{"id":82,"date":"2016-06-01T20:50:17","date_gmt":"2016-06-01T20:50:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=82"},"modified":"2017-12-27T23:54:24","modified_gmt":"2017-12-27T23:54:24","slug":"17-2-1-evaluating-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/17-2-1-evaluating-functions\/","title":{"raw":"Domain and Range","rendered":"Domain and Range"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Define the domain of functions from graphs<\/li>\r\n \t<li>One-to-one functions<\/li>\r\n \t<li>Use the horizontal\u00a0line test to determine whether a function is one-to-one<\/li>\r\n<\/ul>\r\n<\/div>\r\nFunctions are a correspondence between two sets, called the <strong>domain<\/strong> and the <strong>range<\/strong>. When defining a function, you usually state what kind of numbers the domain (<i>x<\/i>) and range (<i>f(x)<\/i>) values can be. But even if you say they are real numbers, that doesn\u2019t mean that <i>all<\/i> real numbers can be used for <i>x<\/i>. It also doesn\u2019t mean that all real numbers can be function values, <i>f<\/i>(<i>x<\/i>). There may be restrictions on the domain and range. The restrictions partly depend on the <i>type<\/i> of function.\r\n\r\nIn this topic, all functions will be restricted to real number values. That is, only real numbers can be used in the domain, and only real numbers can be in the range.\r\n\r\nThere are two main reasons why domains are restricted.\r\n<ul>\r\n \t<li>You can\u2019t divide by 0.<\/li>\r\n \t<li>You can\u2019t take the square (or other even) root of a negative number, as the result will not be a real number.<\/li>\r\n<\/ul>\r\n<h2>Find Domain and Range From a Graph<\/h2>\r\nFinding domain and range of different functions is often a matter of asking yourself, what values can this function <i>not<\/i>\u00a0have? Pictures make it easier to visualize what domain and range are, so we will show how to define the domain and range of functions given their graphs.\r\n\r\nWhat are the domain and range of the real-valued function [latex]f(x)=x+3[\/latex]?\r\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Line for f(x)=x+3\" width=\"322\" height=\"353\" \/>\r\n\r\nAny real number can be substituted for <i>x<\/i> and get a meaningful output. For <i>any<\/i> real number, you can always find an <i>x<\/i> value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range.\r\nThe domain and range are all real numbers.\r\n\r\nFor the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat are the domain and range of the real-valued function [latex]f(x)=\u22123x^{2}+6x+1[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/>\r\n\r\n[reveal-answer q=\"223692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223692\"]This is a <i>quadratic <\/i>function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing that will restrict the domain. Any real number can be used for <i>x<\/i> to get a meaningful output.\r\n\r\nBecause the coefficient of [latex]x^{2}[\/latex] is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value.\r\n\r\nThe vertex, or high\u00a0point, is at (1, 4). From the graph, you can see that [latex]f(x)\\leq4[\/latex].\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers, and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\leq4[\/latex].\r\n\r\nYou can check that the vertex is indeed at (1, 4). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the same <i>y<\/i> value. The vertex must lie on the line of reflection, because it\u2019s the only point that does not have a mirror image!\r\n\r\nIn the previous example, notice that when [latex]x=2[\/latex] and when [latex]x=0[\/latex], the function value is 1. (You can verify this by evaluating <i>f<\/i>(2) and <i>f<\/i>(0).) That is, both (2, 1) and (0, 1) are on the graph. The line of reflection here is [latex]x=1[\/latex], so the vertex must be at the point (1, <i>f<\/i>(1)). Evaluating <i>f<\/i>(1)<i> <\/i>gives [latex]f(1)=4[\/latex], so the vertex is at (1, 4).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat is the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Radical function stemming from negative 5, negative 2.\" width=\"308\" height=\"346\" \/>\r\n\r\n[reveal-answer q=\"231228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231228\"]This is a <i>radical <\/i>function. The domain of a radical function is any <i>x<\/i> value for which the radicand (the value under the radical sign) is not negative. That means [latex]x+5\\geq0[\/latex], so [latex]x\\geq\u22125[\/latex].\r\n\r\nSince the square root must always be positive or 0, [latex] \\displaystyle \\sqrt{x+5}\\ge 0[\/latex]. That means [latex] \\displaystyle -2+\\sqrt{x+5}\\ge -2[\/latex].\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers <i>x<\/i> where [latex]x\\geq\u22125[\/latex], and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\geq\u22122[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n<i>Division by 0<\/i> could happen whenever the function has a variable in the <i>denominator <\/i>of a rational expression. That is, it\u2019s something to look for in <i>rational functions.<\/i> Look at these examples, and note that \u201cdivision by 0\u201d doesn\u2019t necessarily mean that <i>x<\/i> is 0! The following is an example of <i>rational function<i>, w will cover these in detail later.<\/i><\/i>\r\n\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhat is the domain of the real-valued function [latex] \\displaystyle f(x)=\\frac{3x}{x+2}[\/latex]?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232531\/image049.gif\" alt=\"Rational function\" width=\"310\" height=\"321\" \/>\r\n\r\n[reveal-answer q=\"666335\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"666335\"]This is a <i>rational <\/i>function. The domain of a rational function is restricted where the denominator is 0. In this case, [latex]x+2[\/latex] is the denominator, and this is 0 only when [latex]x=\u22122[\/latex].\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers except [latex]\u22122[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show how to define the domain and range of\u00a0functions from their graphs.\r\n\r\nhttps:\/\/youtu.be\/QAxZEelInJc\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Domain and Range from a Graph<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165219\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/>\r\n[reveal-answer q=\"495787\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"495787\"]\r\nWe can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165221\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/>\r\n\r\nThe vertical extent of the graph is 0 to [latex]\u20134[\/latex], so the range is [latex]\\left[-4,0\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n\r\n\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]2316[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Domain and Range from a Graph of Oil Production<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"489\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165223\/CNX_Precalc_Figure_01_02_0092.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\" \/> (credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)[\/caption]\r\n\r\n[reveal-answer q=\"834467\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"834467\"]\r\nThe input quantity along the horizontal axis is \"years,\" which we represent with the variable [latex]t[\/latex] for time. The output quantity is \"thousands of barrels of oil per day,\" which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010[\/latex].\r\n\r\nIn interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.\r\n\r\n[\/hidden-answer]\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the graph, identify the domain and range using interval notation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165225\/CNX_Precalc_Figure_01_02_0102.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\" \/>\r\n[reveal-answer q=\"186149\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"186149\"]\r\n\r\nDomain = [1950, 2002] \u00a0 Range = [47,000,000, 89,000,000]\r\n\r\n[\/hidden-answer]\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Identify a One-to-One Function<\/h2>\r\nRemember that in a function, the input value must have one and only one value for the output.\u00a0There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the <b>domain of the function<\/b>. And the set of output values is called the <b>range of the function<\/b>.\r\n\r\nIn the first example we remind you how to define domain and range using a table of values.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the domain and range for the function.\r\n<table style=\"width: 20%\">\r\n<thead>\r\n<tr>\r\n<th>\r\n<p align=\"center\"><i>x<\/i><\/p>\r\n<\/th>\r\n<th>\r\n<p align=\"center\"><i>y<\/i><\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p align=\"center\">\u22125<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">\u22126<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">\u22122<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">\u22121<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">\u22121<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">0<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">0<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">3<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p align=\"center\">5<\/p>\r\n<\/td>\r\n<td>\r\n<p align=\"center\">15<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"130987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"130987\"]\r\nThe domain is the set of inputs or <i>x<\/i>-coordinates.\r\n<p align=\"center\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\r\nThe range is the set of outputs of <i>y<\/i>-coordinates.\r\n<p align=\"center\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\begin{array}{l}\\text{Domain}:\\{\u22125,\u22122,\u22121,0,5\\}\\\\\\text{Range}:\\{\u22126,\u22121,0,3,15\\}\\end{array}\\\\[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of finding domain and range from tabular data.\r\n\r\nhttps:\/\/youtu.be\/GPBq18fCEv4\r\n\r\nSome functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" \/>\r\n<p id=\"fs-id1165135678633\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in.<\/p>\r\n\r\n<table style=\"width: 20%\" summary=\"Two columns and five rows. The first column is labeled,\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th>Letter grade<\/th>\r\n<th>Grade point average<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>A<\/td>\r\n<td>4.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B<\/td>\r\n<td>3.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C<\/td>\r\n<td>2.0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<\/td>\r\n<td>1.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\r\nTo visualize this concept, let\u2019s look again at the two simple functions sketched in (a)and (b) of Figure 10.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/> <b>Figure 10<\/b>[\/caption]\r\n\r\nThe function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.\r\n<div class=\"textbox\">\r\n<h3>A General Note: One-to-One Function<\/h3>\r\nA one-to-one function is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhich table represents a one-to-one function?\r\n\r\na)\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td>input<\/td>\r\n<td>output<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>12<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>-1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-5<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nb)\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td>input<\/td>\r\n<td>output<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>16<\/td>\r\n<td>32<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>32<\/td>\r\n<td>64<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>64<\/td>\r\n<td>128<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"945171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"945171\"]\r\n\r\nTable a) maps the output value 2 to two different input values, therefore\u00a0this is NOT a one-to-one function.\r\n\r\nTable b) maps each output to one unique input, therefore this IS a one-to-one function.\r\n<h4>Answer<\/h4>\r\nTable b) is one-to-one\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show an example of using tables of values to determine whether a function is one-to-one.\r\n\r\nhttps:\/\/youtu.be\/QFOJmevha_Y\r\n<h2 style=\"text-align: left\">Using the Horizontal Line Test<\/h2>\r\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function\u00a0is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. \u00a0To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\r\n\r\n<div class=\"note precalculus howto textbox\">\r\n<h3 style=\"text-align: center\"><strong>How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/strong><\/h3>\r\n<ol id=\"fs-id1165137611853\">\r\n \t<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\r\n \t<li>If there is any such line, determine that the function is not one-to-one.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nFor\u00a0the following graphs, determine which represent one-to-one functions.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/>\r\n[reveal-answer q=\"783411\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"783411\"]\r\n<p id=\"fs-id1165135185190\">The function in (a) is\u00a0not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\r\n\r\n<figure id=\"Figure_01_01_010\" class=\"small\"><\/figure>\r\n<figure id=\"Figure_01_01_010\" class=\"small\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"487\" height=\"445\" \/><\/figure>\r\nThe function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.\r\n\r\n<img class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\" \/>\r\n\r\nThe function (c) is not one-to-one, and is in fact not a function.\r\n\r\n<img class=\" wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.\r\nhttps:\/\/youtu.be\/tbSGdcSN8RE\r\n<h2>Summary<\/h2>\r\nIn real life and in algebra, different variables are often linked. When a change in value of one variable causes a change in the value of another variable, their interaction is called a relation. A relation has an input value which corresponds to an output value. When each input value has one and only one output value, that relation is a function. Functions can be written as ordered pairs, tables, or graphs. The set of input values is called the domain, and the set of output values is called the range.\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Define the domain of functions from graphs<\/li>\n<li>One-to-one functions<\/li>\n<li>Use the horizontal\u00a0line test to determine whether a function is one-to-one<\/li>\n<\/ul>\n<\/div>\n<p>Functions are a correspondence between two sets, called the <strong>domain<\/strong> and the <strong>range<\/strong>. When defining a function, you usually state what kind of numbers the domain (<i>x<\/i>) and range (<i>f(x)<\/i>) values can be. But even if you say they are real numbers, that doesn\u2019t mean that <i>all<\/i> real numbers can be used for <i>x<\/i>. It also doesn\u2019t mean that all real numbers can be function values, <i>f<\/i>(<i>x<\/i>). There may be restrictions on the domain and range. The restrictions partly depend on the <i>type<\/i> of function.<\/p>\n<p>In this topic, all functions will be restricted to real number values. That is, only real numbers can be used in the domain, and only real numbers can be in the range.<\/p>\n<p>There are two main reasons why domains are restricted.<\/p>\n<ul>\n<li>You can\u2019t divide by 0.<\/li>\n<li>You can\u2019t take the square (or other even) root of a negative number, as the result will not be a real number.<\/li>\n<\/ul>\n<h2>Find Domain and Range From a Graph<\/h2>\n<p>Finding domain and range of different functions is often a matter of asking yourself, what values can this function <i>not<\/i>\u00a0have? Pictures make it easier to visualize what domain and range are, so we will show how to define the domain and range of functions given their graphs.<\/p>\n<p>What are the domain and range of the real-valued function [latex]f(x)=x+3[\/latex]?<br \/>\nThis is a <i>linear <\/i>function. Remember that linear functions are lines that continue forever in each direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232525\/image046.gif\" alt=\"Line for f(x)=x+3\" width=\"322\" height=\"353\" \/><\/p>\n<p>Any real number can be substituted for <i>x<\/i> and get a meaningful output. For <i>any<\/i> real number, you can always find an <i>x<\/i> value that gives you that number for the output. Unless a linear function is a constant, such as [latex]f(x)=2[\/latex], there is no restriction on the range.<br \/>\nThe domain and range are all real numbers.<\/p>\n<p>For the examples that follow, try to figure out the domain and range of the graphs before you look at the answer.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What are the domain and range of the real-valued function [latex]f(x)=\u22123x^{2}+6x+1[\/latex]?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232527\/image047.gif\" alt=\"Downward-opening parabola with vertex of 1, 4.\" width=\"323\" height=\"348\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223692\">Show Solution<\/span><\/p>\n<div id=\"q223692\" class=\"hidden-answer\" style=\"display: none\">This is a <i>quadratic <\/i>function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing that will restrict the domain. Any real number can be used for <i>x<\/i> to get a meaningful output.<\/p>\n<p>Because the coefficient of [latex]x^{2}[\/latex] is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value.<\/p>\n<p>The vertex, or high\u00a0point, is at (1, 4). From the graph, you can see that [latex]f(x)\\leq4[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers, and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\leq4[\/latex].<\/p>\n<p>You can check that the vertex is indeed at (1, 4). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the same <i>y<\/i> value. The vertex must lie on the line of reflection, because it\u2019s the only point that does not have a mirror image!<\/p>\n<p>In the previous example, notice that when [latex]x=2[\/latex] and when [latex]x=0[\/latex], the function value is 1. (You can verify this by evaluating <i>f<\/i>(2) and <i>f<\/i>(0).) That is, both (2, 1) and (0, 1) are on the graph. The line of reflection here is [latex]x=1[\/latex], so the vertex must be at the point (1, <i>f<\/i>(1)). Evaluating <i>f<\/i>(1)<i> <\/i>gives [latex]f(1)=4[\/latex], so the vertex is at (1, 4).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What is the domain and range of the real-valued function [latex]f(x)=-2+\\sqrt{x+5}[\/latex]?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232529\/image048.gif\" alt=\"Radical function stemming from negative 5, negative 2.\" width=\"308\" height=\"346\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q231228\">Show Solution<\/span><\/p>\n<div id=\"q231228\" class=\"hidden-answer\" style=\"display: none\">This is a <i>radical <\/i>function. The domain of a radical function is any <i>x<\/i> value for which the radicand (the value under the radical sign) is not negative. That means [latex]x+5\\geq0[\/latex], so [latex]x\\geq\u22125[\/latex].<\/p>\n<p>Since the square root must always be positive or 0, [latex]\\displaystyle \\sqrt{x+5}\\ge 0[\/latex]. That means [latex]\\displaystyle -2+\\sqrt{x+5}\\ge -2[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers <i>x<\/i> where [latex]x\\geq\u22125[\/latex], and the range is all real numbers <i>f<\/i>(<i>x<\/i>) such that [latex]f(x)\\geq\u22122[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><i>Division by 0<\/i> could happen whenever the function has a variable in the <i>denominator <\/i>of a rational expression. That is, it\u2019s something to look for in <i>rational functions.<\/i> Look at these examples, and note that \u201cdivision by 0\u201d doesn\u2019t necessarily mean that <i>x<\/i> is 0! The following is an example of <i>rational function<i>, w will cover these in detail later.<\/i><\/i><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>What is the domain of the real-valued function [latex]\\displaystyle f(x)=\\frac{3x}{x+2}[\/latex]?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232531\/image049.gif\" alt=\"Rational function\" width=\"310\" height=\"321\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q666335\">Show Solution<\/span><\/p>\n<div id=\"q666335\" class=\"hidden-answer\" style=\"display: none\">This is a <i>rational <\/i>function. The domain of a rational function is restricted where the denominator is 0. In this case, [latex]x+2[\/latex] is the denominator, and this is 0 only when [latex]x=\u22122[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers except [latex]\u22122[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show how to define the domain and range of\u00a0functions from their graphs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Determine the Domain and Range of the Graph of a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Domain and Range from a Graph<\/h3>\n<p>Find the domain and range of the function [latex]f[\/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\/2862\/2017\/12\/26165219\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q495787\">Solution<\/span><\/p>\n<div id=\"q495787\" class=\"hidden-answer\" style=\"display: none\">\nWe can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/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\/2862\/2017\/12\/26165221\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/><\/p>\n<p>The vertical extent of the graph is 0 to [latex]\u20134[\/latex], so the range is [latex]\\left[-4,0\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2316\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2316&theme=oea&iframe_resize_id=ohm2316&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Domain and Range from a Graph of Oil Production<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex].<\/p>\n<div style=\"width: 499px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165223\/CNX_Precalc_Figure_01_02_0092.jpg\" alt=\"Graph of the Alaska Crude Oil Production where the y-axis is thousand barrels per day and the -axis is the years.\" width=\"489\" height=\"329\" \/><\/p>\n<p class=\"wp-caption-text\">(credit: modification of work by the <a href=\"http:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=PET&amp;s=MCRFPAK2&amp;f=A.\" target=\"_blank\" rel=\"noopener\">U.S. Energy Information Administration<\/a>)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834467\">Solution<\/span><\/p>\n<div id=\"q834467\" class=\"hidden-answer\" style=\"display: none\">\nThe input quantity along the horizontal axis is &#8220;years,&#8221; which we represent with the variable [latex]t[\/latex] for time. The output quantity is &#8220;thousands of barrels of oil per day,&#8221; which we represent with the variable [latex]b[\/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\\le t\\le 2008[\/latex] and the range as approximately [latex]180\\le b\\le 2010[\/latex].<\/p>\n<p>In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the graph, identify the domain and range using interval notation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165225\/CNX_Precalc_Figure_01_02_0102.jpg\" alt=\"Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.\" width=\"487\" height=\"333\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186149\">Solution<\/span><\/p>\n<div id=\"q186149\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [1950, 2002] \u00a0 Range = [47,000,000, 89,000,000]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<h2>Identify a One-to-One Function<\/h2>\n<p>Remember that in a function, the input value must have one and only one value for the output.\u00a0There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the <b>domain of the function<\/b>. And the set of output values is called the <b>range of the function<\/b>.<\/p>\n<p>In the first example we remind you how to define domain and range using a table of values.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the domain and range for the function.<\/p>\n<table style=\"width: 20%\">\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center;\"><i>x<\/i><\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\"><i>y<\/i><\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\n<p style=\"text-align: center;\">\u22125<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">\u22126<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">\u22122<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">\u22121<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">\u22121<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td>\n<p style=\"text-align: center;\">15<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q130987\">Show Solution<\/span><\/p>\n<div id=\"q130987\" class=\"hidden-answer\" style=\"display: none\">\nThe domain is the set of inputs or <i>x<\/i>-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22125,\u22122,\u22121,0,5\\}[\/latex]<\/p>\n<p>The range is the set of outputs of <i>y<\/i>-coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\{\u22126,\u22121,0,3,15\\}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\begin{array}{l}\\text{Domain}:\\{\u22125,\u22122,\u22121,0,5\\}\\\\\\text{Range}:\\{\u22126,\u22121,0,3,15\\}\\end{array}\\\\[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show another example of finding domain and range from tabular data.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Give the Domain and Range Given the Points in a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GPBq18fCEv4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Some functions have a given output value that corresponds to two or more input values. For example, in the following stock chart the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200506\/CNX_Precalc_Figure_01_00_001n2.jpg\" alt=\"Figure of a bull and a graph of market prices.\" width=\"975\" height=\"307\" \/><\/p>\n<p id=\"fs-id1165135678633\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in.<\/p>\n<table style=\"width: 20%\" summary=\"Two columns and five rows. The first column is labeled,\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th>Letter grade<\/th>\n<th>Grade point average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>A<\/td>\n<td>4.0<\/td>\n<\/tr>\n<tr>\n<td>B<\/td>\n<td>3.0<\/td>\n<\/tr>\n<tr>\n<td>C<\/td>\n<td>2.0<\/td>\n<\/tr>\n<tr>\n<td>D<\/td>\n<td>1.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137561844\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/p>\n<p>To visualize this concept, let\u2019s look again at the two simple functions sketched in (a)and (b) of Figure 10.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200453\/CNX_Precalc_Figure_01_01_0012.jpg\" alt=\"Three relations that demonstrate what constitute a function.\" width=\"975\" height=\"243\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p>The function in part (a) shows a relationship that is not a one-to-one function because inputs [latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n[\/latex]. The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: One-to-One Function<\/h3>\n<p>A one-to-one function is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Which table represents a one-to-one function?<\/p>\n<p>a)<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td>input<\/td>\n<td>output<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>-1<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>-5<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>b)<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td>input<\/td>\n<td>output<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>16<\/td>\n<td>32<\/td>\n<\/tr>\n<tr>\n<td>32<\/td>\n<td>64<\/td>\n<\/tr>\n<tr>\n<td>64<\/td>\n<td>128<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q945171\">Show Solution<\/span><\/p>\n<div id=\"q945171\" class=\"hidden-answer\" style=\"display: none\">\n<p>Table a) maps the output value 2 to two different input values, therefore\u00a0this is NOT a one-to-one function.<\/p>\n<p>Table b) maps each output to one unique input, therefore this IS a one-to-one function.<\/p>\n<h4>Answer<\/h4>\n<p>Table b) is one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show an example of using tables of values to determine whether a function is one-to-one.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Determine if a Relation Given as a Table is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QFOJmevha_Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 style=\"text-align: left\">Using the Horizontal Line Test<\/h2>\n<p id=\"fs-id1165137871503\">An easy way to determine whether a function\u00a0is a one-to-one function is to use the <strong>horizontal line test <\/strong>on the graph of the function. \u00a0To do this, draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\n<div class=\"note precalculus howto textbox\">\n<h3 style=\"text-align: center\"><strong>How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/strong><\/h3>\n<ol id=\"fs-id1165137611853\">\n<li>Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\n<li>If there is any such line, determine that the function is not one-to-one.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>For\u00a0the following graphs, determine which represent one-to-one functions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200511\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783411\">Show Answer<\/span><\/p>\n<div id=\"q783411\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135185190\">The function in (a) is\u00a0not one-to-one. Using the horizontal line test, as shown below, it intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)<\/p>\n<figure id=\"Figure_01_01_010\" class=\"small\"><\/figure>\n<figure id=\"Figure_01_01_010\" class=\"small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200515\/CNX_Precalc_Figure_01_01_010.jpg\" alt=\"\" width=\"487\" height=\"445\" \/><\/figure>\n<p>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2697 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212200\/Screen-Shot-2016-07-16-at-2.21.48-PM-287x300.png\" alt=\"Graph of a line with three dashed horizontal lines passing through it.\" width=\"287\" height=\"300\" \/><\/p>\n<p>The function (c) is not one-to-one, and is in fact not a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16212527\/Screen-Shot-2016-07-16-at-2.25.36-PM-237x300.png\" alt=\"Graph of a circle with two dashed lines passing through horizontally\" width=\"279\" height=\"353\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of using the horizontal line test to determine whether a graph represents a one-to-one function.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 1:  Determine if the Graph of a Relation is a One-to-One Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tbSGdcSN8RE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>In real life and in algebra, different variables are often linked. When a change in value of one variable causes a change in the value of another variable, their interaction is called a relation. A relation has an input value which corresponds to an output value. When each input value has one and only one output value, that relation is a function. Functions can be written as ordered pairs, tables, or graphs. The set of input values is called the domain, and the set of output values is called the range.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-82\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for Free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/li><li>Unit 17: Functions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Determine the Domain and Range of the Graph of a Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QAxZEelInJc\">https:\/\/youtu.be\/QAxZEelInJc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Determine if a Relation Given as a Table is a One-to-One Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QFOJmevha_Y\">https:\/\/youtu.be\/QFOJmevha_Y<\/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>Question ID#2316. <strong>Authored by<\/strong>: Lippman,David. <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":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for Free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\"},{\"type\":\"cc\",\"description\":\"Unit 17: Functions, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Determine the Domain and Range of the Graph of a Function\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QAxZEelInJc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Determine if a Relation Given as a Table is a One-to-One Function\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QFOJmevha_Y\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID#2316\",\"author\":\"Lippman,David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"11a010d2-7d5a-48da-907f-4afb1fc4a315","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-82","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/82","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":29,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/82\/revisions"}],"predecessor-version":[{"id":4867,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/82\/revisions\/4867"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/82\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=82"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=82"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=82"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=82"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}