{"id":587,"date":"2025-07-11T20:33:02","date_gmt":"2025-07-11T20:33:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=587"},"modified":"2026-03-18T02:36:47","modified_gmt":"2026-03-18T02:36:47","slug":"domain-and-range-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/domain-and-range-fresh-take\/","title":{"raw":"Domain and Range: Fresh Take","rendered":"Domain and Range: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the domain of a function by looking at its equation<\/li>\r\n \t<li>Find the domain of a function by look at its graph<\/li>\r\n \t<li>Sketch graphs of piecewise functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Domain and Range<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Domain:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set of all possible input values (x-values)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Often represented using interval notation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set of all possible output values (y-values)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determined by the function's behavior<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interval Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Uses brackets [ ] for inclusive endpoints<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Uses parentheses ( ) for exclusive endpoints<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex](0, 100][\/latex] means more than 0 and less than or equal to 100<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain Restrictions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Denominators: Exclude values making denominator zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Even roots: Exclude values making radicand negative<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider function's context (e.g., real-world limitations)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range Analysis:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Examine function behavior<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider limitations on output values<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/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\">Finding Domain:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For simple functions: Consider all real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For fractions: Set denominator to zero, solve for x, exclude those values<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For even roots: Set radicand \u2265 0, solve for x<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Finding Range:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Analyze function behavior<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider minimum\/maximum possible outputs<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use graphing or tables to confirm<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Special Cases:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Polynomials: Usually all real numbers for domain and range<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rational functions: Exclude zeros in denominator for domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Root functions: Ensure non-negative radicand for domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Find the domain of the function: [latex]f(x)=5-x+{x}^{3}[\/latex].[reveal-answer q=\"237099\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"237099\"][latex](-\\infty ,\\infty )[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\">Watch the following video to see more examples of how to find the domain of a rational function (one with a fraction).\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fahdaceg-v0IhvIzCc_I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fahdaceg-v0IhvIzCc_I\" 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=6454925&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fahdaceg-v0IhvIzCc_I&amp;vembed=0&amp;video_id=v0IhvIzCc_I&amp;video_target=tpm-plugin-fahdaceg-v0IhvIzCc_I\" 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+-+The+Domain+of+Rational+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: The Domain of Rational Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\">Find the domain of the function [latex]f(x)=\\sqrt{5+2x}[\/latex].[reveal-answer q=\"643325\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"643325\"]\r\n[latex][-\\frac{5}{2},\\infty )[\/latex]\r\n[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]92940[\/ohm_question]<\/section><section class=\"textbox watchIt\">The next video gives more examples of how to define the domain of a function that contains an even root.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-baecechb-lj_JB8sfyIM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/lj_JB8sfyIM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-baecechb-lj_JB8sfyIM\" 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=6454926&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-baecechb-lj_JB8sfyIM&amp;vembed=0&amp;video_id=lj_JB8sfyIM&amp;video_target=tpm-plugin-baecechb-lj_JB8sfyIM\" 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+-+Domain+and+Range+of+Square+Root+Functions_transcript.txt\" rel=\"noopener\">transcript for \u201cEx: Domain and Range of Square Root Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\">Find the domain and range of [latex]f(x)=\\dfrac{2}{x+1}[\/latex].[reveal-answer q=\"71516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"71516\"]We cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex](-\\infty ,-1)\\cup (-1,\\infty )[\/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex](-\\infty ,0)\\cup (0,\\infty )[\/latex].[\/hidden-answer]<\/section><section class=\"textbox example\">Find the domain and range of [latex]f(x)=-\\sqrt{2-x}[\/latex].[reveal-answer q=\"336525\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"336525\"]Domain: [latex](-\\infty ,2][\/latex] \u00a0 Range: [latex](-\\infty ,0][\/latex][\/hidden-answer]<\/section>\r\n<h2>Determine Domain and Range from a Graph<\/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\">Visual Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All input values (x-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: All output values (y-axis)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph Extent:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal extent determines domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical extent determines range<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interval Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Write domain and range from left to right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use appropriate brackets\/parentheses<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graph Limitations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Consider unseen portions of the graph<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Be aware of potential continuations beyond visible area<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/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\">Analyzing Continuous Graphs:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify leftmost and rightmost x-values for domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify lowest and highest y-values for range<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyzing Discrete Graphs:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Consider individual points for domain and range<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pay attention to gaps or jumps in the graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Interpreting Asymptotes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal asymptotes affect range<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptotes affect domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reading Scales:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Note axis labels and units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Estimate values between gridlines when necessary<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Given the graph, identify the domain and range using interval notation.\r\n\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\/18193558\/CNX_Precalc_Figure_01_02_0102.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\" \/> Graph of World Population Increase[\/caption]\r\n\r\n[reveal-answer q=\"186149\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186149\"]Domain = [latex][1950, 2002][\/latex]\u00a0 \u00a0Range = [latex][47,000,000, 89,000,000][\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hfcdfbce-QAxZEelInJc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hfcdfbce-QAxZEelInJc\" 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=12844420&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hfcdfbce-QAxZEelInJc&amp;vembed=0&amp;video_id=QAxZEelInJc&amp;video_target=tpm-plugin-hfcdfbce-QAxZEelInJc\" 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+1+-+Determine+the+Domain+and+Range+of+the+Graph+of+a+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1 - Determine the Domain and Range of the Graph of a Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Domain and Range of Toolkit 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\">Constant Function: [latex]f(x) = c[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex]{c}[\/latex] or [latex][c, c][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity Function: [latex]f(x) = x[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: All real numbers<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Value Function: [latex]f(x) = |x|[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex][0, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic Function: [latex]f(x) = x^2[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex][0, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cubic Function: [latex]f(x) = x^3[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: All real numbers<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reciprocal Function: [latex]f(x) = \\frac{1}{x}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers except 0<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: All real numbers except 0<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reciprocal Squared Function: [latex]f(x) = \\frac{1}{x^2}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers except 0<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex](0, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Square Root Function: [latex]f(x) = \\sqrt{x}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex][0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex][0, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cube Root Function: [latex]f(x) = \\sqrt[3]{x}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: All real numbers<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/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\">Analyzing Domain:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Consider restrictions (e.g., division by zero, even roots of negative numbers)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify any x-values that produce undefined results<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyzing Range:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Consider the function's behavior for all valid inputs<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify any y-values that cannot be achieved<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using Interval Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use square brackets [ ] for inclusive endpoints<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use parentheses ( ) for exclusive endpoints<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use infinity symbols when there's no upper or lower bound<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Analysis:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Visualize the function to confirm domain and range<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pay attention to asymptotes and end behavior<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Find the domain and range of [latex]f(x)=2x^3-x[\/latex].[reveal-answer q=\"427873\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"427873\"]There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.The domain is [latex](-\\infty , \\infty )[\/latex] and the range is also [latex](-\\infty , \\infty )[\/latex].[\/hidden-answer]<\/section><section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bcehdadh-xOqzcRqr7Os\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/xOqzcRqr7Os?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bcehdadh-xOqzcRqr7Os\" 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=12779139&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bcehdadh-xOqzcRqr7Os&amp;vembed=0&amp;video_id=xOqzcRqr7Os&amp;video_target=tpm-plugin-bcehdadh-xOqzcRqr7Os\" 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\/1.2.h+Domain+and+Range+of+Toolkit+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c1.2.h Domain and Range of Toolkit Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Piecewise-Defined 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\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Functions defined by different formulas over different parts of the domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Notation uses curly braces and if-statements<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Absolute Value Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Classic example of a piecewise function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]|x| = \\begin{cases} x &amp; \\text{if } x \\geq 0 \\ -x &amp; \\text{if } x &lt; 0 \\end{cases}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">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 is the union of all piece domains<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range depends on the specific functions used<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphing:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Combine graphs of individual pieces<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pay attention to endpoints and continuity<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/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\">Writing Piecewise Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify intervals for different rules<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine formulas for each interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use proper notation with curly braces<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluating Piecewise Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determine which piece applies to the input<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the corresponding formula<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphing Piecewise Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Graph each piece on its interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use open\/closed circles for endpoints<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ensure the function passes the vertical line test<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]93008[\/ohm_question]<\/section><section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-achgddge-B1jfpiI-QQ8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/B1jfpiI-QQ8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-achgddge-B1jfpiI-QQ8\" 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=12844421&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-achgddge-B1jfpiI-QQ8&amp;vembed=0&amp;video_id=B1jfpiI-QQ8&amp;video_target=tpm-plugin-achgddge-B1jfpiI-QQ8\" 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+2+-+Graph+a+Piecewise+Defined+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Graph a Piecewise Defined Function\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>Find the domain of a function by looking at its equation<\/li>\n<li>Find the domain of a function by look at its graph<\/li>\n<li>Sketch graphs of piecewise functions<\/li>\n<\/ul>\n<\/section>\n<h2>Domain and Range<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Domain:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set of all possible input values (x-values)<\/li>\n<li class=\"whitespace-normal break-words\">Often represented using interval notation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set of all possible output values (y-values)<\/li>\n<li class=\"whitespace-normal break-words\">Determined by the function&#8217;s behavior<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interval Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Uses brackets [ ] for inclusive endpoints<\/li>\n<li class=\"whitespace-normal break-words\">Uses parentheses ( ) for exclusive endpoints<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex](0, 100][\/latex] means more than 0 and less than or equal to 100<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain Restrictions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Denominators: Exclude values making denominator zero<\/li>\n<li class=\"whitespace-normal break-words\">Even roots: Exclude values making radicand negative<\/li>\n<li class=\"whitespace-normal break-words\">Consider function&#8217;s context (e.g., real-world limitations)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Range Analysis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Examine function behavior<\/li>\n<li class=\"whitespace-normal break-words\">Consider limitations on output values<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Finding Domain:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For simple functions: Consider all real numbers<\/li>\n<li class=\"whitespace-normal break-words\">For fractions: Set denominator to zero, solve for x, exclude those values<\/li>\n<li class=\"whitespace-normal break-words\">For even roots: Set radicand \u2265 0, solve for x<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Finding Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Analyze function behavior<\/li>\n<li class=\"whitespace-normal break-words\">Consider minimum\/maximum possible outputs<\/li>\n<li class=\"whitespace-normal break-words\">Use graphing or tables to confirm<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Special Cases:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Polynomials: Usually all real numbers for domain and range<\/li>\n<li class=\"whitespace-normal break-words\">Rational functions: Exclude zeros in denominator for domain<\/li>\n<li class=\"whitespace-normal break-words\">Root functions: Ensure non-negative radicand for domain<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Find the domain of the function: [latex]f(x)=5-x+{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q237099\">Show Solution<\/button><\/p>\n<div id=\"q237099\" class=\"hidden-answer\" style=\"display: none\">[latex](-\\infty ,\\infty )[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">Watch the following video to see more examples of how to find the domain of a rational function (one with a fraction).<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fahdaceg-v0IhvIzCc_I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fahdaceg-v0IhvIzCc_I\" 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=6454925&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fahdaceg-v0IhvIzCc_I&amp;vembed=0&amp;video_id=v0IhvIzCc_I&amp;video_target=tpm-plugin-fahdaceg-v0IhvIzCc_I\" 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+-+The+Domain+of+Rational+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: The Domain of Rational Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Find the domain of the function [latex]f(x)=\\sqrt{5+2x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q643325\">Show Solution<\/button><\/p>\n<div id=\"q643325\" class=\"hidden-answer\" style=\"display: none\">\n[latex][-\\frac{5}{2},\\infty )[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm92940\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92940&theme=lumen&iframe_resize_id=ohm92940&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\">The next video gives more examples of how to define the domain of a function that contains an even root.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-baecechb-lj_JB8sfyIM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/lj_JB8sfyIM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<div id=\"3p-plugin-target-baecechb-lj_JB8sfyIM\" 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=6454926&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-baecechb-lj_JB8sfyIM&amp;vembed=0&amp;video_id=lj_JB8sfyIM&amp;video_target=tpm-plugin-baecechb-lj_JB8sfyIM\" 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+-+Domain+and+Range+of+Square+Root+Functions_transcript.txt\" rel=\"noopener\">transcript for \u201cEx: Domain and Range of Square Root Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Find the domain and range of [latex]f(x)=\\dfrac{2}{x+1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q71516\">Show Solution<\/button><\/p>\n<div id=\"q71516\" class=\"hidden-answer\" style=\"display: none\">We cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex](-\\infty ,-1)\\cup (-1,\\infty )[\/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex](-\\infty ,0)\\cup (0,\\infty )[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Find the domain and range of [latex]f(x)=-\\sqrt{2-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q336525\">Show Solution<\/button><\/p>\n<div id=\"q336525\" class=\"hidden-answer\" style=\"display: none\">Domain: [latex](-\\infty ,2][\/latex] \u00a0 Range: [latex](-\\infty ,0][\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Determine Domain and Range from a Graph<\/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\">Visual Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All input values (x-axis)<\/li>\n<li class=\"whitespace-normal break-words\">Range: All output values (y-axis)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graph Extent:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal extent determines domain<\/li>\n<li class=\"whitespace-normal break-words\">Vertical extent determines range<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interval Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Write domain and range from left to right<\/li>\n<li class=\"whitespace-normal break-words\">Use appropriate brackets\/parentheses<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graph Limitations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Consider unseen portions of the graph<\/li>\n<li class=\"whitespace-normal break-words\">Be aware of potential continuations beyond visible area<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Analyzing Continuous Graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify leftmost and rightmost x-values for domain<\/li>\n<li class=\"whitespace-normal break-words\">Identify lowest and highest y-values for range<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Analyzing Discrete Graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Consider individual points for domain and range<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to gaps or jumps in the graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpreting Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal asymptotes affect range<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptotes affect domain<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reading Scales:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Note axis labels and units<\/li>\n<li class=\"whitespace-normal break-words\">Estimate values between gridlines when necessary<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Given the graph, identify the domain and range using interval notation.<\/p>\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\/18193558\/CNX_Precalc_Figure_01_02_0102.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\" \/><figcaption class=\"wp-caption-text\">Graph of World Population Increase<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q186149\">Show Solution<\/button><\/p>\n<div id=\"q186149\" class=\"hidden-answer\" style=\"display: none\">Domain = [latex][1950, 2002][\/latex]\u00a0 \u00a0Range = [latex][47,000,000, 89,000,000][\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hfcdfbce-QAxZEelInJc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hfcdfbce-QAxZEelInJc\" 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=12844420&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hfcdfbce-QAxZEelInJc&amp;vembed=0&amp;video_id=QAxZEelInJc&amp;video_target=tpm-plugin-hfcdfbce-QAxZEelInJc\" 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+1+-+Determine+the+Domain+and+Range+of+the+Graph+of+a+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1 &#8211; Determine the Domain and Range of the Graph of a Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Domain and Range of Toolkit 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\">Constant Function: [latex]f(x) = c[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex]{c}[\/latex] or [latex][c, c][\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identity Function: [latex]f(x) = x[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Range: All real numbers<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Value Function: [latex]f(x) = |x|[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex][0, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic Function: [latex]f(x) = x^2[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex][0, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Cubic Function: [latex]f(x) = x^3[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Range: All real numbers<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reciprocal Function: [latex]f(x) = \\frac{1}{x}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers except 0<\/li>\n<li class=\"whitespace-normal break-words\">Range: All real numbers except 0<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reciprocal Squared Function: [latex]f(x) = \\frac{1}{x^2}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers except 0<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex](0, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Square Root Function: [latex]f(x) = \\sqrt{x}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex][0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex][0, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Cube Root Function: [latex]f(x) = \\sqrt[3]{x}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Range: All real numbers<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Analyzing Domain:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Consider restrictions (e.g., division by zero, even roots of negative numbers)<\/li>\n<li class=\"whitespace-normal break-words\">Identify any x-values that produce undefined results<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Analyzing Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Consider the function&#8217;s behavior for all valid inputs<\/li>\n<li class=\"whitespace-normal break-words\">Identify any y-values that cannot be achieved<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Using Interval Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use square brackets [ ] for inclusive endpoints<\/li>\n<li class=\"whitespace-normal break-words\">Use parentheses ( ) for exclusive endpoints<\/li>\n<li class=\"whitespace-normal break-words\">Use infinity symbols when there&#8217;s no upper or lower bound<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Analysis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Visualize the function to confirm domain and range<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to asymptotes and end behavior<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Find the domain and range of [latex]f(x)=2x^3-x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q427873\">Show Answer<\/button><\/p>\n<div id=\"q427873\" class=\"hidden-answer\" style=\"display: none\">There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.The domain is [latex](-\\infty , \\infty )[\/latex] and the range is also [latex](-\\infty , \\infty )[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bcehdadh-xOqzcRqr7Os\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/xOqzcRqr7Os?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bcehdadh-xOqzcRqr7Os\" 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=12779139&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bcehdadh-xOqzcRqr7Os&amp;vembed=0&amp;video_id=xOqzcRqr7Os&amp;video_target=tpm-plugin-bcehdadh-xOqzcRqr7Os\" 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\/1.2.h+Domain+and+Range+of+Toolkit+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c1.2.h Domain and Range of Toolkit Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Piecewise-Defined 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\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Functions defined by different formulas over different parts of the domain<\/li>\n<li class=\"whitespace-normal break-words\">Notation uses curly braces and if-statements<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Value Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Classic example of a piecewise function<\/li>\n<li class=\"whitespace-normal break-words\">[latex]|x| = \\begin{cases} x & \\text{if } x \\geq 0 \\ -x & \\text{if } x < 0 \\end{cases}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain and Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain is the union of all piece domains<\/li>\n<li class=\"whitespace-normal break-words\">Range depends on the specific functions used<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphing:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combine graphs of individual pieces<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to endpoints and continuity<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Writing Piecewise Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify intervals for different rules<\/li>\n<li class=\"whitespace-normal break-words\">Determine formulas for each interval<\/li>\n<li class=\"whitespace-normal break-words\">Use proper notation with curly braces<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluating Piecewise Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determine which piece applies to the input<\/li>\n<li class=\"whitespace-normal break-words\">Use the corresponding formula<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphing Piecewise Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Graph each piece on its interval<\/li>\n<li class=\"whitespace-normal break-words\">Use open\/closed circles for endpoints<\/li>\n<li class=\"whitespace-normal break-words\">Ensure the function passes the vertical line test<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm93008\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93008&theme=lumen&iframe_resize_id=ohm93008&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox watchIt\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-achgddge-B1jfpiI-QQ8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/B1jfpiI-QQ8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-achgddge-B1jfpiI-QQ8\" 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=12844421&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-achgddge-B1jfpiI-QQ8&amp;vembed=0&amp;video_id=B1jfpiI-QQ8&amp;video_target=tpm-plugin-achgddge-B1jfpiI-QQ8\" 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+2+-+Graph+a+Piecewise+Defined+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Graph a Piecewise Defined Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":13,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: The Domain of Rational Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/v0IhvIzCc_I\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Domain and Range of Square Root Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/lj_JB8sfyIM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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