{"id":130,"date":"2023-06-21T13:22:36","date_gmt":"2023-06-21T13:22:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/find-domain-and-range-from-a-graph\/"},"modified":"2023-08-21T22:03:24","modified_gmt":"2023-08-21T22:03:24","slug":"find-domain-and-range-from-a-graph","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/find-domain-and-range-from-a-graph\/","title":{"raw":"\u25aa   Determine Domain and Range from a Graph","rendered":"\u25aa   Determine Domain and Range from a Graph"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find domain and range from a graph, and an equation.<\/li>\r\n \t<li>Give the domain and range of the toolkit functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAnother way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[\/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[\/latex]-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193549\/CNX_Precalc_Figure_01_02_0062.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\" \/>\r\n\r\nWe can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Domain and Range from a Graph<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193551\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/>\r\n[reveal-answer q=\"495787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"495787\"]\r\n\r\nWe can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193553\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/>\r\n\r\nThe vertical extent of the graph is 0 to [latex]\u20134[\/latex], so the range is [latex]\\left[-4,0\\right][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=QAxZEelInJc\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Domain and Range from a Graph of Oil Production<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"489\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193556\/CNX_Precalc_Figure_01_02_0092.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\" \/> (credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)[\/caption]\r\n\r\n[reveal-answer q=\"834467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834467\"]\r\n\r\nThe input quantity along the horizontal axis is \"years,\" which we represent with the variable [latex]t[\/latex] for time. The output quantity is \"thousands of barrels of oil per day,\" which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010[\/latex].\r\n\r\nIn interval notation, the domain is [latex][1973, 2008][\/latex], and the range is about [latex][180, 2010][\/latex]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the graph, identify the domain and range using interval notation.\r\n\r\n<img class=\"aligncenter\" 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\" \/>\r\n[reveal-answer q=\"186149\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186149\"]\r\n\r\nDomain = [latex][1950, 2002][\/latex]\u00a0 \u00a0Range = [latex][47,000,000, 89,000,000][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"510\"]3765[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can a function\u2019s domain and range be the same?<\/strong>\r\n\r\n<em>Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em>\r\n\r\n<\/div>\r\n<h2>Domain and Range of Toolkit Functions<\/h2>\r\nWe will now return to our set of toolkit functions to determine the domain and range of each.\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\/18193601\/CNX_Precalc_Figure_01_02_0112.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/> For the <strong>constant function<\/strong> [latex]f\\left(x\\right)=c[\/latex], the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c[\/latex], so the range is the set [latex]\\left\\{c\\right\\}[\/latex] that contains this single element. In interval notation, this is written as [latex]\\left[c,c\\right][\/latex], the interval that both begins and ends with [latex]c[\/latex].[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193604\/CNX_Precalc_Figure_01_02_0122.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/> For the <strong>identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193606\/CNX_Precalc_Figure_01_02_0132.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/> For the <strong>absolute value function<\/strong> [latex]f\\left(x\\right)=|x|[\/latex], there is no restriction on [latex]x[\/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193609\/CNX_Precalc_Figure_01_02_0142.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/> For the <strong>quadratic function<\/strong> [latex]f\\left(x\\right)={x}^{2}[\/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193611\/CNX_Precalc_Figure_01_02_0152.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/> For the <strong>cubic function<\/strong> [latex]f\\left(x\\right)={x}^{3}[\/latex], the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193614\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/> For the <strong>reciprocal function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193617\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/> For the <strong>reciprocal squared function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193620\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/> For the <strong>square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex].[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193622\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/> For the <strong>cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).[\/caption]\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nWhen evaluating functions to plot points, remember to wrap the variable in parentheses to be sure you handle negative input appropriately.\r\n\r\nEx. For the function [latex]f(x) = x^2[\/latex], evaluate [latex]f(-3)[\/latex].\r\n\r\n[reveal-answer q=\"257996\"]more[\/reveal-answer]\r\n[hidden-answer a=\"257996\"]\r\n\r\n[latex]\\begin{align}f(-3) &amp;= (-3)^2 \\\\ &amp;= 9\\end{align}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"660\"]47481[\/ohm_question]\r\n\r\n[ohm_question height=\"460\"]47483[\/ohm_question]\r\n\r\n[ohm_question height=\"460\"]47484[\/ohm_question]\r\n\r\n[ohm_question height=\"440\"]47487[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find domain and range from a graph, and an equation.<\/li>\n<li>Give the domain and range of the toolkit functions.<\/li>\n<\/ul>\n<\/div>\n<p>Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the [latex]x[\/latex]-axis. The range is the set of possible output values, which are shown on the [latex]y[\/latex]-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193549\/CNX_Precalc_Figure_01_02_0062.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\" \/><\/p>\n<p>We can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Domain and Range from a Graph<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193551\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q495787\">Show Solution<\/span><\/p>\n<div id=\"q495787\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193553\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/><\/p>\n<p>The vertical extent of the graph is 0 to [latex]\u20134[\/latex], so the range is [latex]\\left[-4,0\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Determine the Domain and Range of the Graph of a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Domain and Range from a Graph of Oil Production<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex].<\/p>\n<div style=\"width: 499px\" 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\/18193556\/CNX_Precalc_Figure_01_02_0092.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\" \/><\/p>\n<p class=\"wp-caption-text\">(credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834467\">Show Solution<\/span><\/p>\n<div id=\"q834467\" class=\"hidden-answer\" style=\"display: none\">\n<p>The input quantity along the horizontal axis is &#8220;years,&#8221; which we represent with the variable [latex]t[\/latex] for time. The output quantity is &#8220;thousands of barrels of oil per day,&#8221; which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010[\/latex].<\/p>\n<p>In interval notation, the domain is [latex][1973, 2008][\/latex], and the range is about [latex][180, 2010][\/latex]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the graph, identify the domain and range using interval notation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186149\">Show Solution<\/span><\/p>\n<div id=\"q186149\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex][1950, 2002][\/latex]\u00a0 \u00a0Range = [latex][47,000,000, 89,000,000][\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3765\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3765&theme=oea&iframe_resize_id=ohm3765&show_question_numbers\" width=\"100%\" height=\"510\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can a function\u2019s domain and range be the same?<\/strong><\/p>\n<p><em>Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em><\/p>\n<\/div>\n<h2>Domain and Range of Toolkit Functions<\/h2>\n<p>We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\n<div style=\"width: 497px\" 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\/18193601\/CNX_Precalc_Figure_01_02_0112.jpg\" alt=\"Constant function f(x)=c.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>constant function<\/strong> [latex]f\\left(x\\right)=c[\/latex], the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c[\/latex], so the range is the set [latex]\\left\\{c\\right\\}[\/latex] that contains this single element. In interval notation, this is written as [latex]\\left[c,c\\right][\/latex], the interval that both begins and ends with [latex]c[\/latex].<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193604\/CNX_Precalc_Figure_01_02_0122.jpg\" alt=\"Identity function f(x)=x.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193606\/CNX_Precalc_Figure_01_02_0132.jpg\" alt=\"Absolute function f(x)=|x|.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>absolute value function<\/strong> [latex]f\\left(x\\right)=|x|[\/latex], there is no restriction on [latex]x[\/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193609\/CNX_Precalc_Figure_01_02_0142.jpg\" alt=\"Quadratic function f(x)=x^2.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>quadratic function<\/strong> [latex]f\\left(x\\right)={x}^{2}[\/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193611\/CNX_Precalc_Figure_01_02_0152.jpg\" alt=\"Cubic function f(x)-x^3.\" width=\"487\" height=\"436\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>cubic function<\/strong> [latex]f\\left(x\\right)={x}^{3}[\/latex], the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193614\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>reciprocal function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193617\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>reciprocal squared function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193620\/CNX_Precalc_Figure_01_02_0182.jpg\" alt=\"Square root function f(x)=sqrt(x).\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex].<\/p>\n<\/div>\n<div style=\"width: 497px\" 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\/18193622\/CNX_Precalc_Figure_01_02_0192.jpg\" alt=\"Cube root function f(x)=x^(1\/3).\" width=\"487\" height=\"433\" \/><\/p>\n<p class=\"wp-caption-text\">For the <strong>cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>When evaluating functions to plot points, remember to wrap the variable in parentheses to be sure you handle negative input appropriately.<\/p>\n<p>Ex. For the function [latex]f(x) = x^2[\/latex], evaluate [latex]f(-3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q257996\">more<\/span><\/p>\n<div id=\"q257996\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}f(-3) &= (-3)^2 \\\\ &= 9\\end{align}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm47481\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47481&theme=oea&iframe_resize_id=ohm47481&show_question_numbers\" width=\"100%\" height=\"660\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm47483\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47483&theme=oea&iframe_resize_id=ohm47483&show_question_numbers\" width=\"100%\" height=\"460\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm47484\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47484&theme=oea&iframe_resize_id=ohm47484&show_question_numbers\" width=\"100%\" height=\"460\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm47487\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=47487&theme=oea&iframe_resize_id=ohm47487&show_question_numbers\" width=\"100%\" height=\"440\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-130\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 47487. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 3765. <strong>Authored by<\/strong>: Jenck,Michael. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 47481, 47483, 47484. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 3765\",\"author\":\"Jenck,Michael\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 47481, 47483, 47484\",\"author\":\"Day, Alyson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Question ID 47487\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"417bef86-be86-4ce4-af0a-71660837a3f2","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-130","chapter","type-chapter","status-publish","hentry"],"part":115,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/130\/revisions"}],"predecessor-version":[{"id":1368,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/130\/revisions\/1368"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/115"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/130\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=130"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=130"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=130"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}