{"id":16631,"date":"2019-10-03T20:16:56","date_gmt":"2019-10-03T20:16:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/determining-domain-and-range\/"},"modified":"2024-05-02T15:55:09","modified_gmt":"2024-05-02T15:55:09","slug":"determining-domain-and-range","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/determining-domain-and-range\/","title":{"raw":"Finding Domain and Range From a Graph","rendered":"Finding Domain and Range From a Graph"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Find the domain and range of a function from its graph<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe have learned that the set of input values for a function is called the\u00a0<b>domain of the function<\/b> and the set of output values is called the\u00a0<b>range of the function<\/b>.\u00a0 We have also learned how to algebraically determine the domain of a function.\u00a0 It is important to be aware that the domain and range of a function are not necessarily the same. For example, the function [latex]f\\left(x\\right)=-\\dfrac{1}{\\sqrt{x}}[\/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.\r\n\r\nFinding domain and range of different functions is often a matter of asking yourself, what values can this function <i>not<\/i>\u00a0have? Pictures make it easier to visualize what the domain and range are, so we will show how to define the domain and range of functions given their graphs.\r\n\r\nWhat are the domain and range of the real-valued function [latex]f(x)=x+3[\/latex]?\r\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Line for f of x equals x plus 3 plotted in the xy coordinate plane. The x axis extends from negative 6 to 8, and the y axis extends from negative 4 to 10. \" width=\"322\" height=\"353\" \/>\r\n\r\nAny real number can be substituted for <i>x<\/i> and get a meaningful output. For <i>any<\/i> real number, you can always find an <i>x<\/i> value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range.\r\nThe domain and range are all real numbers.\r\n\r\nFor the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat are the domain and range of the real-valued function [latex]f(x)=\u22123x^{2}+6x+1[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/>\r\n\r\n[reveal-answer q=\"223692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223692\"]\r\n\r\nThis is a <i>quadratic <\/i>function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing to restrict from the domain. Any real number can be used for <i>x<\/i> to get a meaningful output.\r\n\r\nBecause the coefficient of [latex]x^{2}[\/latex] is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value or a minimum (least) value. In this case, there is a maximum value.\r\n\r\nThe vertex, or high\u00a0point, is at ([latex]1, 4[\/latex]). From the graph, you can see that [latex]f(x)\\leq4[\/latex].\r\n\r\nThe domain is all real numbers, and the range is all real numbers [latex]f(x)[\/latex] such that [latex]f(x)\\leq4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat is the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Radical function stemming from negative 5, negative 2.\" width=\"308\" height=\"346\" \/>\r\n\r\n[reveal-answer q=\"231228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231228\"]\r\n\r\nThis is a <i>radical <\/i>function. The domain of a radical function is any <i>x<\/i> value for which the radicand (the value under the radical sign) is not negative. That means [latex]x+5\\geq0[\/latex], so [latex]x\\geq\u22125[\/latex].\r\n\r\nSince the square root must always be positive or\u00a0[latex]0[\/latex], [latex] \\displaystyle \\sqrt{x+5}\\ge 0[\/latex]. That means [latex] \\displaystyle -2+\\sqrt{x+5}\\ge -2[\/latex].\r\n\r\nThe domain is all real numbers <i>x<\/i> where [latex]x\\geq\u22125[\/latex], and the range is all real numbers [latex]f(x)[\/latex] such that [latex]f(x)\\geq\u22122[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat is the domain of the real-valued function [latex] \\displaystyle f(x)=\\frac{3x}{x+2}[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232531\/image049.gif\" alt=\"A graph of the function f of x equals 3 x divided by the quantity x plus 2. Both axes extend from negative 12 to 12.\" width=\"310\" height=\"321\" \/>\r\n\r\n[reveal-answer q=\"666335\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"666335\"]\r\n\r\nThis is a <i>rational <\/i>function. The domain of a rational function is restricted where the denominator is\u00a0[latex]0[\/latex]. In this case, [latex]x+2[\/latex] is the denominator, and this is\u00a0[latex]0[\/latex] only when [latex]x=\u22122[\/latex].\r\n\r\nThe domain is all real numbers except [latex]\u22122[\/latex]\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show how to define the domain and range of\u00a0functions from their graphs.\r\n\r\nhttps:\/\/youtu.be\/QAxZEelInJc\r\n<h2>Summary<\/h2>\r\nAlthough a function may be given as \u201creal valued,\u201d it may be that the function has restrictions to its domain and range. There may be some real numbers that cannot be part of the domain or part of the range. This is particularly true with rational and radical functions which can have restrictions to their domain, range, or both. Other functions, such as quadratic functions and polynomial functions of even degree, can also have restrictions to their range.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Find the domain and range of a function from its graph<\/li>\n<\/ul>\n<\/div>\n<p>We have learned that the set of input values for a function is called the\u00a0<b>domain of the function<\/b> and the set of output values is called the\u00a0<b>range of the function<\/b>.\u00a0 We have also learned how to algebraically determine the domain of a function.\u00a0 It is important to be aware that the domain and range of a function are not necessarily the same. For example, the function [latex]f\\left(x\\right)=-\\dfrac{1}{\\sqrt{x}}[\/latex] has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function\u2019s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/p>\n<p>Finding domain and range of different functions is often a matter of asking yourself, what values can this function <i>not<\/i>\u00a0have? Pictures make it easier to visualize what the domain and range are, so we will show how to define the domain and range of functions given their graphs.<\/p>\n<p>What are the domain and range of the real-valued function [latex]f(x)=x+3[\/latex]?<br \/>\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Line for f of x equals x plus 3 plotted in the xy coordinate plane. The x axis extends from negative 6 to 8, and the y axis extends from negative 4 to 10.\" width=\"322\" height=\"353\" \/><\/p>\n<p>Any real number can be substituted for <i>x<\/i> and get a meaningful output. For <i>any<\/i> real number, you can always find an <i>x<\/i> value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range.<br \/>\nThe domain and range are all real numbers.<\/p>\n<p>For the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What are the domain and range of the real-valued function [latex]f(x)=\u22123x^{2}+6x+1[\/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\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223692\">Show Solution<\/span><\/p>\n<div id=\"q223692\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a <i>quadratic <\/i>function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing to restrict from the domain. Any real number can be used for <i>x<\/i> to get a meaningful output.<\/p>\n<p>Because the coefficient of [latex]x^{2}[\/latex] is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value or a minimum (least) value. In this case, there is a maximum value.<\/p>\n<p>The vertex, or high\u00a0point, is at ([latex]1, 4[\/latex]). From the graph, you can see that [latex]f(x)\\leq4[\/latex].<\/p>\n<p>The domain is all real numbers, and the range is all real numbers [latex]f(x)[\/latex] such that [latex]f(x)\\leq4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What is the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}[\/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\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Radical function stemming from negative 5, negative 2.\" width=\"308\" height=\"346\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q231228\">Show Solution<\/span><\/p>\n<div id=\"q231228\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a <i>radical <\/i>function. The domain of a radical function is any <i>x<\/i> value for which the radicand (the value under the radical sign) is not negative. That means [latex]x+5\\geq0[\/latex], so [latex]x\\geq\u22125[\/latex].<\/p>\n<p>Since the square root must always be positive or\u00a0[latex]0[\/latex], [latex]\\displaystyle \\sqrt{x+5}\\ge 0[\/latex]. That means [latex]\\displaystyle -2+\\sqrt{x+5}\\ge -2[\/latex].<\/p>\n<p>The domain is all real numbers <i>x<\/i> where [latex]x\\geq\u22125[\/latex], and the range is all real numbers [latex]f(x)[\/latex] such that [latex]f(x)\\geq\u22122[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What is the domain of the real-valued function [latex]\\displaystyle f(x)=\\frac{3x}{x+2}[\/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\/121\/2016\/04\/18232531\/image049.gif\" alt=\"A graph of the function f of x equals 3 x divided by the quantity x plus 2. Both axes extend from negative 12 to 12.\" width=\"310\" height=\"321\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q666335\">Show Solution<\/span><\/p>\n<div id=\"q666335\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a <i>rational <\/i>function. The domain of a rational function is restricted where the denominator is\u00a0[latex]0[\/latex]. In this case, [latex]x+2[\/latex] is the denominator, and this is\u00a0[latex]0[\/latex] only when [latex]x=\u22122[\/latex].<\/p>\n<p>The domain is all real numbers except [latex]\u22122[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show how to define the domain and range of\u00a0functions from their graphs.<\/p>\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<h2>Summary<\/h2>\n<p>Although a function may be given as \u201creal valued,\u201d it may be that the function has restrictions to its domain and range. There may be some real numbers that cannot be part of the domain or part of the range. This is particularly true with rational and radical functions which can have restrictions to their domain, range, or both. Other functions, such as quadratic functions and polynomial functions of even degree, can also have restrictions to their range.<\/p>\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-16631\">\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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Determine the Domain and Range of the Graph of a Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QAxZEelInJc\">https:\/\/youtu.be\/QAxZEelInJc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 17: Functions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":169554,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Determine the Domain and Range of the Graph of a Function\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QAxZEelInJc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 17: Functions, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"52cea8e435fd4e929118e848314a4c32, 94c4b690d99c442e883b0d9eca97c3eb","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16631","chapter","type-chapter","status-publish","hentry"],"part":16203,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16631","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16631\/revisions"}],"predecessor-version":[{"id":20519,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16631\/revisions\/20519"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16203"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16631\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16631"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16631"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16631"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}