{"id":58892,"date":"2020-04-24T17:58:45","date_gmt":"2020-04-24T17:58:45","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=58892"},"modified":"2026-03-25T11:53:02","modified_gmt":"2026-03-25T16:53:02","slug":"piecewise-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/piecewise-functions\/","title":{"rendered":"Piecewise Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_7G6MZoz9T7A\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_7G6MZoz9T7A\" data-source-videoID=\"7G6MZoz9T7A\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Piecewise Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Piecewise Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_7G6MZoz9T7A:hover {cursor:pointer;} img#videoThumbnailImage_7G6MZoz9T7A {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/329-piecewise-functions-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_7G6MZoz9T7A\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_7G6MZoz9T7A\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Piecewise Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_7G6MZoz9T7A\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_7G6MZoz9T7A\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_7G6MZoz9T7A\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction vGh_Function() {\n  var x = document.getElementById(\"vGh\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"vGh_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"vGh\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Function_%E2%80%93_Definition\" class=\"smooth-scroll\">Function \u2013 Definition<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Absolute_Value_Functions\" class=\"smooth-scroll\">Absolute Value Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Other_Piecewise_Functions\" class=\"smooth-scroll\">Other Piecewise Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Naming_Piecewise_Functions\" class=\"smooth-scroll\">Naming Piecewise Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Piecewise_Function_Practice_Questions\" class=\"smooth-scroll\">Piecewise Function Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about piecewise functions! In this video, we will explore:<\/p>\n<ul>\n<li>What piecewise functions are<\/li>\n<li>How piecewise functions are defined<\/li>\n<li>And how piecewise functions can be used<\/li>\n<\/ul>\n<h2>Function &#8211; Definition<\/h2>\n<p>\nFirst, let\u2019s look at the definition of a function. A <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/common-functions\/\">function<\/a> is a relationship where a single output is assigned to each input.<\/p>\n<p>Many functions belong to <strong>function families<\/strong> because their equations and graphs all have similar characteristics.<\/p>\n<p>For example, functions in the <strong>linear family<\/strong> have equations that resemble \\(f(x)= mx+b\\) and their graphs are straight lines. Functions in the <strong>quadratic family<\/strong> have equations that look like \\(f(x) = ax^2\\), and their graphs are parabolas.<\/p>\n<p>Piecewise functions are not considered a function family on their own. As the name suggests, they are functions comprised of pieces of other functions.<\/p>\n<h2><span id=\"Absolute_Value_Functions\" class=\"m-toc-anchor\"><\/span>Absolute Value Functions<\/h2>\n<p>\nThe first piece we\u2019re going to look at is the absolute value function.<\/p>\n<p>Functions in the absolute value family have equations that resemble \\(f(x) = |x|\\) and their graphs all have a characteristic \u201cV\u201d shape. This is the graph and a section of the table of values of \\(f(x) =|x|\\).<\/p>\n<p>Instead of a single V, \\(f(x)\\) can also be visualized as pieces of two linear functions. On the left, \\(f(x) =-x\\), and on the right, \\(f(x)=x\\).<\/p>\n<p>As we \u201cread\u201d the graph from left to right, we are on the function \\(f(x) =-x\\) until \\(x=0\\). At that point, the function definition changes to \\(f(x)=x\\).<\/p>\n<p>The domain of the absolute value function is all real numbers. Normally, both \\(f(x) =-x\\) and \\(f(x) =x\\) also have domains of all real numbers, but if we were to graph them together, the graph would look like this and we would no longer have a function.<\/p>\n<p>So each piece needs to be defined on a section of its domain in order to define a piecewise function. If the left function is only defined for negative \\(x\\)-values and the right is only defined for positive \\(x\\)-values (and we put the 0 into one of them\u2014more on that in a minute), we can define this as a single piecewise function.<\/p>\n<p>One way to visualize this is to graph both <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/linear-functions\/\">linear<\/a> functions and erase the sections that are not part of the absolute value function.<\/p>\n<h2><span id=\"Other_Piecewise_Functions\" class=\"m-toc-anchor\"><\/span>Other Piecewise Functions<\/h2>\n<p>\nLet\u2019s take a quick break and practice describing a couple of piecewise functions in terms of what their pieces look like and where those pieces are defined.<\/p>\n<h3><span id=\"Integer_Function\" class=\"m-toc-anchor\"><\/span>Integer Function<\/h3>\n<p>\nFirst up is the greatest integer function, \\(f(x)=\\lfloor x \\rfloor\\). This is also called the floor function or stair step function.<\/p>\n<p>This function is made up of pieces of constant functions that are 1 unit wide.<\/p>\n<h3><span id=\"Sawtooth_Function\" class=\"m-toc-anchor\"><\/span>Sawtooth Function<\/h3>\n<p>\nNext, we have the sawtooth function, \\(f(x) = x &#8211; \\lfloor x \\rfloor\\). This function is also called the castle rim function.<\/p>\n<p>This function is made up of pieces of parallel linear functions that are one unit long<\/p>\n<h2><span id=\"Naming_Piecewise_Functions\" class=\"m-toc-anchor\"><\/span>Naming Piecewise Functions<\/h2>\n<p>\nThere are two criteria for naming piecewise functions:<\/p>\n<ol>\n<li>Reading the pieces from top-to-bottom in the list corresponds to reading the graph from left-to-right. So the first piece in the list is the left piece on the graph.<\/li>\n<li>The domains of the pieces must \u201cadd up to\u201d the domain of the entire function.<\/li>\n<\/ol>\n<p>Let\u2019s start by rewriting our absolute value function in this form.<\/p>\n<p>Here\u2019s the list of expressions that define each piece from left to right:<\/p>\n<p>\\[<br \/>\nf(x) =<br \/>\n\\begin{cases}<br \/>\n-x \\\\<br \/>\nx<br \/>\n\\end{cases}<br \/>\n\\]<\/p>\n<p>Now let\u2019s take a minute to consider the domain of each piece.<\/p>\n<p>\\(f(x) =|x|\\) has a domain of all real numbers. We can input any number we want and get a single output, but we have to be careful with our piecewise definition when we consider \\(x=0\\). Why? Because 0 is a defined point on both pieces, but we only need to include it once.<\/p>\n<p>If we write the function like this:<\/p>\n<p>\\[<br \/>\nf(x) =<br \/>\n\\begin{cases}<br \/>\n-x, &#038; \\text{if } x \\lt 0 \\\\<br \/>\nx, &#038; \\text{if } x \\gt 0<br \/>\n\\end{cases}<br \/>\n\\]<\/p>\n<p>Neither piece includes 0 and the domain is incomplete.<\/p>\n<p>If we write it like this:<\/p>\n<p>\\[<br \/>\nf(x) =<br \/>\n\\begin{cases}<br \/>\n-x, &#038; \\text{if } x \\leq 0 \\\\<br \/>\nx, &#038; \\text{if } x \\geq 0<br \/>\n\\end{cases}<br \/>\n\\]<\/p>\n<p>We are not defining a function, because we\u2019re saying that \\(f(x)\\) equals both \\(-x\\) and \\(x\\) at \\(x=0\\) (even though the output would technically be the same in this case).<\/p>\n<p>Since 0 is in the domain of both pieces, we simply choose which piece to put it in. Both of these equations correctly define the function.<\/p>\n<p>All of our examples so far have depicted piecewise functions with domains of all real numbers. The pieces do not need to connect and they do not need to extend to plus or minus infinity.<\/p>\n<p>Take a look at this function and try to define it with an equation (and yes that single point is a part of it!):<\/p>\n<p>\\[<br \/>\nf(x) =<br \/>\n\\begin{cases}<br \/>\nx, &#038; -7 < x \\leq -3 \\\\<br \/>\n1, &#038; x = -1 \\\\<br \/>\n\\sqrt{x}, &#038; 0 \\leq x < 4 \\\\<br \/>\nx, &#038; 6 < x \\leq 7<br \/>\n\\end{cases}<br \/>\n\\]<\/p>\n<p>Like other functions, piecewise functions can be used to tell stories. This is the story of a car journey Bob took last week:<\/p>\n<p>\\[<br \/>\nd(t) =<br \/>\n\\begin{cases}<br \/>\n50t, &#038; 0 \\leq t \\leq 3 \\\\<br \/>\n150, &#038; 3 < t < 3.5 \\\\<br \/>\n25t + 62.5, &#038; 3.5 \\leq t < 7.5 \\\\<br \/>\n250, &#038; 7.5 \\leq t < 8 \\\\<br \/>\n-62.5t, &#038; 8 \\leq t \\leq 12<br \/>\n\\end{cases}<br \/>\n\\]<\/p>\n<p>Let\u2019s see how many of these review questions we can answer about Bob\u2019s journey before we go:<\/p>\n<p>How long did the journey last? <strong>12 hrs<\/strong><br \/>\nWhat\u2019s the farthest distance Bob drove from his home? <strong>250 mi<\/strong><br \/>\nHow long did it take him to get there? <strong>7.5 hrs<\/strong><br \/>\nWhat was Bob doing from 3-3.5 hrs? <strong>not moving<\/strong><br \/>\nWhen did Bob turn toward home? at the <strong>8 hr<\/strong> mark<br \/>\nWhen was Bob driving the fastest? from hr 8 to hr 12 he traveled <strong>62.5 mph<\/strong><br \/>\nWhat was Bob\u2019s average speed from 0 to 8 hrs? <strong>31.25 mph<\/strong><\/p>\n<p>Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Piecewise_Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Piecewise Function Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe following piecewise function shows the speed of a car as a function of time. Daniel says that the flat top portion of the graph shows that the car came to a stop. Is Daniel correct? <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Piecewise-Function-Graph-Example-1.svg\" alt=\"Line graph showing speed versus time, with speed increasing, maintaining a constant level, then decreasing back to zero, forming a trapezoid shape.\" width=\"306.475\" height=\"260.13\" class=\"aligncenter size-full wp-image-287219\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">Daniel is correct.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">Daniel is incorrect.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Each \u201cpiece\u201d of the graph shows the car traveling at a different speed. The first piece of the graph shows the car gradually increasing speed as time elapses. The second piece of the graph shows the car maintaining a steady speed as time elapses. The third piece of the graph shows the car steadily reducing speed. <\/p>\n<p>Even though the flat piece of the graph seems to show the car stopping, the speed is maintained as time is passing.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat kind of piecewise function is graphed below?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Piecewise-Function-Graph-Example-2.svg\" alt=\"Graph of the absolute value function y = |x|, showing a V shape with its vertex at the origin.\" width=\"291.72\" height=\"266.76\" class=\"aligncenter size-full wp-image-287222\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">Linear<\/div><div class=\"PQ\"  id=\"PQ-2-2\">Quadratic<\/div><div class=\"PQ\"  id=\"PQ-2-3\">Step<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-4\">Absolute Value<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>An absolute value function will form a V shape. This type of function is essentially two \u201cpieces\u201d of linear functions. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDefine the following piecewise function.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Piecewise-Function-Graph-Example-3.svg\" alt=\"A piecewise graph with a horizontal line at h(x)=2 for x&lt;1, a closed dot at (1,2), and a sloped line starting at (1,1) with an open dot, increasing as x increases.\" width=\"290.16\" height=\"297.96\" class=\"aligncenter size-full wp-image-287225\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\(h(x)=\\begin{cases}2, &#038; x \\le 3 \\\\x, &#038; x \\gt -1\\end{cases}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">\\(h(x)=\\begin{cases}2, &#038; x \\le 1 \\\\x, &#038; x \\gt 1\\end{cases}\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\(h(x)=\\begin{cases}2, &#038; x \\le 9 \\\\x, &#038; x \\gt 3\\end{cases}\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\(h(x)=\\begin{cases}1, &#038; x \\le 1 \\\\x, &#038; x \\gt -3\\end{cases}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The graph shows that when \\(x\\) is less than or equal to 1, then \\(h(x)\\) is equal to 2. For example, when \\(x\\) is equal to 0, \\(h(x)\\) equals 2. This is defined as \\(h(x)=2\\text{ if } x \\gt 1\\). <\/p>\n<p>When x is greater than 1, then \\(h(x)\\) is equal to \\(x\\). For example, when \\(x\\) is equal to 4, \\(h(x)\\) is also equal to 4. This is defined as \\(h(x)=x\\text{ if } x \\gt 1\\). <\/p>\n<p>These two pieces define the piecewise function for the graph:<\/p>\n<p style=\"text-align:center; line-height: 35px\">\\(h(x)=\\begin{cases}2, &#038; x \\le 1 \\\\x, &#038; x \\gt 1\\end{cases}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph below shows the relationship between time in minutes (\\(x\\)-axis) and water in gallons (\\(y\\)-axis). The graph shows the amount of water that was used (in gallons) when George gave his new puppy a bath.<\/p>\n<p>When did George turn the water faucet off? How much water was in the bath when he bathed the puppy?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Piecewise-Function-Graph-Example-4.svg\" alt=\"Line graph showing gallons of water over time; water increases to 24 gallons by 6 minutes, stays level until 18 minutes, then decreases to 0 gallons by 30 minutes.\" width=\"361.92\" height=\"351\" class=\"aligncenter size-full wp-image-287228\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">George turned the water off at 30 minutes. There were 24 gallons of water in the tub when he bathed the puppy.<\/div><div class=\"PQ\"  id=\"PQ-4-2\">George turned the water off at 12 minutes. There were 18 gallons of water in the tub when he bathed the puppy.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-3\">George turned the water off at 6 minutes. There were 24 gallons of water in the tub when he bathed the puppy.<\/div><div class=\"PQ\"  id=\"PQ-4-4\">George turned the water off at 24 minutes. There were 6 gallons of water in the tub when he bathed the puppy. <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The amount of water in gallons stops increasing when George turns the water faucet off. The \\(x\\)-axis shows us that when six minutes have passed, the amount of water stops increasing. This means that George turns off the water after six minutes, and begins the puppy bath. The \\(y\\)-axis indicates that at this time the water is currently at 24 gallons.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDefine the following piecewise function. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Piecewise-Function-Graph-Example-5.svg\" alt=\"A graph showing two line segments and four points on a coordinate plane, with both closed and open circles, intersecting at different locations.\" width=\"271.44\" height=\"273\" class=\"aligncenter size-full wp-image-287216\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">\\(f(x)=\\begin{cases}x+2, &#038; -5 \\le x \\le -1 \\\\-x, &#038; 1 \\lt x \\le 3\\end{cases}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(f(x)=\\begin{cases}x+5, &#038; -7 \\le x \\le 1 \\\\-x, &#038; 1 \\lt x \\le 3\\end{cases}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">\\(f(x)=\\begin{cases}x+1, &#038; -4 \\le x \\le 1 \\\\-x, &#038; 1 \\lt x \\le 3\\end{cases}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\(f(x)=\\begin{cases}x+3, &#038; -5 \\le x \\le 1 \\\\-x-4, &#038; 1 \\lt x \\le 2\\end{cases}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The graphed line on the left side represents \\(x\\)-values that are greater than or equal to \u22124 and less than or equal to 1. This region can be described as \\(-4\\leq x\\leq 1\\).<\/p>\n<p>In this region, the \\(f(x)\\)-value, or \\(y\\)-value, is always one more than the \\(x\\)-value. For example, when \\(x\\) is 1, \\(y\\) is 2. When \\(x\\) is \u22123, \\(y\\) is \u22122. This means that in the region of \\(-4\\leq x\\leq 1\\), \\(y\\) is always one more than \\(x\\). This is defined as:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(f(x)=x+1\\)<br \/>\n\\(-4\\leq x\\leq 1\\)<\/p>\n<p>The graphed line on the right side represents \\(x\\)-values that are greater than 1 and less than or equal to 3. This region can be described as \\(1 \\lt x \\leq 3\\).<\/p>\n<p>In this region, the \\(f(x)\\)-value, or the \\(y\\)-value, is always the \\(x\\)-value negated. For example, when \\(x\\) is 2, \\(y\\) is \u22122. When \\(x\\) is 3, \\(y\\) is \u22123. This means that in the region of \\(1 \\lt x \\leq 3\\), \\(y\\) is always the negative value of \\(x\\). This is defined as:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(f(x)=-x\\)<br \/>\n\\(1 \\lt x \\leq 3\\)<\/p>\n<p>When both \u201cpieces\u201d are put together, they define the piecewise function as:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(f(x)=\\begin{cases}x+1, &#038; -4 \\le x \\le 1 \\\\-x, &#038; 1 \\lt x \\le 3\\end{cases}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-ii\/\">Return to Algebra II Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra II Videos<\/p>\n","protected":false},"author":1,"featured_media":164168,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-58892","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-functions-and-their-graphs-videos","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/58892","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=58892"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/58892\/revisions"}],"predecessor-version":[{"id":281822,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/58892\/revisions\/281822"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/164168"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=58892"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}