{"id":4127,"date":"2022-08-01T22:50:07","date_gmt":"2022-08-01T22:50:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=4127"},"modified":"2026-01-26T20:18:04","modified_gmt":"2026-01-26T20:18:04","slug":"1-1-1-relations-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/1-1-1-relations-2\/","title":{"raw":"1.1.1: Relations","rendered":"1.1.1: Relations"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Define a relation using ordered pairs, a mapping, a graph, and an equation<\/li>\r\n \t<li>Determine the domain and range of a relation<\/li>\r\n \t<li>Write the domain and range in interval notation<\/li>\r\n \t<li>Write the domain and range in set-builder notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Parentheses, Brackets, and Braces<\/h2>\r\n<em><strong>Parentheses<\/strong><\/em> ( ), <em><strong>brackets<\/strong><\/em> [ ], and <em><strong>braces<\/strong><\/em> { } play an important role in mathematics and have several different uses. Parentheses can be used to denote multiplication: [latex]3(4) = 3\\cdot 4=12[\/latex]. They can also be used along with brackets and braces to denote grouping and order of operations: [latex]\\{3+[2-(3x-1)]\\}[\/latex]. Parentheses are also used to denote ordered pairs. An <em><strong>ordered pair<\/strong><\/em> consists of two numbers (or more generally two mathematical objects) enclosed by a pair of parentheses, where the order of the numbers in the pair of parentheses matters.\u00a0An ordered pair has many meanings in mathematics.\r\n<h3>Coordinates<\/h3>\r\nAn ordered pair may refer to the <em><strong>coordinates<\/strong><\/em> of a point on a coordinate plane. A <em><strong>rectangular coordinate plane<\/strong><\/em> (or <em><strong>Cartesian plane<\/strong><\/em>) is composed of two <em><strong>number lines<\/strong><\/em> or two <em><strong>axes<\/strong><\/em> and the plane that they form. The horizontal number line is called the [latex]x[\/latex]-axis and the vertical number line is called the [latex]y[\/latex]-axis. The two axes cross at a point where both the [latex]x[\/latex]- and [latex]y[\/latex]-values are zero, called the <em><strong>origin<\/strong><\/em>. The two axes form a grid, called a rectangular\u00a0coordinate plane, and each intersection on the grid is a <em><strong>point <\/strong><\/em>(Figure 1). The points on the grid are not only integers. There are an infinite number of real number intersections that make up the grid.\r\n\r\n[caption id=\"attachment_4206\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-4206 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-300x300.png\" alt=\"Graph showing the quadrants and axis' described above.\" width=\"300\" height=\"300\" \/> Figure 1. Rectangular coordinate plane[\/caption]\r\n\r\nThere are an infinite number of points on the coordinate plane. The location of a point may be specified using the values on the [latex]x[\/latex] and [latex]y[\/latex]-axis. The ordered pair (3, \u20132) refers to a point on the coordinate plane where the location is 3 relative to the horizontal number line (or [latex]x[\/latex]-axis) and \u20132 relative to the vertical number line (or [latex]y[\/latex]-axis). We call 3 the [latex]x[\/latex]-coordinate and \u20132 the\u00a0[latex]y[\/latex]-coordinate. Therefore, the ordered pair\u00a0[latex](3, -2)[\/latex] may represent a pair of coordinates on the coordinate plane (Figure 2).\r\n\r\n[caption id=\"attachment_3191\" align=\"aligncenter\" width=\"350\"]<img class=\"wp-image-3191\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/06230146\/1-1-1-OrderedPair-300x300.png\" alt=\"The point (3,-2) on the coordinate plane described above.\" width=\"350\" height=\"350\" \/> Figure 2. The point (3, \u20132) on the coordinate plane[\/caption]\r\n\r\nAny point on the coordinate plane [latex](x, y)[\/latex] is an ordered pair because it is two numbers (a pair) written in the specific order of [latex]x[\/latex]-coordinate first, [latex]y[\/latex]-coordinate second.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-3359 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-300x300.png\" alt=\"A graph showing the points described below.\" width=\"300\" height=\"300\" \/><\/p>\r\n\r\n<h4>Solution<\/h4>\r\nPoint [latex]A[\/latex] is right above [latex]3[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]2[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]A=(3, 2)[\/latex].\r\n\r\nPoint [latex]B[\/latex] lies on [latex]4[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]0[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]B=(4, 0)[\/latex].\r\n\r\nPoint [latex]C[\/latex] is right below [latex]0[\/latex] on the [latex]x[\/latex]-axis and lies on [latex]-2[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]C=(0, -2)[\/latex].\r\n\r\nPoint [latex]D[\/latex] is right below [latex]-4[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]-3[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]D=(-4, -3)[\/latex].\r\n\r\nPoint [latex]E[\/latex] is right above [latex]-2[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]1[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]E=(-2, 1)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nState the coordinates of the points on the graph:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-3358 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-300x300.png\" alt=\"6 points on a coordinate plane. A is 3 units right and four units up, B is 7 right and 3 down, C is two down, D is 4 left and 4 down, E is 8 left, and F is 2 left and 7 down.\" width=\"300\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm305\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm305\"]\r\n\r\n[latex]A=(3,4),\\;B=(7,-3),\\;C=(0,-2),\\;D=(-4,-4),\\;E=(-8,0),\\;F=(-2,7)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Intervals<\/h3>\r\nAn ordered pair may also refer to an <em><strong>interval<\/strong> <\/em>with a starting value and an ending value. The pair of numbers are ordered with the smaller number written first. For example, [latex](-1, 3)[\/latex] means an interval on a number line starting at \u20131 and ending at 3. The parentheses means both of the two numbers \u20131 and 3 are not included in the interval, but are <em><strong>boundaries<\/strong><\/em> of the interval. Figure 3 shows this interval on a number line. We use an empty circle to show that an end point is <span style=\"text-decoration: underline;\">not<\/span> included in an interval. Since the two end points \u20131 and 3 are not included in the interval, they are both empty circles on the number line. Since this interval is for [latex]x[\/latex], it is equivalent to the inequality [latex]1 &lt; x &lt; 3[\/latex].\r\n\r\n[caption id=\"attachment_3371\" align=\"aligncenter\" width=\"247\"]<img class=\"wp-image-3371 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/1-3.png\" alt=\"interval between -1 and 3 described above.\" width=\"247\" height=\"39\" \/> Figure 3. Interval (\u20131, 3)[\/caption]\r\n\r\n[latex](-2, 1][\/latex] represents an interval on a number line starting at \u20132 and ending at 1. The parentheses next to \u20132 tells us that \u20132 is not included in the interval, while the bracket next to 1 tells us that 1 is included in the interval. Figure 4 shows the interval on a number line. We use a solid circle to show that an end point is included in an interval. This interval is equivalent to [latex] -2 &lt; x \u2264 1 [\/latex].\r\n\r\n[caption id=\"attachment_3372\" align=\"aligncenter\" width=\"247\"]<img class=\"wp-image-3372 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/22003726\/2-1.png\" alt=\"Interval (-2, 1] described above.\" width=\"247\" height=\"39\" \/> Figure 4. Interval (\u20132, 1][\/caption]\r\n\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nWrite the interval.\r\n<ol>\r\n \t<li>All values between [latex]-1[\/latex] and [latex]4[\/latex].<\/li>\r\n \t<li>All values between [latex]-1[\/latex] and [latex]4[\/latex], including [latex]4[\/latex].<\/li>\r\n \t<li>All values between [latex]-1[\/latex] and [latex]4[\/latex], including [latex]-1[\/latex] and [latex]4[\/latex].<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]-1[\/latex] and [latex]4[\/latex] are not included so we use parentheses: [latex](-1,4)[\/latex]<\/li>\r\n \t<li>[latex]-1[\/latex] is not included so we use parentheses on that end, and [latex]4[\/latex] is included so we use a bracket on that end: [latex](-1,4][\/latex]<\/li>\r\n \t<li>\u00a0Both\u00a0[latex]-1[\/latex] and [latex]4[\/latex] are included so we use brackets: [latex][-1,4][\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nWrite the interval.\r\n<ol>\r\n \t<li>\u00a0All values between [latex]-7[\/latex] and [latex]5[\/latex].<\/li>\r\n \t<li>\u00a0All values between [latex]-7[\/latex] and [latex]5[\/latex], including [latex]-7[\/latex].<\/li>\r\n \t<li>\u00a0All values between [latex]-7[\/latex] and [latex]5[\/latex], including [latex]-7[\/latex] and [latex]5[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm595\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm595\"]\r\n<ol>\r\n \t<li>[latex](-7,5)[\/latex]<\/li>\r\n \t<li>[latex][-7,5)[\/latex]<\/li>\r\n \t<li>[latex][-7,5][\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nSketch the interval on a number line.\r\n<ol>\r\n \t<li>\u00a0(\u20133, 2)<\/li>\r\n \t<li>\u00a0[4, 7]<\/li>\r\n \t<li>\u00a0[latex](-\\infty,0][\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n\r\n1. The parentheses tell us that neither \u20133 nor 2 are included in the interval, so we use open circles on both ends:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-3363 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/32.png\" alt=\"Interval with a line connecting an open circle on -3 and an open circle on 2\" width=\"247\" height=\"39\" \/><\/p>\r\n2. The brackets tell us that 4 and 7 are included in the interval, so we use closed circles on both ends:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-3364 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/4-7.png\" alt=\"[4,7] interval. A line connecting a filled in circle at 4 and a filled in circle at 7.\" width=\"247\" height=\"39\" \/><\/p>\r\n3. Negative infinity can never be reached so it has a parenthesis to show that it is not included in the interval. We will use an arrow on that end of the line to show that it goes on forever. The bracket next to 0 tells us that 0 is included in the interval, so we will use a closed circle on that end:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-4101 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/inft0.png\" alt=\"A ray starting at zero and moving to the left forever. An arrow at the end of the line makes it a ray.\" width=\"262\" height=\"53\" \/><\/p>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nWrite the interval shown on the number line.\r\n\r\n1.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-4102\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/01201235\/infty-5.png\" alt=\"line starting at 5 and moving left with arrow on left. Or a ray starting at 5 and moving left.\" width=\"247\" height=\"50\" \/><\/p>\r\n2.\u00a0<img class=\"aligncenter wp-image-3367 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/31.png\" alt=\"A line segment joining a closed circle at -3 with a closed circle at 1.\" width=\"247\" height=\"39\" \/>\r\n\r\n3.\u00a0<img class=\"aligncenter wp-image-3368 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/49.png\" alt=\"interval between 4 and 9. A line segment joining an open circle at 4 to an open circle at 9.\" width=\"247\" height=\"39\" \/>\r\n\r\n[reveal-answer q=\"hjm282\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm282\"]\r\n<ol>\r\n \t<li>[latex](-\\infty,5][\/latex]<\/li>\r\n \t<li>[\u20133, 1]<\/li>\r\n \t<li>(4, 9)<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Input \u2013 Output<\/h3>\r\nAn ordered pair may also refer to a relation or mapping where the first object in the pair is a source or input in a mapping and the second object in the pair is a target or output in a mapping. For example, we may define the relation <em>double the input value to get the output value<\/em> using the ordered pair idea where the second number is double the first number. Therefore, [latex](1, 2), (2, 4), (3, 6)...[\/latex] are specific examples of the relation. In general, the relation is [latex](x, 2x)[\/latex]. We will explore this further under <strong>Relations<\/strong>.\r\n<h3>Sets<\/h3>\r\nBraces { } are used when we talk about sets. A <em><strong>set<\/strong><\/em> is simply a group or collection of objects. We use braces { } to enclose the group of objects. Each object in a set is called an\u00a0<em><strong>element<\/strong><\/em>. For example, {blue, red, yellow} is the set of primary colors. Blue, red, and yellow are the elements of the set. The set {2, 4, 6, 8, ...} is the set of even whole numbers. The set {(1, 2), (3, 8), (4, 9)} is a set of three ordered pairs. A set that contains no elements, [latex]\\{\\;\\}[\/latex], is called the <em><strong>empty set<\/strong> <\/em>and is notated [latex]\\varnothing[\/latex]. We will look further at sets in <strong>set builder notation<\/strong>.\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\n<ol>\r\n \t<li>\u00a0Write the set of all integers between \u20132 and 5.<\/li>\r\n \t<li>\u00a0How many elements are in the set of currently printed US currency notes?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>The integers between \u20132 and 5 are {\u20131, 0, 1, 2, 3, 4}.<\/li>\r\n \t<li>The set of currently printed US currency notes is {$1, $5, $10, $20, $50, $100} so there are 6 elements in this set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\n<ol>\r\n \t<li>\u00a0Write the set of US currency coins intended for circulation.<\/li>\r\n \t<li>\u00a0How many elements are in the set of proper fractions between 0 and 1?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm071\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm071\"]\r\n<ol>\r\n \t<li>[latex]\\{1\u00a2,\\;5\u00a2,\\;10\u00a2,\\;25\u00a2\\}[\/latex]<\/li>\r\n \t<li>There are an infinite number of proper fractions between zero and 1.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Relations<\/h2>\r\nA <em><strong>relation<\/strong><\/em> is a way in which two or more objects are connected. For example, every person has a date of birth, so there is a relation between the person and their date of birth. Indeed, there is a relation between the set of all people and the set of all birth dates. Every perfect square has a whole number square root, so there is a relation between a perfect square and its whole number square root. Further, there is a relation between the set of perfect squares and the set of whole number square roots.\r\n\r\nMathematically, a <em><strong>binary relation<\/strong><\/em> is defined as a connection between <em>two<\/em> sets of objects. That relation can be represented in multiple ways: a set of ordered pairs; a mapping; a graph; or an equation.\r\n\r\nFor example, the relation between two whole numbers where\u00a0the second number is double the first number, may be represented in different ways:\r\n<ul>\r\n \t<li>A <em><strong>set of ordered pairs<\/strong><\/em> shows the first number followed by the second number, or the source followed by the target: {(0, 0), (1, 2), (2, 4), (3, 6), (4, 8),...}. The ellipses ... show that the established pattern continues forever.<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li>A <em><strong>mapping<\/strong><\/em> shows the association of the first numbers (the set of whole numbers) with an arrow that points to the corresponding second number (double the first number):<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-704 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-300x226.png\" alt=\"mapping of doubles. A mapping of the ordered pairs described above using arrows from the domain (x) value to the range (y) value.\" width=\"300\" height=\"226\" \/><\/p>\r\n\r\n<ul>\r\n \t<li>A <em><strong>graph<\/strong><\/em> shows the set of ordered pairs plotted on the rectangular (Cartesian) coordinate plane. If the first number is represented by [latex]x[\/latex] and the second number by [latex]y[\/latex], the graph shows the plotted points [latex](x, y)[\/latex]. Obviously, the graph does not show all possible coordinate pairs as the whole numbers start at zero and go to infinity. Notice also, that the plotted pairs are not joined by a line, as the line that joins them represents real numbers, not whole numbers.<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-703 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-300x300.png\" alt=\"A cartesian coordinate plane containing the points contained in the mapping above.\" width=\"300\" height=\"300\" \/><\/p>\r\n\r\n<ul>\r\n \t<li>An <em><strong>equation<\/strong><\/em>: If the first number is represented by [latex]x[\/latex] and the second number by [latex]y[\/latex], then since [latex]y[\/latex] is double the value of [latex]x[\/latex], the relation can be represented by the equation [latex]y = 2x[\/latex] where [latex]x[\/latex] is a whole number.<\/li>\r\n<\/ul>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nThe relation given by the set of ordered pairs {(Juan, Brown), (Ian, Blue), (Bailey, Brown), (Kirsty, Hazel), (Jean, Grey), (Freddy, Blue)} is the eye color of each named person. Create a mapping to show this relation.\r\n<h4>Solution<\/h4>\r\nWe start by listing the elements of each set inside a border, then draw arrows from the person to their respective eye color.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-701 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-300x224.png\" alt=\"Mapping of the eye color ordered pairs described above with the domain the persons name, and the range the persons eye color.\" width=\"300\" height=\"224\" \/><\/p>\r\nNotice that there are more people than colors, since some people share the same eye color.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nWhat relation is shown in the mapping?\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-702 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-300x237.png\" alt=\"Mapping from the domain { -3, 3, 2.5, -two thirds, square root of 3} to the range { 9, 6.25. four ninths, 3} representing the rule &quot;square the domain value&quot;.\" width=\"300\" height=\"237\" \/><\/p>\r\n[reveal-answer q=\"hjm887\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm887\"]\r\n\r\nIn words, each [latex]x[\/latex]-value is mapped to its square ([latex]y[\/latex]-value).\r\n\r\nAs an equation: [latex]y=x^2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nGraph the relation [latex]y=-x+3[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nThis relation is given in the form of a linear equation. To graph an equation we can use a table of values. Choose any [latex]x[\/latex]-value and find the corresponding [latex]y[\/latex]-value, then plot the ordered pairs.\r\n<table style=\"border-collapse: collapse; width: 3.20261%; height: 149px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 10.8696%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 179.959%; text-align: center;\">[latex]y=-x+3[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.8696%; text-align: center;\">-2<\/td>\r\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(-2)+3=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.8696%; text-align: center;\">-1<\/td>\r\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(-1)+3=4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.8696%; text-align: center;\">0<\/td>\r\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(0)+3=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.8696%; text-align: center;\">1<\/td>\r\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(1)+3=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.8696%; text-align: center;\">2<\/td>\r\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(2)+3=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter wp-image-713 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-300x300.png\" alt=\"Graph of y=-x+3.  A point for each row of the table above with a solid line drawn through them.\" width=\"300\" height=\"300\" \/>\r\n<p style=\"text-align: left;\">The equation [latex]y=-x+3[\/latex] had no limitations on the values of [latex]x[\/latex] so we can assume that it includes all real numbers. This means that we can join the dots to form a line that continues forever in both directions.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nUse the graph to create a mapping from the set of real numbers to the set of real numbers.\r\n\r\n<img class=\"aligncenter wp-image-4457 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-300x300.png\" alt=\"A smooth curve through the points (-3,14), (-2,-7), (-1,0), (0,5), (1,2), and (2,9). Use the points to make a mapping.\" width=\"300\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"hjm913\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm913\"]\r\n\r\n<img class=\"aligncenter wp-image-715 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-300x210.png\" alt=\"Mapping using points from the graph above.\" width=\"300\" height=\"210\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Domain and Range<\/h2>\r\nThe word <em><strong>domain<\/strong><\/em> is used to refer to the set of elements of the source in a relation. The word <em><strong>range<\/strong><\/em> is used to refer to the set of elements of the target in a relation. For example, the domain in the relation between two whole numbers where\u00a0the second number is double the first number is the set {0, 1, 2, 3, 4....}. The range in this relation is the set {0, 2, 4, 6, 8,...}.\r\n\r\nWhen the relation is given as a set of ordered pairs, where the first element in the pair is mapped to the second element in the pair, the first element in the pair belongs to the domain, while the second element belongs to the range. For example, the pair (3, 5) means the element 3 in the domain is mapped to the element 5 in the range. For the set of ordered pairs {(1, 2), (4, 8), (5, -3), (4, 6)}, the domain is the set {1, 4, 5}, while the range is the set {2, 8, -3, 6}.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nState the domain and the range of the relation {(-3, 2), (0, 3), (4, 2), (-3, 0)}.\r\n\r\n[reveal-answer q=\"hjm651\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm651\"]\r\n\r\nDomain = {-3, 0, 4}\r\n\r\nRange = (0, 2, 3}\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nEvery graph on a coordinate plane is composed of points which are ordered pairs. Therefore, the domain of the graph will be the set of the [latex]x[\/latex]-coordinates of all the points on the graph and the range will be the set of the [latex]y[\/latex]-coordinates of all the points on the graph. However, depending on the graph, it may be impossible to list all the [latex]x[\/latex]-coordinates for the domain and all the [latex]y[\/latex]-coordinates for the range (see figure 1). In these situations, there are alternative ways to help us describe the domain and range.\r\n\r\n[caption id=\"attachment_659\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-659 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-300x300.png\" alt=\"A smooth continuous curve representing a function. This graph hits all x values between negative infinity and positive infinity, and all y values above -2.\" width=\"300\" height=\"300\" \/> Figure 1.[\/caption]\r\n<h3>Interval Notation:<\/h3>\r\nSince the domain of a graph is a set of [latex]x[\/latex]-values, we describe the domain of a graph using an interval. Basically, the interval describes the coverage of the graph along a number line. The covered area of the [latex]x[\/latex]-coordinates of the graph in figure 1 is from negative infinity to positive infinity. Therefore, the domain is\r\n<p style=\"text-align: center;\">[latex](-\\infty, +\\infty)[\/latex]<\/p>\r\nWe can do the same with the [latex]y[\/latex]-coordinates of the graph to get the range. The covered area of the [latex]y[\/latex]-coordinates is from the lowest value of\u00a0[latex]y[\/latex] where the graph turns at \u20132 to positive infinity. Therefore, the range is\r\n<p style=\"text-align: center;\">[latex][-2,+\\infty)[\/latex]<\/p>\r\nIn interval notation a parenthesis ( or ) means the number next to it is not included. On the other hand, a bracket [ or ] means the number next to it is included.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 8<\/h3>\r\nWrite the domain and range of the graphed relation in interval notation:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-706 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-300x300.png\" alt=\"Graph of upside down parabola reaching a high point at (1,1).\" width=\"300\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm475\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm475\"]\r\n\r\nDomain = [latex](-\\infty, \\infty)[\/latex]\r\n\r\nRange = [latex](-\\infty, 1][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 9<\/h3>\r\nWrite the domain and range of the graphed relation, in interval notation:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter size-medium wp-image-717\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/10202809\/circle-radius-6-300x300.png\" alt=\"Graph of a circle centered at the origin with a radius of 6\" width=\"300\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm993\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm993\"]\r\n\r\nDomain = [latex][-6, 6][\/latex]\r\n\r\nRange = [latex][-6, 6][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Set-Builder Notation:<\/h3>\r\nAnother way to describe the domain and range is using set-builder notation. Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. Set-builder notation is composed of three elements, the braces (curly brackets), the variable being described, and the conditions on the variable. For example, { [latex]x[\/latex] | [latex]x\\in\\mathbb{R}\\text{ and }x &gt; 3[\/latex]\u00a0}. The braces enclose elements of a set. The vertical bar is read, \"such that\" and the condition on the variable is placed after the vertical bar. It describes and constrains the elements of the set. [latex]x\\in\\mathbb{R}\\text{ and }x &gt; 3[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span><span style=\"font-size: 1em; text-align: initial;\">tells us that the elements of the set are real numbers greater than 3.<\/span><span style=\"font-size: 1em; text-align: initial;\">\u00a0<\/span>\r\n\r\nThe set-builder notation for the domain of the graph in figure 1 is { [latex]x\\; | \\;\\;x\\in\\mathbb{R}[\/latex]}. The statement [latex]x\\in\\mathbb{R}[\/latex] is read [latex]x[\/latex] is a real number and is identical to all elements on the [latex]x[\/latex]-axis from negative infinity to positive infinity. The set notation for the range will be [latex] \\{y\\; |\\;\\;y\\in\\mathbb{R}\\text{ and }y \u2265 -2 \\}[\/latex].\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 10<\/h3>\r\nWrite the domain and range of the graphed relation in set-builder notation:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-706 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09212529\/y-x%5E22x-300x300.png\" alt=\"Graph of upside down parabola reaching a high point of (1,1)\" width=\"300\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm476\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm476\"]\r\n\r\nDomain = [latex]\\{x\\;|\\;x\\in\\mathbb{R}\\}[\/latex]\r\n\r\nRange = [latex]\\{y\\;|\\;y\\in\\mathbb{R}\\text{ and }y\\leq 1\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 11<\/h3>\r\nWrite the domain and range of the graphed relation in set-builder notation:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-719 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-300x300.png\" alt=\"graph of an ellipse (an horizontally elongated circle). Four important points are (-3,-2), (9,-2), (3,1), and (3,-5).\" width=\"300\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm233\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm233\"]\r\n\r\nDomain = [latex]\\{x\\;|\\;x\\in\\mathbb{R}\\text{ and } -3 \\leq x \\leq 9 \\}[\/latex]\r\n\r\nRange = [latex]\\{y\\;|\\;y\\in\\mathbb{R}\\text{ and } -5 \\leq y \\leq 1 \\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Define a relation using ordered pairs, a mapping, a graph, and an equation<\/li>\n<li>Determine the domain and range of a relation<\/li>\n<li>Write the domain and range in interval notation<\/li>\n<li>Write the domain and range in set-builder notation<\/li>\n<\/ul>\n<\/div>\n<h2>Parentheses, Brackets, and Braces<\/h2>\n<p><em><strong>Parentheses<\/strong><\/em> ( ), <em><strong>brackets<\/strong><\/em> [ ], and <em><strong>braces<\/strong><\/em> { } play an important role in mathematics and have several different uses. Parentheses can be used to denote multiplication: [latex]3(4) = 3\\cdot 4=12[\/latex]. They can also be used along with brackets and braces to denote grouping and order of operations: [latex]\\{3+[2-(3x-1)]\\}[\/latex]. Parentheses are also used to denote ordered pairs. An <em><strong>ordered pair<\/strong><\/em> consists of two numbers (or more generally two mathematical objects) enclosed by a pair of parentheses, where the order of the numbers in the pair of parentheses matters.\u00a0An ordered pair has many meanings in mathematics.<\/p>\n<h3>Coordinates<\/h3>\n<p>An ordered pair may refer to the <em><strong>coordinates<\/strong><\/em> of a point on a coordinate plane. A <em><strong>rectangular coordinate plane<\/strong><\/em> (or <em><strong>Cartesian plane<\/strong><\/em>) is composed of two <em><strong>number lines<\/strong><\/em> or two <em><strong>axes<\/strong><\/em> and the plane that they form. The horizontal number line is called the [latex]x[\/latex]-axis and the vertical number line is called the [latex]y[\/latex]-axis. The two axes cross at a point where both the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-values are zero, called the <em><strong>origin<\/strong><\/em>. The two axes form a grid, called a rectangular\u00a0coordinate plane, and each intersection on the grid is a <em><strong>point <\/strong><\/em>(Figure 1). The points on the grid are not only integers. There are an infinite number of real number intersections that make up the grid.<\/p>\n<div id=\"attachment_4206\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4206\" class=\"wp-image-4206 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-300x300.png\" alt=\"Graph showing the quadrants and axis' described above.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/08\/desmos-graph-2022-08-22T224600.0011.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-4206\" class=\"wp-caption-text\">Figure 1. Rectangular coordinate plane<\/p>\n<\/div>\n<p>There are an infinite number of points on the coordinate plane. The location of a point may be specified using the values on the [latex]x[\/latex] and [latex]y[\/latex]-axis. The ordered pair (3, \u20132) refers to a point on the coordinate plane where the location is 3 relative to the horizontal number line (or [latex]x[\/latex]-axis) and \u20132 relative to the vertical number line (or [latex]y[\/latex]-axis). We call 3 the [latex]x[\/latex]-coordinate and \u20132 the\u00a0[latex]y[\/latex]-coordinate. Therefore, the ordered pair\u00a0[latex](3, -2)[\/latex] may represent a pair of coordinates on the coordinate plane (Figure 2).<\/p>\n<div id=\"attachment_3191\" style=\"width: 360px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3191\" class=\"wp-image-3191\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/06230146\/1-1-1-OrderedPair-300x300.png\" alt=\"The point (3,-2) on the coordinate plane described above.\" width=\"350\" height=\"350\" \/><\/p>\n<p id=\"caption-attachment-3191\" class=\"wp-caption-text\">Figure 2. The point (3, \u20132) on the coordinate plane<\/p>\n<\/div>\n<p>Any point on the coordinate plane [latex](x, y)[\/latex] is an ordered pair because it is two numbers (a pair) written in the specific order of [latex]x[\/latex]-coordinate first, [latex]y[\/latex]-coordinate second.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3359 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-300x300.png\" alt=\"A graph showing the points described below.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T134733.520.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>Point [latex]A[\/latex] is right above [latex]3[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]2[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]A=(3, 2)[\/latex].<\/p>\n<p>Point [latex]B[\/latex] lies on [latex]4[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]0[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]B=(4, 0)[\/latex].<\/p>\n<p>Point [latex]C[\/latex] is right below [latex]0[\/latex] on the [latex]x[\/latex]-axis and lies on [latex]-2[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]C=(0, -2)[\/latex].<\/p>\n<p>Point [latex]D[\/latex] is right below [latex]-4[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]-3[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]D=(-4, -3)[\/latex].<\/p>\n<p>Point [latex]E[\/latex] is right above [latex]-2[\/latex] on the [latex]x[\/latex]-axis and across from\u00a0[latex]1[\/latex] on the [latex]y[\/latex]-axis. Therefore, [latex]E=(-2, 1)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>State the coordinates of the points on the graph:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3358 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-300x300.png\" alt=\"6 points on a coordinate plane. A is 3 units right and four units up, B is 7 right and 3 down, C is two down, D is 4 left and 4 down, E is 8 left, and F is 2 left and 7 down.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-20T133850.248.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm305\">Show Answer<\/span><\/p>\n<div id=\"qhjm305\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]A=(3,4),\\;B=(7,-3),\\;C=(0,-2),\\;D=(-4,-4),\\;E=(-8,0),\\;F=(-2,7)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Intervals<\/h3>\n<p>An ordered pair may also refer to an <em><strong>interval<\/strong> <\/em>with a starting value and an ending value. The pair of numbers are ordered with the smaller number written first. For example, [latex](-1, 3)[\/latex] means an interval on a number line starting at \u20131 and ending at 3. The parentheses means both of the two numbers \u20131 and 3 are not included in the interval, but are <em><strong>boundaries<\/strong><\/em> of the interval. Figure 3 shows this interval on a number line. We use an empty circle to show that an end point is <span style=\"text-decoration: underline;\">not<\/span> included in an interval. Since the two end points \u20131 and 3 are not included in the interval, they are both empty circles on the number line. Since this interval is for [latex]x[\/latex], it is equivalent to the inequality [latex]1 < x < 3[\/latex].\n\n\n\n<div id=\"attachment_3371\" style=\"width: 257px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3371\" class=\"wp-image-3371 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/1-3.png\" alt=\"interval between -1 and 3 described above.\" width=\"247\" height=\"39\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/1-3.png 247w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/1-3-65x10.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/1-3-225x36.png 225w\" sizes=\"auto, (max-width: 247px) 100vw, 247px\" \/><\/p>\n<p id=\"caption-attachment-3371\" class=\"wp-caption-text\">Figure 3. Interval (\u20131, 3)<\/p>\n<\/div>\n<p>[latex](-2, 1][\/latex] represents an interval on a number line starting at \u20132 and ending at 1. The parentheses next to \u20132 tells us that \u20132 is not included in the interval, while the bracket next to 1 tells us that 1 is included in the interval. Figure 4 shows the interval on a number line. We use a solid circle to show that an end point is included in an interval. This interval is equivalent to [latex]-2 < x \u2264 1[\/latex].\n\n\n\n<div id=\"attachment_3372\" style=\"width: 257px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3372\" class=\"wp-image-3372 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/22003726\/2-1.png\" alt=\"Interval (-2, 1] described above.\" width=\"247\" height=\"39\" \/><\/p>\n<p id=\"caption-attachment-3372\" class=\"wp-caption-text\">Figure 4. Interval (\u20132, 1]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Write the interval.<\/p>\n<ol>\n<li>All values between [latex]-1[\/latex] and [latex]4[\/latex].<\/li>\n<li>All values between [latex]-1[\/latex] and [latex]4[\/latex], including [latex]4[\/latex].<\/li>\n<li>All values between [latex]-1[\/latex] and [latex]4[\/latex], including [latex]-1[\/latex] and [latex]4[\/latex].<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]-1[\/latex] and [latex]4[\/latex] are not included so we use parentheses: [latex](-1,4)[\/latex]<\/li>\n<li>[latex]-1[\/latex] is not included so we use parentheses on that end, and [latex]4[\/latex] is included so we use a bracket on that end: [latex](-1,4][\/latex]<\/li>\n<li>\u00a0Both\u00a0[latex]-1[\/latex] and [latex]4[\/latex] are included so we use brackets: [latex][-1,4][\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Write the interval.<\/p>\n<ol>\n<li>\u00a0All values between [latex]-7[\/latex] and [latex]5[\/latex].<\/li>\n<li>\u00a0All values between [latex]-7[\/latex] and [latex]5[\/latex], including [latex]-7[\/latex].<\/li>\n<li>\u00a0All values between [latex]-7[\/latex] and [latex]5[\/latex], including [latex]-7[\/latex] and [latex]5[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm595\">Show Answer<\/span><\/p>\n<div id=\"qhjm595\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex](-7,5)[\/latex]<\/li>\n<li>[latex][-7,5)[\/latex]<\/li>\n<li>[latex][-7,5][\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Sketch the interval on a number line.<\/p>\n<ol>\n<li>\u00a0(\u20133, 2)<\/li>\n<li>\u00a0[4, 7]<\/li>\n<li>\u00a0[latex](-\\infty,0][\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<p>1. The parentheses tell us that neither \u20133 nor 2 are included in the interval, so we use open circles on both ends:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3363 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/32.png\" alt=\"Interval with a line connecting an open circle on -3 and an open circle on 2\" width=\"247\" height=\"39\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/32.png 247w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/32-65x10.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/32-225x36.png 225w\" sizes=\"auto, (max-width: 247px) 100vw, 247px\" \/><\/p>\n<p>2. The brackets tell us that 4 and 7 are included in the interval, so we use closed circles on both ends:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3364 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/4-7.png\" alt=\"[4,7] interval. A line connecting a filled in circle at 4 and a filled in circle at 7.\" width=\"247\" height=\"39\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/4-7.png 247w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/4-7-65x10.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/4-7-225x36.png 225w\" sizes=\"auto, (max-width: 247px) 100vw, 247px\" \/><\/p>\n<p>3. Negative infinity can never be reached so it has a parenthesis to show that it is not included in the interval. We will use an arrow on that end of the line to show that it goes on forever. The bracket next to 0 tells us that 0 is included in the interval, so we will use a closed circle on that end:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4101 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/inft0.png\" alt=\"A ray starting at zero and moving to the left forever. An arrow at the end of the line makes it a ray.\" width=\"262\" height=\"53\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/inft0.png 262w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/inft0-65x13.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/inft0-225x46.png 225w\" sizes=\"auto, (max-width: 262px) 100vw, 262px\" \/><\/p>\n<p style=\"text-align: center;\">\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Write the interval shown on the number line.<\/p>\n<p>1.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4102\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/01201235\/infty-5.png\" alt=\"line starting at 5 and moving left with arrow on left. Or a ray starting at 5 and moving left.\" width=\"247\" height=\"50\" \/><\/p>\n<p>2.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3367 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/31.png\" alt=\"A line segment joining a closed circle at -3 with a closed circle at 1.\" width=\"247\" height=\"39\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/31.png 247w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/31-65x10.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/31-225x36.png 225w\" sizes=\"auto, (max-width: 247px) 100vw, 247px\" \/><\/p>\n<p>3.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3368 size-full\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/49.png\" alt=\"interval between 4 and 9. A line segment joining an open circle at 4 to an open circle at 9.\" width=\"247\" height=\"39\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/49.png 247w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/49-65x10.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/49-225x36.png 225w\" sizes=\"auto, (max-width: 247px) 100vw, 247px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm282\">Show Answer<\/span><\/p>\n<div id=\"qhjm282\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex](-\\infty,5][\/latex]<\/li>\n<li>[\u20133, 1]<\/li>\n<li>(4, 9)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h3>Input \u2013 Output<\/h3>\n<p>An ordered pair may also refer to a relation or mapping where the first object in the pair is a source or input in a mapping and the second object in the pair is a target or output in a mapping. For example, we may define the relation <em>double the input value to get the output value<\/em> using the ordered pair idea where the second number is double the first number. Therefore, [latex](1, 2), (2, 4), (3, 6)...[\/latex] are specific examples of the relation. In general, the relation is [latex](x, 2x)[\/latex]. We will explore this further under <strong>Relations<\/strong>.<\/p>\n<h3>Sets<\/h3>\n<p>Braces { } are used when we talk about sets. A <em><strong>set<\/strong><\/em> is simply a group or collection of objects. We use braces { } to enclose the group of objects. Each object in a set is called an\u00a0<em><strong>element<\/strong><\/em>. For example, {blue, red, yellow} is the set of primary colors. Blue, red, and yellow are the elements of the set. The set {2, 4, 6, 8, &#8230;} is the set of even whole numbers. The set {(1, 2), (3, 8), (4, 9)} is a set of three ordered pairs. A set that contains no elements, [latex]\\{\\;\\}[\/latex], is called the <em><strong>empty set<\/strong> <\/em>and is notated [latex]\\varnothing[\/latex]. We will look further at sets in <strong>set builder notation<\/strong>.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<ol>\n<li>\u00a0Write the set of all integers between \u20132 and 5.<\/li>\n<li>\u00a0How many elements are in the set of currently printed US currency notes?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>The integers between \u20132 and 5 are {\u20131, 0, 1, 2, 3, 4}.<\/li>\n<li>The set of currently printed US currency notes is {$1, $5, $10, $20, $50, $100} so there are 6 elements in this set.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<ol>\n<li>\u00a0Write the set of US currency coins intended for circulation.<\/li>\n<li>\u00a0How many elements are in the set of proper fractions between 0 and 1?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm071\">Show Answer<\/span><\/p>\n<div id=\"qhjm071\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\{1\u00a2,\\;5\u00a2,\\;10\u00a2,\\;25\u00a2\\}[\/latex]<\/li>\n<li>There are an infinite number of proper fractions between zero and 1.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Relations<\/h2>\n<p>A <em><strong>relation<\/strong><\/em> is a way in which two or more objects are connected. For example, every person has a date of birth, so there is a relation between the person and their date of birth. Indeed, there is a relation between the set of all people and the set of all birth dates. Every perfect square has a whole number square root, so there is a relation between a perfect square and its whole number square root. Further, there is a relation between the set of perfect squares and the set of whole number square roots.<\/p>\n<p>Mathematically, a <em><strong>binary relation<\/strong><\/em> is defined as a connection between <em>two<\/em> sets of objects. That relation can be represented in multiple ways: a set of ordered pairs; a mapping; a graph; or an equation.<\/p>\n<p>For example, the relation between two whole numbers where\u00a0the second number is double the first number, may be represented in different ways:<\/p>\n<ul>\n<li>A <em><strong>set of ordered pairs<\/strong><\/em> shows the first number followed by the second number, or the source followed by the target: {(0, 0), (1, 2), (2, 4), (3, 6), (4, 8),&#8230;}. The ellipses &#8230; show that the established pattern continues forever.<\/li>\n<\/ul>\n<ul>\n<li>A <em><strong>mapping<\/strong><\/em> shows the association of the first numbers (the set of whole numbers) with an arrow that points to the corresponding second number (double the first number):<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-704 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-300x226.png\" alt=\"mapping of doubles. A mapping of the ordered pairs described above using arrows from the domain (x) value to the range (y) value.\" width=\"300\" height=\"226\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-300x226.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-768x579.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-1024x772.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-65x49.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-225x170.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles-350x264.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/mapping-doubles.png 1058w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<ul>\n<li>A <em><strong>graph<\/strong><\/em> shows the set of ordered pairs plotted on the rectangular (Cartesian) coordinate plane. If the first number is represented by [latex]x[\/latex] and the second number by [latex]y[\/latex], the graph shows the plotted points [latex](x, y)[\/latex]. Obviously, the graph does not show all possible coordinate pairs as the whole numbers start at zero and go to infinity. Notice also, that the plotted pairs are not joined by a line, as the line that joins them represents real numbers, not whole numbers.<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-703 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-300x300.png\" alt=\"A cartesian coordinate plane containing the points contained in the mapping above.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-black.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<ul>\n<li>An <em><strong>equation<\/strong><\/em>: If the first number is represented by [latex]x[\/latex] and the second number by [latex]y[\/latex], then since [latex]y[\/latex] is double the value of [latex]x[\/latex], the relation can be represented by the equation [latex]y = 2x[\/latex] where [latex]x[\/latex] is a whole number.<\/li>\n<\/ul>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>The relation given by the set of ordered pairs {(Juan, Brown), (Ian, Blue), (Bailey, Brown), (Kirsty, Hazel), (Jean, Grey), (Freddy, Blue)} is the eye color of each named person. Create a mapping to show this relation.<\/p>\n<h4>Solution<\/h4>\n<p>We start by listing the elements of each set inside a border, then draw arrows from the person to their respective eye color.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-701 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-300x224.png\" alt=\"Mapping of the eye color ordered pairs described above with the domain the persons name, and the range the persons eye color.\" width=\"300\" height=\"224\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-300x224.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-768x574.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-1024x766.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-65x49.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-225x168.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color-350x262.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-eye-color.png 1166w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Notice that there are more people than colors, since some people share the same eye color.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>What relation is shown in the mapping?<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-702 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-300x237.png\" alt=\"Mapping from the domain { -3, 3, 2.5, -two thirds, square root of 3} to the range { 9, 6.25. four ninths, 3} representing the rule &quot;square the domain value&quot;.\" width=\"300\" height=\"237\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-300x237.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-768x606.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-1024x808.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-65x51.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-225x178.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares-350x276.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-squares.png 1034w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm887\">Show Answer<\/span><\/p>\n<div id=\"qhjm887\" class=\"hidden-answer\" style=\"display: none\">\n<p>In words, each [latex]x[\/latex]-value is mapped to its square ([latex]y[\/latex]-value).<\/p>\n<p>As an equation: [latex]y=x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Graph the relation [latex]y=-x+3[\/latex]<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>This relation is given in the form of a linear equation. To graph an equation we can use a table of values. Choose any [latex]x[\/latex]-value and find the corresponding [latex]y[\/latex]-value, then plot the ordered pairs.<\/p>\n<table style=\"border-collapse: collapse; width: 3.20261%; height: 149px;\">\n<tbody>\n<tr>\n<th style=\"width: 10.8696%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 179.959%; text-align: center;\">[latex]y=-x+3[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 10.8696%; text-align: center;\">-2<\/td>\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(-2)+3=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.8696%; text-align: center;\">-1<\/td>\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(-1)+3=4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.8696%; text-align: center;\">0<\/td>\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(0)+3=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.8696%; text-align: center;\">1<\/td>\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(1)+3=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.8696%; text-align: center;\">2<\/td>\n<td style=\"width: 179.959%; text-align: center;\">[latex]y=-(2)+3=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-713 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-300x300.png\" alt=\"Graph of y=-x+3.  A point for each row of the table above with a solid line drawn through them.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x3.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: left;\">The equation [latex]y=-x+3[\/latex] had no limitations on the values of [latex]x[\/latex] so we can assume that it includes all real numbers. This means that we can join the dots to form a line that continues forever in both directions.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Use the graph to create a mapping from the set of real numbers to the set of real numbers.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4457 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-300x300.png\" alt=\"A smooth curve through the points (-3,14), (-2,-7), (-1,0), (0,5), (1,2), and (2,9). Use the points to make a mapping.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-768x765.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-1024x1020.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-225x224.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.1.1-Pracice2-350x349.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm913\">Show Answer<\/span><\/p>\n<div id=\"qhjm913\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-715 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-300x210.png\" alt=\"Mapping using points from the graph above.\" width=\"300\" height=\"210\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-300x210.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-768x538.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-1024x717.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-65x46.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-225x158.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping-350x245.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Mapping.png 1162w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Domain and Range<\/h2>\n<p>The word <em><strong>domain<\/strong><\/em> is used to refer to the set of elements of the source in a relation. The word <em><strong>range<\/strong><\/em> is used to refer to the set of elements of the target in a relation. For example, the domain in the relation between two whole numbers where\u00a0the second number is double the first number is the set {0, 1, 2, 3, 4&#8230;.}. The range in this relation is the set {0, 2, 4, 6, 8,&#8230;}.<\/p>\n<p>When the relation is given as a set of ordered pairs, where the first element in the pair is mapped to the second element in the pair, the first element in the pair belongs to the domain, while the second element belongs to the range. For example, the pair (3, 5) means the element 3 in the domain is mapped to the element 5 in the range. For the set of ordered pairs {(1, 2), (4, 8), (5, -3), (4, 6)}, the domain is the set {1, 4, 5}, while the range is the set {2, 8, -3, 6}.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>State the domain and the range of the relation {(-3, 2), (0, 3), (4, 2), (-3, 0)}.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm651\">Show Answer<\/span><\/p>\n<div id=\"qhjm651\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = {-3, 0, 4}<\/p>\n<p>Range = (0, 2, 3}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Every graph on a coordinate plane is composed of points which are ordered pairs. Therefore, the domain of the graph will be the set of the [latex]x[\/latex]-coordinates of all the points on the graph and the range will be the set of the [latex]y[\/latex]-coordinates of all the points on the graph. However, depending on the graph, it may be impossible to list all the [latex]x[\/latex]-coordinates for the domain and all the [latex]y[\/latex]-coordinates for the range (see figure 1). In these situations, there are alternative ways to help us describe the domain and range.<\/p>\n<div id=\"attachment_659\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-659\" class=\"wp-image-659 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-300x300.png\" alt=\"A smooth continuous curve representing a function. This graph hits all x values between negative infinity and positive infinity, and all y values above -2.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Parabola.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-659\" class=\"wp-caption-text\">Figure 1.<\/p>\n<\/div>\n<h3>Interval Notation:<\/h3>\n<p>Since the domain of a graph is a set of [latex]x[\/latex]-values, we describe the domain of a graph using an interval. Basically, the interval describes the coverage of the graph along a number line. The covered area of the [latex]x[\/latex]-coordinates of the graph in figure 1 is from negative infinity to positive infinity. Therefore, the domain is<\/p>\n<p style=\"text-align: center;\">[latex](-\\infty, +\\infty)[\/latex]<\/p>\n<p>We can do the same with the [latex]y[\/latex]-coordinates of the graph to get the range. The covered area of the [latex]y[\/latex]-coordinates is from the lowest value of\u00a0[latex]y[\/latex] where the graph turns at \u20132 to positive infinity. Therefore, the range is<\/p>\n<p style=\"text-align: center;\">[latex][-2,+\\infty)[\/latex]<\/p>\n<p>In interval notation a parenthesis ( or ) means the number next to it is not included. On the other hand, a bracket [ or ] means the number next to it is included.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 8<\/h3>\n<p>Write the domain and range of the graphed relation in interval notation:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-706 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-300x300.png\" alt=\"Graph of upside down parabola reaching a high point at (1,1).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y-x^22x.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm475\">Show Answer<\/span><\/p>\n<div id=\"qhjm475\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex](-\\infty, \\infty)[\/latex]<\/p>\n<p>Range = [latex](-\\infty, 1][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 9<\/h3>\n<p>Write the domain and range of the graphed relation, in interval notation:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-717\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/10202809\/circle-radius-6-300x300.png\" alt=\"Graph of a circle centered at the origin with a radius of 6\" width=\"300\" height=\"300\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm993\">Show Answer<\/span><\/p>\n<div id=\"qhjm993\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex][-6, 6][\/latex]<\/p>\n<p>Range = [latex][-6, 6][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Set-Builder Notation:<\/h3>\n<p>Another way to describe the domain and range is using set-builder notation. Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. Set-builder notation is composed of three elements, the braces (curly brackets), the variable being described, and the conditions on the variable. For example, { [latex]x[\/latex] | [latex]x\\in\\mathbb{R}\\text{ and }x > 3[\/latex]\u00a0}. The braces enclose elements of a set. The vertical bar is read, &#8220;such that&#8221; and the condition on the variable is placed after the vertical bar. It describes and constrains the elements of the set. [latex]x\\in\\mathbb{R}\\text{ and }x > 3[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span><span style=\"font-size: 1em; text-align: initial;\">tells us that the elements of the set are real numbers greater than 3.<\/span><span style=\"font-size: 1em; text-align: initial;\">\u00a0<\/span><\/p>\n<p>The set-builder notation for the domain of the graph in figure 1 is { [latex]x\\; | \\;\\;x\\in\\mathbb{R}[\/latex]}. The statement [latex]x\\in\\mathbb{R}[\/latex] is read [latex]x[\/latex] is a real number and is identical to all elements on the [latex]x[\/latex]-axis from negative infinity to positive infinity. The set notation for the range will be [latex]\\{y\\; |\\;\\;y\\in\\mathbb{R}\\text{ and }y \u2265 -2 \\}[\/latex].<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 10<\/h3>\n<p>Write the domain and range of the graphed relation in set-builder notation:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-706 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09212529\/y-x%5E22x-300x300.png\" alt=\"Graph of upside down parabola reaching a high point of (1,1)\" width=\"300\" height=\"300\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm476\">Show Answer<\/span><\/p>\n<div id=\"qhjm476\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex]\\{x\\;|\\;x\\in\\mathbb{R}\\}[\/latex]<\/p>\n<p>Range = [latex]\\{y\\;|\\;y\\in\\mathbb{R}\\text{ and }y\\leq 1\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 11<\/h3>\n<p>Write the domain and range of the graphed relation in set-builder notation:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-719 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-300x300.png\" alt=\"graph of an ellipse (an horizontally elongated circle). Four important points are (-3,-2), (9,-2), (3,1), and (3,-5).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/ellipse.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm233\">Show Answer<\/span><\/p>\n<div id=\"qhjm233\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex]\\{x\\;|\\;x\\in\\mathbb{R}\\text{ and } -3 \\leq x \\leq 9 \\}[\/latex]<\/p>\n<p>Range = [latex]\\{y\\;|\\;y\\in\\mathbb{R}\\text{ and } -5 \\leq y \\leq 1 \\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/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-4127\">\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>Relations and Functions. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All Examples and Try Its. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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":422608,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Relations and Functions\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All Examples and Try Its\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4127","chapter","type-chapter","status-publish","hentry"],"part":4124,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4127","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":20,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4127\/revisions"}],"predecessor-version":[{"id":4865,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4127\/revisions\/4865"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/4124"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/4127\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=4127"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4127"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=4127"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=4127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}