{"id":27,"date":"2023-11-08T13:27:52","date_gmt":"2023-11-08T13:27:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/defining-function\/"},"modified":"2026-02-05T09:25:42","modified_gmt":"2026-02-05T09:25:42","slug":"1-2-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/1-2-functions\/","title":{"raw":"1.2 Functions","rendered":"1.2 Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>To Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine if a graph is the graph of a function by using the Vertical Line Test.<\/li>\r\n \t<li>Find the value of a function from a graph.<\/li>\r\n \t<li>State the domain and range from the graph of a function in set-builder, inequality, and\/or interval notation.<\/li>\r\n \t<li>Evaluate functions at a given input value using an equation. Also, determine inputs resulting in the given output value<\/li>\r\n \t<li>Using the equation of a function, state the domain in set-builder, inequality, and\/or interval notation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Functions<\/h2>\r\nYou should be familiar with functions from previous math courses. Here, we'll do a quick review. Some people think of functions as \u201cmathematical machines.\u201d Imagine you have a machine that changes a number according to a specific rule such as \u201cmultiply by\u00a0[latex]3[\/latex] and add\u00a0[latex]2[\/latex]\u201d or \u201cdivide by\u00a0[latex]5[\/latex], add\u00a0[latex]25[\/latex], and multiply by [latex]\u22121[\/latex].\u201d If you put a number into the machine, a new number will come out the other end having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.\r\n\r\nYou can also call the machine \u201c[latex]f[\/latex]<i>\u201d <\/i>for function. If you put [latex]x[\/latex]<i> <\/i>into the machine, [latex]f(x)[\/latex] (read \"[latex]f[\/latex] of [latex]x[\/latex]\")<i>, <\/i>comes out. Mathematically speaking, [latex]x[\/latex] is the input, or the <strong>independent variable<\/strong>, and [latex]f(x)[\/latex] is the output, or the <strong>dependent variable<\/strong>, since it depends on the value of [latex]x[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Relations and Functions<\/h3>\r\nIn mathematics, a <strong>relation<\/strong> is a set of ordered pairs. The set of first components of the ordered pairs is called the <strong>domain<\/strong> and the set of second components of the ordered pairs is called the <strong>range<\/strong>.\u00a0In a set of ordered pairs, you find the domain by listing all of the input values (the [latex]x[\/latex]-coordinates). To find the range, list all of the output values (the [latex]y[\/latex]-coordinates).\r\n\r\nA <strong>function<\/strong> is a special type of relation in which each domain value corresponds to exactly one range value.\r\n\r\nIf you find that you need more help <a href=\"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-1-intro-functions\/\">understanding functions<\/a>, please click on the link to learn more about functions.\r\n\r\n<\/div>\r\nConsider the following relation: [latex]\\{(\u22122,0),(0,6),(-2,12),(4,18)\\}[\/latex].\r\n\r\nFor this relation, the domain is [latex]\\{\u22122,0,4\\}[\/latex] and the range is [latex]\\{0,6,12,18\\}[\/latex].\r\n\r\nIs the above relation a function or not?\r\n\r\nThis relation is NOT a function because the input values are not unique, in other words, there is a repeated [latex]x[\/latex]-value that corresponds to different [latex]y[\/latex]-values.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nList the domain and range for the following relation where [latex]x[\/latex] is the input and [latex]f(x)[\/latex] is the output and determine whether the relation given is a function. Remember each pair of [latex]x[\/latex] and [latex]f(x)[\/latex] exists as a point [latex](x,f(x))[\/latex] on the graph of the function .\r\n\r\n&nbsp;\r\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px; font-size: 110%;\">\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">\u00a0[latex]x[\/latex]<\/th>\r\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]\u22123[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]\u22122[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]\u22121[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"594198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594198\"]\r\n\r\nThe domain describes all the inputs, and we can use set notation with brackets { } to make the list. The domain is [latex]\\{-3,-2,-1,2,3\\}[\/latex]. The range describes all the outputs. The range is [latex]\\{1,2,4\\}[\/latex]. Notice that [latex]4[\/latex] is only listed once in the list of elements in the range.\r\n\r\nThis relation is a function because each unique input (no repeated [latex]x[\/latex] values) has a single output. Notice also that [latex]4[\/latex] is a repeated [latex]f(x)[\/latex]-value but that has no effect on whether it is a function or not.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we provide another example of identifying domain and range. Determine whether the table of values represents a function.\r\n\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/GPBq18fCEv4?si=Csbo-rmx3N_t6Ez3\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nIn the above video, each domain value corresponds to exactly one range value. No [latex]x[\/latex]-coordinates were repeated so that relation is a function.\r\n<h2>Function Notation<\/h2>\r\nThe notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex] or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable. When a function is defined by an equation, we usually use function notation.\r\n\r\nFor example, the equation [latex]y=3x+1[\/latex] may also be written as [latex]f(x)=3x+1[\/latex]. [latex]f[\/latex] is the name of the function, [latex]x[\/latex] is an input from the domain of the function, and [latex]f(x)[\/latex] is the function value at [latex]x[\/latex] also called the output.\r\n<h3>Inputs of Functions<\/h3>\r\nA function can be evaluated at different values of [latex]x[\/latex]. Given the function [latex]f(x)=3x+1[\/latex], find [latex]f(-2)[\/latex]. In other words, if [latex]x[\/latex] is [latex]-2[\/latex] find [latex]f(x)[\/latex]. We must substitute [latex]\\color{Green}{-2}[\/latex] in place of [latex]\\color{Green}{x}[\/latex] and evaluate.\r\n[latex]\\begin{align} f(\\color{Green}{-2}\\color{black}{)} &amp;= 3(\\color{Green}{-2}\\color{black}{)+1}\\\\ &amp;=-6+1\\\\ &amp;= -5 \\end{align}[\/latex]\r\n\r\nSo, when [latex]x=-2[\/latex], [latex]f(x)=-5[\/latex] (or [latex]y=-5[\/latex]). This can be written as the ordered pair [latex](-2,-5)[\/latex] and it is a point on the graph of the line [latex]f(x)=3x+1[\/latex].\r\n\r\nIt is important to note that the parentheses that are part of function notation <strong><span style=\"text-decoration: underline;\">do not mean multiply<\/span><\/strong>. The notation [latex]f(x)[\/latex] does not mean [latex]f[\/latex] multiplied by [latex]x[\/latex]. Instead, the notation means \u201c[latex]f\\text{ of } x[\/latex]\u201d or \u201c<em>the function of<\/em> [latex]x[\/latex]<i>.\"<\/i>\u00a0To evaluate the function, take the value given for [latex]x[\/latex]<i>,<\/i>\u00a0and substitute that value in for [latex]x[\/latex] in the expression.\r\n\r\nLet us look at a couple of examples.\r\n<div class=\"textbox exercises\">\r\n<h3>ExAMple<\/h3>\r\nGiven [latex]f(x)=3x\u20134[\/latex],\u00a0evaluate [latex]f(5)[\/latex].\r\n\r\n[reveal-answer q=\"42679\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42679\"]\r\n\r\nSubstitute\u00a0[latex]\\color{Green}{5}[\/latex] in for [latex]\\color{Green}{x}[\/latex]<i> <\/i>in the function. Then simplify the expression on the right side of the equation.\r\n\r\n[latex]\\begin{align} f(\\color{Green}{5}) &amp;= 3(\\color{Green}{5})-4\\\\ &amp;= 15-4\\\\ &amp;= 11 \\end{align}[\/latex]\r\n\r\n&nbsp;\r\n\r\nWe were given [latex]x=5[\/latex] and evaluated and found that [latex]f(5)=11[\/latex], which is the point [latex](5,11)[\/latex] on the graph of the function [latex]f(x)=3x-4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven [latex]p(x)=-2x^{2}+5[\/latex], evaluate [latex]p(\u22123)[\/latex].\r\n\r\n[reveal-answer q=\"489384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489384\"]\r\n\r\nSubstitute [latex]\\color{green}{-3}[\/latex] in for [latex]x[\/latex]<i> <\/i>in the function.\r\n\r\n[latex]\\begin{align} p(\\color{Green}{\u22123}) &amp;= -2(\\color{Green}{\u22123})^{2}+5\\\\ &amp;= -2(9)+5\\\\ &amp;=-18+5\\\\ &amp;= -13\\end{align}[\/latex]\r\n\r\nThis is the point [latex](-3,-13)[\/latex] on the graph of the function [latex]f(x)=-2x^2+5[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven [latex]f(x)=|4x-3|[\/latex], evaluate [latex]f(0)[\/latex], [latex]f(2)[\/latex], and [latex]f(\u22121)[\/latex].\r\n\r\n[reveal-answer q=\"971051\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971051\"]\r\n\r\nTreat these like three separate problems. In each case, you substitute the value in for [latex]x[\/latex] and simplify.\r\n\r\n<strong>Evaluate [latex]f(0)[\/latex]:<\/strong>\r\n\r\nStart with [latex]\\color{green}{x=0}[\/latex] and evaluate [latex]f(x)[\/latex] for [latex]\\color{green}{x=0}[\/latex].\r\n\r\n[latex]\\begin{align} f(\\color{green}{0}) &amp;= |4(\\color{green}{0})-3|\\\\ &amp;= |0-3|\\\\ &amp;= |-3|\\\\ &amp;= 3\\end{align}[\/latex]\r\n\r\n[latex]f(0)=3[\/latex]\r\n\r\n&nbsp;\r\n\r\n<strong>Evaluate [latex]f(2)[\/latex]:<\/strong>\r\n\r\nEvaluate [latex]f(x)[\/latex] for [latex]\\color{green}{x=2}[\/latex].\r\n\r\n[latex]\\begin{align}f(\\color{green}{2}) &amp;= |4(\\color{green}{2})-3|\\\\ &amp;= |8-3|\\\\ &amp;= |5|\\\\ &amp;= 5 \\end{align}[\/latex]\r\n\r\n[latex]f(2)=5[\/latex]\r\n\r\n&nbsp;\r\n\r\n<strong>Evaluate [latex]f(-1)[\/latex]:<\/strong>\r\n\r\nEvaluate [latex]f(x)[\/latex] for [latex]\\color{green}{x=\u22121}[\/latex].\r\n\r\n[latex]\\begin{align} f(\\color{green}{\u22121}) &amp;= |4(\\color{green}{-1})-3|\\\\ &amp;= |-4-3|\\\\ &amp;= |-7|\\\\ &amp;= 7 \\end{align}[\/latex]\r\n\r\n[latex]f(-1)=7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Outputs of Functions<\/h3>\r\nReconsider the equation [latex]y=4x+1[\/latex], which represents a function. If instead of having a given input value, you have a known output value, you can find the input value that would give that output. For example, for an output value of [latex]-7[\/latex], you can find the input value. First substitute [latex]\\color{Green}{y=-7}[\/latex] into the equation and then solve for [latex]x[\/latex].\r\n\r\n[latex]\\begin{align}\\color{Green}{y} &amp;= 4x+1\\\\ \\color{Green}{-7} &amp;= 4x+1\\\\ -7-1 &amp;= 4x\\\\ -8 &amp;= 4x\\\\ \\frac{-8}{4} &amp;= x\\\\ -2 &amp;= x\\\\ x &amp;= -2\\end{align}[\/latex]\r\n\r\nThe output of [latex]-7[\/latex] corresponds to an input of [latex]-2[\/latex]. These values make an ordered pair, [latex](-2, -7)[\/latex], that represents a point on the graph of this function. We will discuss this idea more fully later in this section.\r\n\r\nIn the next example, you will not be given the [latex]x[\/latex]-value but instead you will be given a [latex]y[\/latex]-value and asked to find the [latex]x[\/latex]-value.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven [latex]f(x)=3x\u20134[\/latex],\u00a0find the [latex]x[\/latex]-value that yields an output of [latex]5[\/latex].\r\n\r\n[reveal-answer q=\"42639\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42639\"]\r\n\r\nFirst write the equation with [latex]x[\/latex] and [latex]y[\/latex]\r\n\r\n[latex]y=3x-4[\/latex]\r\n\r\nThen substitute\u00a0[latex]\\color{green}{5}[\/latex] in for [latex]y[\/latex] in the equation.\r\n\r\n[latex]\\color{green}{5}=3x-4[\/latex]\r\n\r\nSolve for [latex]x[\/latex].\r\n\r\n[latex]\\begin{align}5&amp;=3x-4\\\\5+4&amp;=3x\\\\9&amp;=3x\\\\\\frac{9}{3}&amp;=x\\\\3&amp;=x\\\\x&amp;=3\\\\\\end{align}[\/latex]\r\n\r\nFor an output of [latex]5[\/latex], the input is [latex]3[\/latex].\r\n\r\nThese values make an ordered pair, [latex](3, 5)[\/latex], that represents a point on the graph of this function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]288721[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Graphs of Functions and the Vertical Line Test<\/h2>\r\nWhen both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the [latex]x[\/latex]-axis and the dependent value is plotted on the [latex]y[\/latex]-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as\u00a0[latex]y = f(x)[\/latex], or \"[latex]y[\/latex] is a function of\u00a0[latex]x[\/latex]\", we would write ordered pairs\u00a0[latex](x, f(x))[\/latex] using function notation instead of\u00a0[latex](x,y)[\/latex] as you may have seen previously.\r\n\r\nWe can identify whether the graph of a relation represents a function\u00a0because for each [latex]x[\/latex]-coordinate there will be exactly one [latex]y[\/latex]-coordinate. The blue relation graphed below represents a function. Notice that when a vertical line is placed over the graph, the line does not intersect the graph more than once for any value of [latex]x[\/latex]. This indicates that the relation is a function.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/>If instead a graph shows two or more intersections with a vertical line, then an input ([latex]x[\/latex]-coordinate) has more than one output ([latex]y[\/latex]-coordinate), and [latex]y[\/latex] is not a function of [latex]x[\/latex].\r\n\r\nExamining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the <strong>Vertical Line Test<\/strong>.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the vertical line test to determine whether the relation plotted on this graph is a function.\r\n\r\n<img class=\"size-medium wp-image-2680 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16200323\/Screen-Shot-2016-07-16-at-1.02.50-PM-300x275.png\" alt=\"Graph with circle plotted - center at (0,0) radius = 2, more points include (2,0), (-2,0)\" width=\"300\" height=\"275\" \/>\r\n[reveal-answer q=\"28965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"28965\"]\r\n\r\nThis relationship cannot be a function, because some of the [latex]x[\/latex]-coordinates have two corresponding [latex]y[\/latex]-coordinates.\r\n<p align=\"center\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232418\/image002.jpg\" alt=\"A circle with four vertical lines through it. Three of the lines cross the circle at two points, and one line crosses the edge of the circle at one point.\" width=\"340\" height=\"343\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Find the Value of a Function from a Graph<\/h2>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA function is graphed below.\r\n\r\n<img class=\"alignnone wp-image-1601 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-300x300.jpg\" alt=\"Graph of a diagonal line on a coordinate plane that goes through the origin (0,0). Additional points at (2,2) and (negative 2, negative 2). Labeled f of x.\" width=\"300\" height=\"300\" \/>\r\n<ol>\r\n \t<li>For the given function, evaluate [latex]f(2)[\/latex].<\/li>\r\n \t<li>For the given graph of a function, find [latex]x[\/latex] when [latex]f(x)=-2[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"719392\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"719392\"]\r\n<ol>\r\n \t<li>To evaluate [latex]f(2)[\/latex], go to [latex]x=2[\/latex] on the graph, then vertically to the graph of the function. The [latex]y[\/latex]-value at this point is [latex]2[\/latex]. Then [latex]f(2)=2[\/latex].<\/li>\r\n \t<li>To find [latex]x[\/latex] when [latex]f(x)=-2[\/latex] using the graph, go to [latex]y=-2[\/latex], then horizontally to the graph of the function. The [latex]x[\/latex]-value at this point is [latex]-2[\/latex]. Then [latex]f(x)=-2[\/latex] when [latex]x=-2[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>ExAmple<\/h3>\r\nA function is graphed below.\r\n\r\n<img class=\"alignnone wp-image-1606 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-300x273.jpg\" alt=\"A graph of an upward facing absolute function (v-shape) on a coordinate plane with the vertex at (0,0). Additional points shown at (negative 2,2) and (2,2).\" width=\"300\" height=\"273\" \/>\r\n<ol>\r\n \t<li>For the given function, evaluate [latex]f(-1)[\/latex].<\/li>\r\n \t<li>For the given function, evaluate [latex]f(0)[\/latex].<\/li>\r\n \t<li>For the given graph of a function, find [latex]x[\/latex] when [latex]f(x)=2[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"932196\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"932196\"]\r\n<ol>\r\n \t<li>On the graph, go to [latex]x=-1[\/latex], then vertically to the graph of the function. The [latex]y[\/latex]-value at this point is [latex]1[\/latex]. Then [latex]f(-1)=1[\/latex].<\/li>\r\n \t<li>On the graph, go to [latex]x=0[\/latex], then vertically to the graph of the function. The [latex]y[\/latex]-value at this point is [latex]0[\/latex]. Then [latex]f(0)=0[\/latex].<\/li>\r\n \t<li>On the graph, go to [latex]y=2[\/latex], then horizontally to the graph of the function. Notice that there are two [latex]x[\/latex]-values that have this associated [latex]y[\/latex]-value: [latex]x=-2[\/latex] and [latex]x=2[\/latex]. Then [latex]f(x)=2[\/latex] when [latex]x=-2[\/latex] or [latex]x=2[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]288724[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Finding the Domain and Range of a Function from its Graph<\/h2>\r\nFinding domain of different functions is often a matter of asking yourself, \"what values of [latex]x[\/latex] can this function <i>not<\/i>\u00a0have?\" Pictures make it easier to visualize what the domain and range are, so we will first show how to define the domain and range of functions given their graphs. What are the domain and range of the function [latex]f(x)=x+3[\/latex]?\r\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"322\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Graph of a diagonal line that goes through (0,3) and (negative 3,0).\" width=\"322\" height=\"353\" \/> Graph of the function [latex]f(x)=x+3[\/latex][\/caption]For this function, any real number can be substituted for [latex]x[\/latex] and you will get a meaningful output. Also, for any real number, you can always find an [latex]x[\/latex] value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range. For the function [latex]f(x)=x+3[\/latex], the domain and range are all real numbers. We can write this domain and range in a variety of formats.\r\n<table style=\"border-collapse: collapse; font-size: 110%; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.333333%;\"><\/td>\r\n<td style=\"width: 33.333333%;\"><strong>Domain<\/strong><\/td>\r\n<td style=\"width: 33.333333%;\"><strong>Range<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.333333%;\"><strong>Set-Builder Notation<\/strong><\/td>\r\n<td style=\"width: 33.333333%;\">[latex]\\{x|\\text{all real numbers}\\}[\/latex]<\/td>\r\n<td style=\"width: 33.333333%;\">[latex]\\{y|\\text{all real numbers}\\}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.333333%;\"><strong>Inequality Notation<\/strong><\/td>\r\n<td style=\"width: 33.333333%;\">[latex]-\\infty&lt;x&lt;\\infty[\/latex]<\/td>\r\n<td style=\"width: 33.333333%;\">[latex]-\\infty&lt;y&lt;\\infty[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.333333%;\"><strong>Interval Notation<\/strong><\/td>\r\n<td style=\"width: 33.333333%;\">[latex](-\\infty,\\infty)[\/latex]<\/td>\r\n<td style=\"width: 33.333333%;\">[latex](-\\infty,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nFor the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the graph to determine the domain and range of the function [latex]f(x)=\u22123x^{2}+6x+1.[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/>\r\n\r\n[reveal-answer q=\"223692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223692\"]\r\n\r\nExamining the graph, the arrows indicate that the function continues for larger and for smaller values of [latex]x[\/latex]. The domain is all real numbers.\r\n\r\nThe vertex, or high\u00a0point of this graph, is at [latex](1,4)[\/latex]. From the graph, you can see that all [latex]y[\/latex]-values of the function are less than or equal to [latex]4[\/latex]. In other words, [latex]f(x)\\leq4[\/latex]. The range of this function is all [latex]y[\/latex] values less than or equal to [latex]4[\/latex].\r\n\r\n&nbsp;\r\n<table style=\"height: 36px; font-size: 110%; width: 380px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 43px;\"><\/td>\r\n<td style=\"height: 12px; width: 140px;\">set-builder notation<\/td>\r\n<td style=\"height: 12px; width: 105px;\">inequality notation<\/td>\r\n<td style=\"height: 12px; width: 92px;\">interval notation<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 43px;\">domain<\/td>\r\n<td style=\"height: 12px; width: 140px;\">[latex]\\{x|\\text{all real numbers}\\}[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 105px;\">[latex]-\\infty&lt;x&lt;\\infty[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 92px;\">[latex](-\\infty,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 43px;\">range<\/td>\r\n<td style=\"height: 12px; width: 140px;\">[latex]\\{y|y\\le4\\}[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 105px;\">[latex]y\\le4[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 92px;\">[latex](-\\infty,4][\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nUse the graph to determine the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}.[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Square root curve stemming from negative 5, negative 2 and increasing slowly to the right.\" width=\"308\" height=\"346\" \/>\r\n\r\n[reveal-answer q=\"231228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231228\"]\r\n\r\nFrom the graph, we see that the smallest [latex]x[\/latex]-value is [latex]x=-5[\/latex]. Then the domain is all real numbers greater than or equal to [latex]-5[\/latex].\r\n\r\nThe smallest [latex]y[\/latex]-value on the graph of the function is [latex]y=-2[\/latex]. The range will be all real numbers greater than or equal to [latex]-2[\/latex].\r\n<table style=\"height: 36px; font-size: 110%; width: 333px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 37px;\"><\/td>\r\n<td style=\"height: 12px; width: 123px;\">set-builder notation<\/td>\r\n<td style=\"height: 12px; width: 92px;\">inequality notation<\/td>\r\n<td style=\"height: 12px; width: 81px;\">interval notation<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 37px;\">domain<\/td>\r\n<td style=\"height: 12px; width: 123px;\">[latex]\\{x|x\\geq\u22125\\}[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 92px;\">[latex]x\\geq\u22125[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 81px;\">[latex][-5,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 37px;\">range<\/td>\r\n<td style=\"height: 12px; width: 123px;\">[latex]\\{y|y\\geq\u22122\\}[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 92px;\">[latex]y\\geq\u22122[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 81px;\">[latex][-2,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nTry the following problem, remembering to look at the graph for the lowest and highest [latex]x[\/latex]-values that appear in the graph to find the domain, then for the smallest and highest [latex]y[\/latex]-values to find the range. Make sure to notice if the graph has arrows or endpoints. That will change how you write your answer.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question sameseed=1]250616[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show how to define the domain and range of\u00a0functions from their graphs.\r\n\r\nhttps:\/\/youtu.be\/QAxZEelInJc\r\n<h2>Finding the Domain and Range from the Equation of a Function<\/h2>\r\nFor this discussion, we will restrict all functions to real number values for the domain and range. Even with this starting point, it does not mean that for any function we discuss that <i>all<\/i> real numbers can be used for [latex]x[\/latex]. It also does not mean that all real numbers can be function output values, [latex]f(x)[\/latex]. There may be restrictions on the domain and range. The restrictions partly depend on the <i>type<\/i> of function.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Domain restrictions<\/h3>\r\nIn our course, there are two main reasons why domains are restricted.\r\n<ul>\r\n \t<li><strong>You cannot divide by\u00a0[latex]0[\/latex].<\/strong>\r\nIf you have a function like [latex]f(x)=\\frac{1}{x-2}[\/latex], the domain has to be restricted so that no inputs make the denominator zero. For this example, [latex]x[\/latex] cannot be [latex]2[\/latex] because if you replace [latex]x[\/latex] with [latex]2[\/latex], you will end up with the denominator as zero which is undefined. The domain would be [latex](-\\infty,2)\\cup(2,\\infty)[\/latex].<\/li>\r\n \t<li><strong>You cannot take the square (or other even) root of a negative number, as the result will not be a real number.<\/strong>\r\nIf you have a function like [latex]f(x)=\\sqrt{x}[\/latex], the domain has to be restricted so that we don't take the square root of a negative number. Recall that the number or expression written under a root symbol is called the radicand. The radicand of a square (or other even) root must not be negative. Then the domain for this example would be [latex]x\\ge0[\/latex] or [latex][0,\\infty)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nState the domain of the function [latex]f(x)=\\dfrac{x+1}{x-3}[\/latex].\r\n\r\n[reveal-answer q=\"114918\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"114918\"]\r\n\r\nWe have to restrict the domain to be sure that we do not divide by zero. We need to exclude from the domain any values of [latex]x[\/latex] that would make the denominator zero. To find those values, set the denominator equal to zero and solve.\r\n\r\n[latex]\\begin{align}x-3&amp;=0\\\\x&amp;=3\\end{align}[\/latex]\r\n\r\nThen the domain is all real values of [latex]x[\/latex] except [latex]3[\/latex]. This is all real numbers where [latex]x&lt;3[\/latex] or [latex]x&gt;3[\/latex]. Use the union symbol, [latex]\\cup [\/latex], to write the two sets in the domain. In interval notation, the domain is [latex](-\\infty,3)\\cup(3,\\infty)[\/latex].\r\n\r\nNotice that we do not need to worry about the numerator of the function being zero. Zero divided by any nonzero number is just zero, not undefined.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nState the domain of the function [latex]f(x)=\\sqrt{2x+1}[\/latex].\r\n\r\n[reveal-answer q=\"890895\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"890895\"]\r\n\r\nWhen we take the square root of a negative number, the result will be a non-real number. The radicand of a square root must not be negative.\u00a0The domain is all the [latex]x[\/latex]-values that would make the radicand non-negative values. In other words, all the [latex]x[\/latex]-values that make the radicand greater than or equal to zero. So we need to solve the inequality [latex]2x+1\\ge0[\/latex].\r\n\r\n[latex]\\begin{align}2x+1&amp;\\ge0\\\\2x&amp;\\ge0-1\\\\2x&amp;\\ge-1\\\\x&amp;\\ge-\\frac{1}{2}\\end{align}[\/latex]\r\n\r\nThen the domain is all real values of [latex]x[\/latex] that are greater than or equal to [latex]-\\dfrac{1}{2}[\/latex]. Using interval notation, this domain is [latex][-\\dfrac{1}{2},\\infty)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4ZWbeESjv4M?si=umpHih1cf4-LbPF5\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nIn the following video, only watch the first example given. We will talk about the other two examples later in the course.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=v0IhvIzCc_I&amp;feature=youtu.be\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nState the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].\r\n\r\n[reveal-answer q=\"275485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"275485\"]\r\n\r\nWhen there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.\r\n\r\nSet the radicand greater than or equal to zero and solve for [latex]x[\/latex].\r\n\r\n[latex]\\begin{align}7-x &amp;\\ge 0\\\\-x&amp;\\ge -7 &amp;&amp; \\color{blue}{\\textsf{remember to reverse the inequality symbol when dividing or multiplying by a negative number}}\\\\x&amp;\\le7\\end{align}[\/latex]\r\n\r\nExclude any number greater than [latex]7[\/latex] from the domain as those inputs would make the radicand negative. The answers are all real numbers less than or equal to [latex]7[\/latex], or [latex]\\left(-\\infty ,7\\right][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>To Learning Outcomes<\/h3>\n<ul>\n<li>Determine if a graph is the graph of a function by using the Vertical Line Test.<\/li>\n<li>Find the value of a function from a graph.<\/li>\n<li>State the domain and range from the graph of a function in set-builder, inequality, and\/or interval notation.<\/li>\n<li>Evaluate functions at a given input value using an equation. Also, determine inputs resulting in the given output value<\/li>\n<li>Using the equation of a function, state the domain in set-builder, inequality, and\/or interval notation.<\/li>\n<\/ul>\n<\/div>\n<h2>Functions<\/h2>\n<p>You should be familiar with functions from previous math courses. Here, we&#8217;ll do a quick review. Some people think of functions as \u201cmathematical machines.\u201d Imagine you have a machine that changes a number according to a specific rule such as \u201cmultiply by\u00a0[latex]3[\/latex] and add\u00a0[latex]2[\/latex]\u201d or \u201cdivide by\u00a0[latex]5[\/latex], add\u00a0[latex]25[\/latex], and multiply by [latex]\u22121[\/latex].\u201d If you put a number into the machine, a new number will come out the other end having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.<\/p>\n<p>You can also call the machine \u201c[latex]f[\/latex]<i>\u201d <\/i>for function. If you put [latex]x[\/latex]<i> <\/i>into the machine, [latex]f(x)[\/latex] (read &#8220;[latex]f[\/latex] of [latex]x[\/latex]&#8220;)<i>, <\/i>comes out. Mathematically speaking, [latex]x[\/latex] is the input, or the <strong>independent variable<\/strong>, and [latex]f(x)[\/latex] is the output, or the <strong>dependent variable<\/strong>, since it depends on the value of [latex]x[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Relations and Functions<\/h3>\n<p>In mathematics, a <strong>relation<\/strong> is a set of ordered pairs. The set of first components of the ordered pairs is called the <strong>domain<\/strong> and the set of second components of the ordered pairs is called the <strong>range<\/strong>.\u00a0In a set of ordered pairs, you find the domain by listing all of the input values (the [latex]x[\/latex]-coordinates). To find the range, list all of the output values (the [latex]y[\/latex]-coordinates).<\/p>\n<p>A <strong>function<\/strong> is a special type of relation in which each domain value corresponds to exactly one range value.<\/p>\n<p>If you find that you need more help <a href=\"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-1-intro-functions\/\">understanding functions<\/a>, please click on the link to learn more about functions.<\/p>\n<\/div>\n<p>Consider the following relation: [latex]\\{(\u22122,0),(0,6),(-2,12),(4,18)\\}[\/latex].<\/p>\n<p>For this relation, the domain is [latex]\\{\u22122,0,4\\}[\/latex] and the range is [latex]\\{0,6,12,18\\}[\/latex].<\/p>\n<p>Is the above relation a function or not?<\/p>\n<p>This relation is NOT a function because the input values are not unique, in other words, there is a repeated [latex]x[\/latex]-value that corresponds to different [latex]y[\/latex]-values.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>List the domain and range for the following relation where [latex]x[\/latex] is the input and [latex]f(x)[\/latex] is the output and determine whether the relation given is a function. Remember each pair of [latex]x[\/latex] and [latex]f(x)[\/latex] exists as a point [latex](x,f(x))[\/latex] on the graph of the function .<\/p>\n<p>&nbsp;<\/p>\n<table style=\"border-collapse: collapse; width: 25%; border: 1px solid black; height: 72px; font-size: 110%;\">\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">\u00a0[latex]x[\/latex]<\/th>\n<th style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]\u22123[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]\u22122[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]\u22121[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 30%; border: 1px solid #999999; height: 12px; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q594198\">Show Solution<\/span><\/p>\n<div id=\"q594198\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain describes all the inputs, and we can use set notation with brackets { } to make the list. The domain is [latex]\\{-3,-2,-1,2,3\\}[\/latex]. The range describes all the outputs. The range is [latex]\\{1,2,4\\}[\/latex]. Notice that [latex]4[\/latex] is only listed once in the list of elements in the range.<\/p>\n<p>This relation is a function because each unique input (no repeated [latex]x[\/latex] values) has a single output. Notice also that [latex]4[\/latex] is a repeated [latex]f(x)[\/latex]-value but that has no effect on whether it is a function or not.\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the following video we provide another example of identifying domain and range. Determine whether the table of values represents a function.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/GPBq18fCEv4?si=Csbo-rmx3N_t6Ez3\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the above video, each domain value corresponds to exactly one range value. No [latex]x[\/latex]-coordinates were repeated so that relation is a function.<\/p>\n<h2>Function Notation<\/h2>\n<p>The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex] or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable. When a function is defined by an equation, we usually use function notation.<\/p>\n<p>For example, the equation [latex]y=3x+1[\/latex] may also be written as [latex]f(x)=3x+1[\/latex]. [latex]f[\/latex] is the name of the function, [latex]x[\/latex] is an input from the domain of the function, and [latex]f(x)[\/latex] is the function value at [latex]x[\/latex] also called the output.<\/p>\n<h3>Inputs of Functions<\/h3>\n<p>A function can be evaluated at different values of [latex]x[\/latex]. Given the function [latex]f(x)=3x+1[\/latex], find [latex]f(-2)[\/latex]. In other words, if [latex]x[\/latex] is [latex]-2[\/latex] find [latex]f(x)[\/latex]. We must substitute [latex]\\color{Green}{-2}[\/latex] in place of [latex]\\color{Green}{x}[\/latex] and evaluate.<br \/>\n[latex]\\begin{align} f(\\color{Green}{-2}\\color{black}{)} &= 3(\\color{Green}{-2}\\color{black}{)+1}\\\\ &=-6+1\\\\ &= -5 \\end{align}[\/latex]<\/p>\n<p>So, when [latex]x=-2[\/latex], [latex]f(x)=-5[\/latex] (or [latex]y=-5[\/latex]). This can be written as the ordered pair [latex](-2,-5)[\/latex] and it is a point on the graph of the line [latex]f(x)=3x+1[\/latex].<\/p>\n<p>It is important to note that the parentheses that are part of function notation <strong><span style=\"text-decoration: underline;\">do not mean multiply<\/span><\/strong>. The notation [latex]f(x)[\/latex] does not mean [latex]f[\/latex] multiplied by [latex]x[\/latex]. Instead, the notation means \u201c[latex]f\\text{ of } x[\/latex]\u201d or \u201c<em>the function of<\/em> [latex]x[\/latex]<i>.&#8221;<\/i>\u00a0To evaluate the function, take the value given for [latex]x[\/latex]<i>,<\/i>\u00a0and substitute that value in for [latex]x[\/latex] in the expression.<\/p>\n<p>Let us look at a couple of examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>ExAMple<\/h3>\n<p>Given [latex]f(x)=3x\u20134[\/latex],\u00a0evaluate [latex]f(5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q42679\">Show Solution<\/span><\/p>\n<div id=\"q42679\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute\u00a0[latex]\\color{Green}{5}[\/latex] in for [latex]\\color{Green}{x}[\/latex]<i> <\/i>in the function. Then simplify the expression on the right side of the equation.<\/p>\n<p>[latex]\\begin{align} f(\\color{Green}{5}) &= 3(\\color{Green}{5})-4\\\\ &= 15-4\\\\ &= 11 \\end{align}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>We were given [latex]x=5[\/latex] and evaluated and found that [latex]f(5)=11[\/latex], which is the point [latex](5,11)[\/latex] on the graph of the function [latex]f(x)=3x-4[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given [latex]p(x)=-2x^{2}+5[\/latex], evaluate [latex]p(\u22123)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q489384\">Show Solution<\/span><\/p>\n<div id=\"q489384\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute [latex]\\color{green}{-3}[\/latex] in for [latex]x[\/latex]<i> <\/i>in the function.<\/p>\n<p>[latex]\\begin{align} p(\\color{Green}{\u22123}) &= -2(\\color{Green}{\u22123})^{2}+5\\\\ &= -2(9)+5\\\\ &=-18+5\\\\ &= -13\\end{align}[\/latex]<\/p>\n<p>This is the point [latex](-3,-13)[\/latex] on the graph of the function [latex]f(x)=-2x^2+5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given [latex]f(x)=|4x-3|[\/latex], evaluate [latex]f(0)[\/latex], [latex]f(2)[\/latex], and [latex]f(\u22121)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q971051\">Show Solution<\/span><\/p>\n<div id=\"q971051\" class=\"hidden-answer\" style=\"display: none\">\n<p>Treat these like three separate problems. In each case, you substitute the value in for [latex]x[\/latex] and simplify.<\/p>\n<p><strong>Evaluate [latex]f(0)[\/latex]:<\/strong><\/p>\n<p>Start with [latex]\\color{green}{x=0}[\/latex] and evaluate [latex]f(x)[\/latex] for [latex]\\color{green}{x=0}[\/latex].<\/p>\n<p>[latex]\\begin{align} f(\\color{green}{0}) &= |4(\\color{green}{0})-3|\\\\ &= |0-3|\\\\ &= |-3|\\\\ &= 3\\end{align}[\/latex]<\/p>\n<p>[latex]f(0)=3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Evaluate [latex]f(2)[\/latex]:<\/strong><\/p>\n<p>Evaluate [latex]f(x)[\/latex] for [latex]\\color{green}{x=2}[\/latex].<\/p>\n<p>[latex]\\begin{align}f(\\color{green}{2}) &= |4(\\color{green}{2})-3|\\\\ &= |8-3|\\\\ &= |5|\\\\ &= 5 \\end{align}[\/latex]<\/p>\n<p>[latex]f(2)=5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Evaluate [latex]f(-1)[\/latex]:<\/strong><\/p>\n<p>Evaluate [latex]f(x)[\/latex] for [latex]\\color{green}{x=\u22121}[\/latex].<\/p>\n<p>[latex]\\begin{align} f(\\color{green}{\u22121}) &= |4(\\color{green}{-1})-3|\\\\ &= |-4-3|\\\\ &= |-7|\\\\ &= 7 \\end{align}[\/latex]<\/p>\n<p>[latex]f(-1)=7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Outputs of Functions<\/h3>\n<p>Reconsider the equation [latex]y=4x+1[\/latex], which represents a function. If instead of having a given input value, you have a known output value, you can find the input value that would give that output. For example, for an output value of [latex]-7[\/latex], you can find the input value. First substitute [latex]\\color{Green}{y=-7}[\/latex] into the equation and then solve for [latex]x[\/latex].<\/p>\n<p>[latex]\\begin{align}\\color{Green}{y} &= 4x+1\\\\ \\color{Green}{-7} &= 4x+1\\\\ -7-1 &= 4x\\\\ -8 &= 4x\\\\ \\frac{-8}{4} &= x\\\\ -2 &= x\\\\ x &= -2\\end{align}[\/latex]<\/p>\n<p>The output of [latex]-7[\/latex] corresponds to an input of [latex]-2[\/latex]. These values make an ordered pair, [latex](-2, -7)[\/latex], that represents a point on the graph of this function. We will discuss this idea more fully later in this section.<\/p>\n<p>In the next example, you will not be given the [latex]x[\/latex]-value but instead you will be given a [latex]y[\/latex]-value and asked to find the [latex]x[\/latex]-value.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given [latex]f(x)=3x\u20134[\/latex],\u00a0find the [latex]x[\/latex]-value that yields an output of [latex]5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q42639\">Show Solution<\/span><\/p>\n<div id=\"q42639\" class=\"hidden-answer\" style=\"display: none\">\n<p>First write the equation with [latex]x[\/latex] and [latex]y[\/latex]<\/p>\n<p>[latex]y=3x-4[\/latex]<\/p>\n<p>Then substitute\u00a0[latex]\\color{green}{5}[\/latex] in for [latex]y[\/latex] in the equation.<\/p>\n<p>[latex]\\color{green}{5}=3x-4[\/latex]<\/p>\n<p>Solve for [latex]x[\/latex].<\/p>\n<p>[latex]\\begin{align}5&=3x-4\\\\5+4&=3x\\\\9&=3x\\\\\\frac{9}{3}&=x\\\\3&=x\\\\x&=3\\\\\\end{align}[\/latex]<\/p>\n<p>For an output of [latex]5[\/latex], the input is [latex]3[\/latex].<\/p>\n<p>These values make an ordered pair, [latex](3, 5)[\/latex], that represents a point on the graph of this function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm288721\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288721&theme=oea&iframe_resize_id=ohm288721&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Graphs of Functions and the Vertical Line Test<\/h2>\n<p>When both the independent quantity (input) and the dependent quantity (output) are real numbers, a function can be represented by a graph in the coordinate plane. The independent value is plotted on the [latex]x[\/latex]-axis and the dependent value is plotted on the [latex]y[\/latex]-axis. The fact that each input value has exactly one output value means graphs of functions have certain characteristics. For each input on the graph, there will be exactly one output. For a function defined as\u00a0[latex]y = f(x)[\/latex], or &#8220;[latex]y[\/latex] is a function of\u00a0[latex]x[\/latex]&#8220;, we would write ordered pairs\u00a0[latex](x, f(x))[\/latex] using function notation instead of\u00a0[latex](x,y)[\/latex] as you may have seen previously.<\/p>\n<p>We can identify whether the graph of a relation represents a function\u00a0because for each [latex]x[\/latex]-coordinate there will be exactly one [latex]y[\/latex]-coordinate. The blue relation graphed below represents a function. Notice that when a vertical line is placed over the graph, the line does not intersect the graph more than once for any value of [latex]x[\/latex]. This indicates that the relation is a function.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/>If instead a graph shows two or more intersections with a vertical line, then an input ([latex]x[\/latex]-coordinate) has more than one output ([latex]y[\/latex]-coordinate), and [latex]y[\/latex] is not a function of [latex]x[\/latex].<\/p>\n<p>Examining the graph of a relation to determine if a vertical line would intersect with more than one point is a quick way to determine if the relation shown by the graph is a function. This method is often called the <strong>Vertical Line Test<\/strong>.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the vertical line test to determine whether the relation plotted on this graph is a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2680 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16200323\/Screen-Shot-2016-07-16-at-1.02.50-PM-300x275.png\" alt=\"Graph with circle plotted - center at (0,0) radius = 2, more points include (2,0), (-2,0)\" width=\"300\" height=\"275\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q28965\">Show Solution<\/span><\/p>\n<div id=\"q28965\" class=\"hidden-answer\" style=\"display: none\">\n<p>This relationship cannot be a function, because some of the [latex]x[\/latex]-coordinates have two corresponding [latex]y[\/latex]-coordinates.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232418\/image002.jpg\" alt=\"A circle with four vertical lines through it. Three of the lines cross the circle at two points, and one line crosses the edge of the circle at one point.\" width=\"340\" height=\"343\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<h2>Find the Value of a Function from a Graph<\/h2>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A function is graphed below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1601 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-300x300.jpg\" alt=\"Graph of a diagonal line on a coordinate plane that goes through the origin (0,0). Additional points at (2,2) and (negative 2, negative 2). Labeled f of x.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-300x300.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-150x150.jpg 150w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-65x65.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity-225x224.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/identity.jpg 320w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol>\n<li>For the given function, evaluate [latex]f(2)[\/latex].<\/li>\n<li>For the given graph of a function, find [latex]x[\/latex] when [latex]f(x)=-2[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q719392\">Show Answer<\/span><\/p>\n<div id=\"q719392\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>To evaluate [latex]f(2)[\/latex], go to [latex]x=2[\/latex] on the graph, then vertically to the graph of the function. The [latex]y[\/latex]-value at this point is [latex]2[\/latex]. Then [latex]f(2)=2[\/latex].<\/li>\n<li>To find [latex]x[\/latex] when [latex]f(x)=-2[\/latex] using the graph, go to [latex]y=-2[\/latex], then horizontally to the graph of the function. The [latex]x[\/latex]-value at this point is [latex]-2[\/latex]. Then [latex]f(x)=-2[\/latex] when [latex]x=-2[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>ExAmple<\/h3>\n<p>A function is graphed below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1606 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-300x273.jpg\" alt=\"A graph of an upward facing absolute function (v-shape) on a coordinate plane with the vertex at (0,0). Additional points shown at (negative 2,2) and (2,2).\" width=\"300\" height=\"273\" srcset=\"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-300x273.jpg 300w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-65x59.jpg 65w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1-225x205.jpg 225w, https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-content\/uploads\/sites\/5871\/2023\/11\/absolute-value-1.jpg 324w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol>\n<li>For the given function, evaluate [latex]f(-1)[\/latex].<\/li>\n<li>For the given function, evaluate [latex]f(0)[\/latex].<\/li>\n<li>For the given graph of a function, find [latex]x[\/latex] when [latex]f(x)=2[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q932196\">Show Answer<\/span><\/p>\n<div id=\"q932196\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>On the graph, go to [latex]x=-1[\/latex], then vertically to the graph of the function. The [latex]y[\/latex]-value at this point is [latex]1[\/latex]. Then [latex]f(-1)=1[\/latex].<\/li>\n<li>On the graph, go to [latex]x=0[\/latex], then vertically to the graph of the function. The [latex]y[\/latex]-value at this point is [latex]0[\/latex]. Then [latex]f(0)=0[\/latex].<\/li>\n<li>On the graph, go to [latex]y=2[\/latex], then horizontally to the graph of the function. Notice that there are two [latex]x[\/latex]-values that have this associated [latex]y[\/latex]-value: [latex]x=-2[\/latex] and [latex]x=2[\/latex]. Then [latex]f(x)=2[\/latex] when [latex]x=-2[\/latex] or [latex]x=2[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm288724\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288724&theme=oea&iframe_resize_id=ohm288724&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Finding the Domain and Range of a Function from its Graph<\/h2>\n<p>Finding domain of different functions is often a matter of asking yourself, &#8220;what values of [latex]x[\/latex] can this function <i>not<\/i>\u00a0have?&#8221; Pictures make it easier to visualize what the domain and range are, so we will first show how to define the domain and range of functions given their graphs. What are the domain and range of the function [latex]f(x)=x+3[\/latex]?<br \/>\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.<\/p>\n<div style=\"width: 332px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Graph of a diagonal line that goes through (0,3) and (negative 3,0).\" width=\"322\" height=\"353\" \/><\/p>\n<p class=\"wp-caption-text\">Graph of the function [latex]f(x)=x+3[\/latex]<\/p>\n<\/div>\n<p>For this function, any real number can be substituted for [latex]x[\/latex] and you will get a meaningful output. Also, for any real number, you can always find an [latex]x[\/latex] value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range. For the function [latex]f(x)=x+3[\/latex], the domain and range are all real numbers. We can write this domain and range in a variety of formats.<\/p>\n<table style=\"border-collapse: collapse; font-size: 110%; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.333333%;\"><\/td>\n<td style=\"width: 33.333333%;\"><strong>Domain<\/strong><\/td>\n<td style=\"width: 33.333333%;\"><strong>Range<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.333333%;\"><strong>Set-Builder Notation<\/strong><\/td>\n<td style=\"width: 33.333333%;\">[latex]\\{x|\\text{all real numbers}\\}[\/latex]<\/td>\n<td style=\"width: 33.333333%;\">[latex]\\{y|\\text{all real numbers}\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.333333%;\"><strong>Inequality Notation<\/strong><\/td>\n<td style=\"width: 33.333333%;\">[latex]-\\infty<x<\\infty[\/latex]<\/td>\n<td style=\"width: 33.333333%;\">[latex]-\\infty<y<\\infty[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.333333%;\"><strong>Interval Notation<\/strong><\/td>\n<td style=\"width: 33.333333%;\">[latex](-\\infty,\\infty)[\/latex]<\/td>\n<td style=\"width: 33.333333%;\">[latex](-\\infty,\\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>For the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the graph to determine the domain and range of the function [latex]f(x)=\u22123x^{2}+6x+1.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223692\">Show Solution<\/span><\/p>\n<div id=\"q223692\" class=\"hidden-answer\" style=\"display: none\">\n<p>Examining the graph, the arrows indicate that the function continues for larger and for smaller values of [latex]x[\/latex]. The domain is all real numbers.<\/p>\n<p>The vertex, or high\u00a0point of this graph, is at [latex](1,4)[\/latex]. From the graph, you can see that all [latex]y[\/latex]-values of the function are less than or equal to [latex]4[\/latex]. In other words, [latex]f(x)\\leq4[\/latex]. The range of this function is all [latex]y[\/latex] values less than or equal to [latex]4[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<table style=\"height: 36px; font-size: 110%; width: 380px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 43px;\"><\/td>\n<td style=\"height: 12px; width: 140px;\">set-builder notation<\/td>\n<td style=\"height: 12px; width: 105px;\">inequality notation<\/td>\n<td style=\"height: 12px; width: 92px;\">interval notation<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 43px;\">domain<\/td>\n<td style=\"height: 12px; width: 140px;\">[latex]\\{x|\\text{all real numbers}\\}[\/latex]<\/td>\n<td style=\"height: 12px; width: 105px;\">[latex]-\\infty<x<\\infty[\/latex]<\/td>\n<td style=\"height: 12px; width: 92px;\">[latex](-\\infty,\\infty)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 43px;\">range<\/td>\n<td style=\"height: 12px; width: 140px;\">[latex]\\{y|y\\le4\\}[\/latex]<\/td>\n<td style=\"height: 12px; width: 105px;\">[latex]y\\le4[\/latex]<\/td>\n<td style=\"height: 12px; width: 92px;\">[latex](-\\infty,4][\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Use the graph to determine the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}.[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Square root curve stemming from negative 5, negative 2 and increasing slowly to the right.\" width=\"308\" height=\"346\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q231228\">Show Solution<\/span><\/p>\n<div id=\"q231228\" class=\"hidden-answer\" style=\"display: none\">\n<p>From the graph, we see that the smallest [latex]x[\/latex]-value is [latex]x=-5[\/latex]. Then the domain is all real numbers greater than or equal to [latex]-5[\/latex].<\/p>\n<p>The smallest [latex]y[\/latex]-value on the graph of the function is [latex]y=-2[\/latex]. The range will be all real numbers greater than or equal to [latex]-2[\/latex].<\/p>\n<table style=\"height: 36px; font-size: 110%; width: 333px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 37px;\"><\/td>\n<td style=\"height: 12px; width: 123px;\">set-builder notation<\/td>\n<td style=\"height: 12px; width: 92px;\">inequality notation<\/td>\n<td style=\"height: 12px; width: 81px;\">interval notation<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 37px;\">domain<\/td>\n<td style=\"height: 12px; width: 123px;\">[latex]\\{x|x\\geq\u22125\\}[\/latex]<\/td>\n<td style=\"height: 12px; width: 92px;\">[latex]x\\geq\u22125[\/latex]<\/td>\n<td style=\"height: 12px; width: 81px;\">[latex][-5,\\infty)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 37px;\">range<\/td>\n<td style=\"height: 12px; width: 123px;\">[latex]\\{y|y\\geq\u22122\\}[\/latex]<\/td>\n<td style=\"height: 12px; width: 92px;\">[latex]y\\geq\u22122[\/latex]<\/td>\n<td style=\"height: 12px; width: 81px;\">[latex][-2,\\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Try the following problem, remembering to look at the graph for the lowest and highest [latex]x[\/latex]-values that appear in the graph to find the domain, then for the smallest and highest [latex]y[\/latex]-values to find the range. Make sure to notice if the graph has arrows or endpoints. That will change how you write your answer.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm250616\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=250616&theme=oea&iframe_resize_id=ohm250616&sameseed=1&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show how to define the domain and range of\u00a0functions from their graphs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Determine the Domain and Range of the Graph of a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Finding the Domain and Range from the Equation of a Function<\/h2>\n<p>For this discussion, we will restrict all functions to real number values for the domain and range. Even with this starting point, it does not mean that for any function we discuss that <i>all<\/i> real numbers can be used for [latex]x[\/latex]. It also does not mean that all real numbers can be function output values, [latex]f(x)[\/latex]. There may be restrictions on the domain and range. The restrictions partly depend on the <i>type<\/i> of function.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Domain restrictions<\/h3>\n<p>In our course, there are two main reasons why domains are restricted.<\/p>\n<ul>\n<li><strong>You cannot divide by\u00a0[latex]0[\/latex].<\/strong><br \/>\nIf you have a function like [latex]f(x)=\\frac{1}{x-2}[\/latex], the domain has to be restricted so that no inputs make the denominator zero. For this example, [latex]x[\/latex] cannot be [latex]2[\/latex] because if you replace [latex]x[\/latex] with [latex]2[\/latex], you will end up with the denominator as zero which is undefined. The domain would be [latex](-\\infty,2)\\cup(2,\\infty)[\/latex].<\/li>\n<li><strong>You cannot take the square (or other even) root of a negative number, as the result will not be a real number.<\/strong><br \/>\nIf you have a function like [latex]f(x)=\\sqrt{x}[\/latex], the domain has to be restricted so that we don&#8217;t take the square root of a negative number. Recall that the number or expression written under a root symbol is called the radicand. The radicand of a square (or other even) root must not be negative. Then the domain for this example would be [latex]x\\ge0[\/latex] or [latex][0,\\infty)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>State the domain of the function [latex]f(x)=\\dfrac{x+1}{x-3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q114918\">Show Answer<\/span><\/p>\n<div id=\"q114918\" class=\"hidden-answer\" style=\"display: none\">\n<p>We have to restrict the domain to be sure that we do not divide by zero. We need to exclude from the domain any values of [latex]x[\/latex] that would make the denominator zero. To find those values, set the denominator equal to zero and solve.<\/p>\n<p>[latex]\\begin{align}x-3&=0\\\\x&=3\\end{align}[\/latex]<\/p>\n<p>Then the domain is all real values of [latex]x[\/latex] except [latex]3[\/latex]. This is all real numbers where [latex]x<3[\/latex] or [latex]x>3[\/latex]. Use the union symbol, [latex]\\cup[\/latex], to write the two sets in the domain. In interval notation, the domain is [latex](-\\infty,3)\\cup(3,\\infty)[\/latex].<\/p>\n<p>Notice that we do not need to worry about the numerator of the function being zero. Zero divided by any nonzero number is just zero, not undefined.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>State the domain of the function [latex]f(x)=\\sqrt{2x+1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q890895\">Show Answer<\/span><\/p>\n<div id=\"q890895\" class=\"hidden-answer\" style=\"display: none\">\n<p>When we take the square root of a negative number, the result will be a non-real number. The radicand of a square root must not be negative.\u00a0The domain is all the [latex]x[\/latex]-values that would make the radicand non-negative values. In other words, all the [latex]x[\/latex]-values that make the radicand greater than or equal to zero. So we need to solve the inequality [latex]2x+1\\ge0[\/latex].<\/p>\n<p>[latex]\\begin{align}2x+1&\\ge0\\\\2x&\\ge0-1\\\\2x&\\ge-1\\\\x&\\ge-\\frac{1}{2}\\end{align}[\/latex]<\/p>\n<p>Then the domain is all real values of [latex]x[\/latex] that are greater than or equal to [latex]-\\dfrac{1}{2}[\/latex]. Using interval notation, this domain is [latex][-\\dfrac{1}{2},\\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4ZWbeESjv4M?si=umpHih1cf4-LbPF5\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following video, only watch the first example given. We will talk about the other two examples later in the course.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  The Domain of Rational Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>State the domain of the function [latex]f\\left(x\\right)=\\sqrt{7-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q275485\">Show Solution<\/span><\/p>\n<div id=\"q275485\" class=\"hidden-answer\" style=\"display: none\">\n<p>When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.<\/p>\n<p>Set the radicand greater than or equal to zero and solve for [latex]x[\/latex].<\/p>\n<p>[latex]\\begin{align}7-x &\\ge 0\\\\-x&\\ge -7 && \\color{blue}{\\textsf{remember to reverse the inequality symbol when dividing or multiplying by a negative number}}\\\\x&\\le7\\end{align}[\/latex]<\/p>\n<p>Exclude any number greater than [latex]7[\/latex] from the domain as those inputs would make the radicand negative. The answers are all real numbers less than or equal to [latex]7[\/latex], or [latex]\\left(-\\infty ,7\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/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-27\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Find Domain and Range of Ordered Pairs, Function or Not. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/kzgLfwgxE8g\">https:\/\/youtu.be\/kzgLfwgxE8g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface.%20\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface.%20<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Give the Domain and Range Given the Points in a Table. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/GPBq18fCEv4\">https:\/\/youtu.be\/GPBq18fCEv4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Determine if a Table of Values Represents a Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/y2TqnP_6M1s\">https:\/\/youtu.be\/y2TqnP_6M1s<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Find Domain and Range of Ordered Pairs, Function or Not\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/kzgLfwgxE8g\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\" http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface. \",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Give the Domain and Range Given the Points in a Table\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/GPBq18fCEv4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine if a Table of Values Represents a Function\",\"author\":\"James Sousa (Mathispower4u.com) 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