{"id":16030,"date":"2019-09-26T17:24:05","date_gmt":"2019-09-26T17:24:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-graph-piecewise-functions\/"},"modified":"2024-05-02T15:55:52","modified_gmt":"2024-05-02T15:55:52","slug":"read-graph-piecewise-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-graph-piecewise-functions\/","title":{"raw":"Graphing Piecewise Functions","rendered":"Graphing Piecewise Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write a piecewise function given an application<\/li>\r\n \t<li>Graph a piecewise function using domain<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this section, we will plot piecewise functions. The function plotted below represents the cost to transfer data for a given cell phone company.\u00a0We can see where the function changes from a constant to a line with a positive slope\u00a0at [latex]g=2[\/latex]. When we plot piecewise functions, it is important to make sure each formula is applied on its proper domain. [latex]C\\left(g\\right)=\\begin{cases}{25} \\text{ if }{ 0 }&lt;{ g }&lt;{ 2 }\\\\10g+5\\text{ if }{ g}\\ge{ 2 }\\end{cases}[\/latex]\r\n\r\nIn this case, the output is\u00a0[latex]25[\/latex] for any input between\u00a0[latex]0[\/latex] and\u00a0[latex]2[\/latex]. For values equal to or greater than\u00a0[latex]2[\/latex], the output is defined as [latex]10g+5[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200643\/CNX_Precalc_Figure_01_02_0222.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3 id=\"fs-id1165135532516\">How To: Given a piecewise function, sketch a graph<\/h3>\r\n<ol id=\"fs-id1165137588539\">\r\n \t<li>Indicate on the <em>x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\r\n \t<li>For each piece of the domain, graph the corresponding equation on that interval. Do not graph two functions over one interval because it would violate the criteria of a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSketch a graph of the function and state its domain and range.\r\n\r\nGiven the piecewise definition [latex]f(x)=\\begin{cases}\u2212x \u2212 3\\text{ if }x &lt; \u22123\\\\ x + 3\\text{ if } x \\ge \u22123\\end{cases}[\/latex]\r\n\r\n[reveal-answer q=\"895830\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"895830\"]\r\n\r\nFirst, graph the line [latex]f(x) = \u2212x\u22123[\/latex] erasing the part where x is greater than\u00a0[latex]-3[\/latex]. Place an open circle at\u00a0[latex](-3,0)[\/latex].\r\n\r\n<img class=\"size-medium wp-image-2032 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01000356\/Screen-Shot-2016-06-30-at-5.02.40-PM-300x212.png\" alt=\"Graph of the line f(x)=-3-x with the restriction x&lt;-3\" width=\"300\" height=\"212\" \/>\r\n\r\nNow place the line [latex]f(x) = x+3[\/latex] on the graph starting at the point where [latex]x=-3[\/latex]. Note that for this portion of the graph the point\u00a0[latex](-3,0)[\/latex] is included so you can remove the open circle.\r\n\r\n<img class=\"size-medium wp-image-2034 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01001119\/Screen-Shot-2016-06-30-at-5.10.38-PM-300x240.png\" alt=\"graph of the line f(x)=-x-3 and f(x) = x+3 \" width=\"300\" height=\"240\" \/>\r\n\r\nNotice the two pieces of the graph meet at the point\u00a0[latex](-3,0)[\/latex].\u00a0 This will not be the case for all piecewise functions.\r\n\r\nThe domain of this function is all real numbers because\u00a0[latex](-3,0)[\/latex] is not included as the endpoint of [latex]f(x) = \u2212x\u22123[\/latex], but it is included as the endpoint for\u00a0[latex]f(x) = x+3[\/latex].\r\n\r\nThe range of this function starts at [latex]f(x)=0[\/latex] and includes\u00a0[latex]0[\/latex], \u00a0and goes to infinity, so we would write this as [latex]y\\ge0[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example, we will graph a piecewise-defined function that\u00a0models the cost of shipping for an online comic book retailer.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAn on-line comic book retailer charges shipping costs\u00a0of $2.50 plus an additional $1.50 for every comic book purchased for up to 14 books, but if 15 or more comic books are purchased then shipping is free. The following formula represents the cost of shipping, where <em>n<\/em> is the number of comic books:\r\n\r\n[latex]S(n)=\\begin{cases}1.5n+2.5\\text{ if }1\\le{n}\\le14\\\\0\\text{ if }n\\ge15\\end{cases}[\/latex]\r\n\r\nDraw a graph of the cost function.\r\n[reveal-answer q=\"688588\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688588\"]\r\n\r\n&nbsp;\r\n\r\nFirst, draw the line\u00a0[latex]S(n)=1.5n+2.5[\/latex]. \u00a0We can use transformations: this is a vertical stretch by a factor of\u00a0[latex]1.5[\/latex] and a vertical shift by\u00a0[latex]2.5[\/latex].\r\n\r\n<img class=\"wp-image-2265 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07200231\/Screen-Shot-2016-07-07-at-12.58.26-PM-283x300.png\" alt=\"S(n)=1.5n+2.5\" width=\"415\" height=\"440\" \/>\r\n\r\nNow we can eliminate the portions of the graph that are not in the domain. This leaves us with the portion of the graph at [latex]1\\le{n}\\le14[\/latex].\r\n\r\n<img class=\"wp-image-2266 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07200537\/Screen-Shot-2016-07-07-at-1.05.03-PM-261x300.png\" alt=\"S(n) = 1.5n+2.5 for 1&lt;=n&lt;=14\" width=\"424\" height=\"488\" \/>\r\n\r\n&nbsp;\r\n\r\nLast, add the constant function [latex]S(n)=0[\/latex] for inputs greater than or equal to\u00a0[latex]15[\/latex]. Place closed dots on the ends of the graph to indicate the\u00a0inclusion of the endpoints.\r\n\r\n<img class=\"wp-image-2268 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07201300\/Screen-Shot-2016-07-07-at-1.09.58-PM-272x300.png\" alt=\"Screen Shot 2016-07-07 at 1.09.58 PM\" width=\"439\" height=\"484\" \/>\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 graph a piecewise-defined function which is linear over both domains.\r\n\r\nhttps:\/\/youtu.be\/B1jfpiI-QQ8\r\n<h2>Summary<\/h2>\r\nTo graph piecewise functions, first identify where the domain is partitioned by boundary values. Graph functions on the domain using tools such as plotting points or transformations. Be sure to use open or closed circles on the endpoints of each domain based on whether the endpoint is included.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write a piecewise function given an application<\/li>\n<li>Graph a piecewise function using domain<\/li>\n<\/ul>\n<\/div>\n<p>In this section, we will plot piecewise functions. The function plotted below represents the cost to transfer data for a given cell phone company.\u00a0We can see where the function changes from a constant to a line with a positive slope\u00a0at [latex]g=2[\/latex]. When we plot piecewise functions, it is important to make sure each formula is applied on its proper domain. [latex]C\\left(g\\right)=\\begin{cases}{25} \\text{ if }{ 0 }<{ g }<{ 2 }\\\\10g+5\\text{ if }{ g}\\ge{ 2 }\\end{cases}[\/latex]\n\nIn this case, the output is\u00a0[latex]25[\/latex] for any input between\u00a0[latex]0[\/latex] and\u00a0[latex]2[\/latex]. For values equal to or greater than\u00a0[latex]2[\/latex], the output is defined as [latex]10g+5[\/latex].\n\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200643\/CNX_Precalc_Figure_01_02_0222.jpg\" alt=\"Graph of C(g)\" width=\"487\" height=\"296\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3 id=\"fs-id1165135532516\">How To: Given a piecewise function, sketch a graph<\/h3>\n<ol id=\"fs-id1165137588539\">\n<li>Indicate on the <em>x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\n<li>For each piece of the domain, graph the corresponding equation on that interval. Do not graph two functions over one interval because it would violate the criteria of a function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Sketch a graph of the function and state its domain and range.<\/p>\n<p>Given the piecewise definition [latex]f(x)=\\begin{cases}\u2212x \u2212 3\\text{ if }x < \u22123\\\\ x + 3\\text{ if } x \\ge \u22123\\end{cases}[\/latex]\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q895830\">Show Solution<\/span><\/p>\n<div id=\"q895830\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, graph the line [latex]f(x) = \u2212x\u22123[\/latex] erasing the part where x is greater than\u00a0[latex]-3[\/latex]. Place an open circle at\u00a0[latex](-3,0)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2032 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01000356\/Screen-Shot-2016-06-30-at-5.02.40-PM-300x212.png\" alt=\"Graph of the line f(x)=-3-x with the restriction x&lt;-3\" width=\"300\" height=\"212\" \/><\/p>\n<p>Now place the line [latex]f(x) = x+3[\/latex] on the graph starting at the point where [latex]x=-3[\/latex]. Note that for this portion of the graph the point\u00a0[latex](-3,0)[\/latex] is included so you can remove the open circle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2034 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/01001119\/Screen-Shot-2016-06-30-at-5.10.38-PM-300x240.png\" alt=\"graph of the line f(x)=-x-3 and f(x) = x+3\" width=\"300\" height=\"240\" \/><\/p>\n<p>Notice the two pieces of the graph meet at the point\u00a0[latex](-3,0)[\/latex].\u00a0 This will not be the case for all piecewise functions.<\/p>\n<p>The domain of this function is all real numbers because\u00a0[latex](-3,0)[\/latex] is not included as the endpoint of [latex]f(x) = \u2212x\u22123[\/latex], but it is included as the endpoint for\u00a0[latex]f(x) = x+3[\/latex].<\/p>\n<p>The range of this function starts at [latex]f(x)=0[\/latex] and includes\u00a0[latex]0[\/latex], \u00a0and goes to infinity, so we would write this as [latex]y\\ge0[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the next example, we will graph a piecewise-defined function that\u00a0models the cost of shipping for an online comic book retailer.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>An on-line comic book retailer charges shipping costs\u00a0of $2.50 plus an additional $1.50 for every comic book purchased for up to 14 books, but if 15 or more comic books are purchased then shipping is free. The following formula represents the cost of shipping, where <em>n<\/em> is the number of comic books:<\/p>\n<p>[latex]S(n)=\\begin{cases}1.5n+2.5\\text{ if }1\\le{n}\\le14\\\\0\\text{ if }n\\ge15\\end{cases}[\/latex]<\/p>\n<p>Draw a graph of the cost function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q688588\">Show Solution<\/span><\/p>\n<div id=\"q688588\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>First, draw the line\u00a0[latex]S(n)=1.5n+2.5[\/latex]. \u00a0We can use transformations: this is a vertical stretch by a factor of\u00a0[latex]1.5[\/latex] and a vertical shift by\u00a0[latex]2.5[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2265 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07200231\/Screen-Shot-2016-07-07-at-12.58.26-PM-283x300.png\" alt=\"S(n)=1.5n+2.5\" width=\"415\" height=\"440\" \/><\/p>\n<p>Now we can eliminate the portions of the graph that are not in the domain. This leaves us with the portion of the graph at [latex]1\\le{n}\\le14[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2266 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07200537\/Screen-Shot-2016-07-07-at-1.05.03-PM-261x300.png\" alt=\"S(n) = 1.5n+2.5 for 1&lt;=n&lt;=14\" width=\"424\" height=\"488\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Last, add the constant function [latex]S(n)=0[\/latex] for inputs greater than or equal to\u00a0[latex]15[\/latex]. Place closed dots on the ends of the graph to indicate the\u00a0inclusion of the endpoints.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2268 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07201300\/Screen-Shot-2016-07-07-at-1.09.58-PM-272x300.png\" alt=\"Screen Shot 2016-07-07 at 1.09.58 PM\" width=\"439\" height=\"484\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show how to graph a piecewise-defined function which is linear over both domains.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2:  Graph a Piecewise Defined Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/B1jfpiI-QQ8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>To graph piecewise functions, first identify where the domain is partitioned by boundary values. Graph functions on the domain using tools such as plotting points or transformations. Be sure to use open or closed circles on the endpoints of each domain based on whether the endpoint is included.<\/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-16030\">\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>College Algebra, Unit 1.4 Function Notation. <strong>Authored by<\/strong>: Carl Stitz and Jeff Zeager. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.stitz-zeager.com\/szca07042013.pdf\">http:\/\/www.stitz-zeager.com\/szca07042013.pdf<\/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>Ex 2: Graph a Piecewise Defined Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/B1jfpiI-QQ8\">https:\/\/youtu.be\/B1jfpiI-QQ8<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\"},{\"type\":\"cc\",\"description\":\"College Algebra, Unit 1.4 Function Notation\",\"author\":\"Carl Stitz and Jeff Zeager\",\"organization\":\"\",\"url\":\"http:\/\/www.stitz-zeager.com\/szca07042013.pdf\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Graph a Piecewise Defined Function\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/B1jfpiI-QQ8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"e4c9f564245c451db0a1948ba5fbda8d, 941cc4b4167b46e0aef3072488c97e67, bcb37fe909b54e41909f6a0a426210de 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