id":2045,"date":"2024-07-03T00:01:30","date_gmt":"2024-07-03T00:01:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2045"},"modified":"2025-01-19T04:55:56","modified_gmt":"2025-01-19T04:55:56","slug":"radical-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/radical-functions-fresh-take\/","title":{"raw":"Inverses and Radical Functions: Fresh Take","rendered":"Inverses and Radical Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Learn how to find the inverse (or \"reverse\") of a polynomial function when it's possible<\/li>\r\n \t<li>Figure out how to limit the domain of a polynomial function so you can find its inverse<\/li>\r\n \t<li>Use radical functions to solve real-world problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Radical Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Radical Functions:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = \\sqrt[n]{g(x)}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is a positive integer (root degree)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x)[\/latex] is any function of [latex]x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Inverse Functions:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Notation: [latex]f^{-1}(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Property: [latex]f^{-1}(f(x)) = x[\/latex] and [latex]f(f^{-1}(x)) = x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical relationship: Symmetric about [latex]y = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>One-to-One Functions:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Definition: Each output corresponds to a unique input<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Test: Horizontal line test<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Importance: Only one-to-one functions have inverses that are functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Types of Radical 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\"><strong>Square Root Function:<\/strong> [latex]f(x) = \\sqrt{x}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Cube Root Function:<\/strong> [latex]f(x) = \\sqrt[3]{x}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Higher-Order Root Functions:<\/strong> [latex]f(x) = \\sqrt[n]{x}[\/latex], where [latex]n \\geq 3[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Steps to Finding 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\">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 function [latex]f^{-1}(x)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Show that [latex]f\\left(x\\right)=\\dfrac{1}{x+1}[\/latex] and [latex]{f}^{-1}\\left(x\\right)=\\dfrac{1}{x}-1[\/latex] are inverses, for [latex]x\\ne 0,-1[\/latex] .[reveal-answer q=\"631376\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"631376\"]We must show that [latex]{f}^{-1}\\left(f\\left(x\\right)\\right)=x[\/latex] and [latex]f\\left({f}^{-1}\\left(x\\right)\\right)=x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{f}^{-1}\\left(f\\left(x\\right)\\right)&amp;={f}^{-1}\\left(\\frac{1}{x+1}\\right) \\\\[1mm] &amp;=\\dfrac{1}{\\frac{1}{x+1}}-1 \\\\[1mm] &amp;=\\left(x+1\\right)-1 \\\\[1mm] &amp;=x \\\\[5mm] f\\left({f}^{-1}\\left(x\\right)\\right)&amp;=f\\left(\\frac{1}{x}-1\\right) \\\\[1mm] &amp;=\\dfrac{1}{\\left(\\frac{1}{x}-1\\right)+1} \\\\[1mm] &amp;=\\dfrac{1}{\\frac{1}{x}} \\\\[1mm] &amp;=x \\end{align}[\/latex]<\/p>\r\nTherefore, [latex]f\\left(x\\right)=\\dfrac{1}{x+1}[\/latex]\u00a0and [latex]{f}^{-1}\\left(x\\right)=\\dfrac{1}{x}-1[\/latex] are inverses.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the inverse function of [latex]f\\left(x\\right)=\\sqrt[3]{x+4}[\/latex].[reveal-answer q=\"987592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"987592\"][latex]{f}^{-1}\\left(x\\right)={x}^{3}-4[\/latex][\/hidden-answer]<\/section>\r\n<h2>Domains of Radical 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\">The domain of a radical function depends on the index [latex]n[\/latex] of the root:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For even roots (e.g., square roots), the expression inside the radical must be non-negative.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For odd roots (e.g., cube roots), the expression inside the radical can be any real number.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">To find the domain of a radical function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set up an inequality ensuring the radicand is non-negative (for even roots).<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify critical points where the expression could change sign.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Test intervals and determine where the inequality holds.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not all functions are one-to-one. To find an inverse of a non-one-to-one function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Restrict the domain to make the function one-to-one.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the inverse on this restricted domain.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Steps to find the inverse of a function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\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 function [latex]f^{-1}(x)[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ensure the domain of [latex]f^{-1}(x)[\/latex] corresponds to the range of [latex]f(x)[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For radical functions, when finding the inverse:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determine the range of the original function.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This range becomes the domain restriction for the inverse function.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphically, inverse functions are reflections of each other over the line [latex]y = x[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The domain of [latex]f(x)[\/latex] becomes the range of [latex]f^{-1}(x)[\/latex], and vice versa.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When graphing radical functions and their inverses:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Points of intersection will always lie on the line [latex]y = x[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex](a, b)[\/latex] is on the graph of [latex]f(x)[\/latex], then [latex](b, a)[\/latex] is on the graph of [latex]f^{-1}(x)[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the inverse of the function [latex]f\\left(x\\right)={x}^{2}+1[\/latex], on the domain [latex]x\\ge 0[\/latex].[reveal-answer q=\"758215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"758215\"][latex]{f}^{-1}\\left(x\\right)=\\sqrt{x - 1}[\/latex][\/hidden-answer]<\/section>Watch the following video to see more examples of how to restrict the domain of a quadratic function to find it's inverse.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-agbggced-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-agbggced-rsJ14O5-KDw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844444&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-agbggced-rsJ14O5-KDw&vembed=0&video_id=rsJ14O5-KDw&video_target=tpm-plugin-agbggced-rsJ14O5-KDw'><\/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>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Learn how to find the inverse (or &#8220;reverse&#8221;) of a polynomial function when it&#8217;s possible<\/li>\n<li>Figure out how to limit the domain of a polynomial function so you can find its inverse<\/li>\n<li>Use radical functions to solve real-world problems<\/li>\n<\/ul>\n<\/section>\n<h2>Radical Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Radical Functions:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = \\sqrt[n]{g(x)}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is a positive integer (root degree)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]g(x)[\/latex] is any function of [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Inverse Functions:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Notation: [latex]f^{-1}(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Property: [latex]f^{-1}(f(x)) = x[\/latex] and [latex]f(f^{-1}(x)) = x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Graphical relationship: Symmetric about [latex]y = x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>One-to-One Functions:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Definition: Each output corresponds to a unique input<\/li>\n<li class=\"whitespace-normal break-words\">Test: Horizontal line test<\/li>\n<li class=\"whitespace-normal break-words\">Importance: Only one-to-one functions have inverses that are functions<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Types of Radical Functions<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Square Root Function:<\/strong> [latex]f(x) = \\sqrt{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Cube Root Function:<\/strong> [latex]f(x) = \\sqrt[3]{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Higher-Order Root Functions:<\/strong> [latex]f(x) = \\sqrt[n]{x}[\/latex], where [latex]n \\geq 3[\/latex]<\/li>\n<\/ol>\n<p class=\"font-600 text-lg font-bold\"><strong>Steps to Finding Inverse Functions:<\/strong><\/p>\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 function [latex]f^{-1}(x)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Show that [latex]f\\left(x\\right)=\\dfrac{1}{x+1}[\/latex] and [latex]{f}^{-1}\\left(x\\right)=\\dfrac{1}{x}-1[\/latex] are inverses, for [latex]x\\ne 0,-1[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q631376\">Show Solution<\/button><\/p>\n<div id=\"q631376\" class=\"hidden-answer\" style=\"display: none\">We must show that [latex]{f}^{-1}\\left(f\\left(x\\right)\\right)=x[\/latex] and [latex]f\\left({f}^{-1}\\left(x\\right)\\right)=x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{f}^{-1}\\left(f\\left(x\\right)\\right)&={f}^{-1}\\left(\\frac{1}{x+1}\\right) \\\\[1mm] &=\\dfrac{1}{\\frac{1}{x+1}}-1 \\\\[1mm] &=\\left(x+1\\right)-1 \\\\[1mm] &=x \\\\[5mm] f\\left({f}^{-1}\\left(x\\right)\\right)&=f\\left(\\frac{1}{x}-1\\right) \\\\[1mm] &=\\dfrac{1}{\\left(\\frac{1}{x}-1\\right)+1} \\\\[1mm] &=\\dfrac{1}{\\frac{1}{x}} \\\\[1mm] &=x \\end{align}[\/latex]<\/p>\n<p>Therefore, [latex]f\\left(x\\right)=\\dfrac{1}{x+1}[\/latex]\u00a0and [latex]{f}^{-1}\\left(x\\right)=\\dfrac{1}{x}-1[\/latex] are inverses.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the inverse function of [latex]f\\left(x\\right)=\\sqrt[3]{x+4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q987592\">Show Solution<\/button><\/p>\n<div id=\"q987592\" class=\"hidden-answer\" style=\"display: none\">[latex]{f}^{-1}\\left(x\\right)={x}^{3}-4[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Domains of Radical 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\">The domain of a radical function depends on the index [latex]n[\/latex] of the root:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For even roots (e.g., square roots), the expression inside the radical must be non-negative.<\/li>\n<li class=\"whitespace-normal break-words\">For odd roots (e.g., cube roots), the expression inside the radical can be any real number.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">To find the domain of a radical function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set up an inequality ensuring the radicand is non-negative (for even roots).<\/li>\n<li class=\"whitespace-normal break-words\">Identify critical points where the expression could change sign.<\/li>\n<li class=\"whitespace-normal break-words\">Test intervals and determine where the inequality holds.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Not all functions are one-to-one. To find an inverse of a non-one-to-one function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Restrict the domain to make the function one-to-one.<\/li>\n<li class=\"whitespace-normal break-words\">Find the inverse on this restricted domain.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Steps to find the inverse of a function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\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 function [latex]f^{-1}(x)[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Ensure the domain of [latex]f^{-1}(x)[\/latex] corresponds to the range of [latex]f(x)[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For radical functions, when finding the inverse:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determine the range of the original function.<\/li>\n<li class=\"whitespace-normal break-words\">This range becomes the domain restriction for the inverse function.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphically, inverse functions are reflections of each other over the line [latex]y = x[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">The domain of [latex]f(x)[\/latex] becomes the range of [latex]f^{-1}(x)[\/latex], and vice versa.<\/li>\n<li class=\"whitespace-normal break-words\">When graphing radical functions and their inverses:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Points of intersection will always lie on the line [latex]y = x[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">If [latex](a, b)[\/latex] is on the graph of [latex]f(x)[\/latex], then [latex](b, a)[\/latex] is on the graph of [latex]f^{-1}(x)[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the inverse of the function [latex]f\\left(x\\right)={x}^{2}+1[\/latex], on the domain [latex]x\\ge 0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q758215\">Show Solution<\/button><\/p>\n<div id=\"q758215\" class=\"hidden-answer\" style=\"display: none\">[latex]{f}^{-1}\\left(x\\right)=\\sqrt{x - 1}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video to see more examples of how to restrict the domain of a quadratic function to find it&#8217;s inverse.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-agbggced-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-agbggced-rsJ14O5-KDw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844444&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-agbggced-rsJ14O5-KDw&#38;vembed=0&#38;video_id=rsJ14O5-KDw&#38;video_target=tpm-plugin-agbggced-rsJ14O5-KDw\"><\/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","protected":false},"author":12,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"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\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":232,"module-header":"fresh_take","content_attributions":[{"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"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844444&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-agbggced-rsJ14O5-KDw&vembed=0&video_id=rsJ14O5-KDw&video_target=tpm-plugin-agbggced-rsJ14O5-KDw'><\/script>\n","media_targets":["tpm-plugin-agbggced-rsJ14O5-KDw"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2045"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2045\/revisions"}],"predecessor-version":[{"id":7275,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2045\/revisions\/7275"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/232"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2045\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2045"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2045"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2045"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2045"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}