{"id":71,"date":"2016-06-01T20:50:18","date_gmt":"2016-06-01T20:50:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=71"},"modified":"2016-10-03T21:04:26","modified_gmt":"2016-10-03T21:04:26","slug":"17-1-1-identifying-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tallahassee-intermediatealgebra\/chapter\/17-1-1-identifying-functions\/","title":{"raw":"Functions and Their Notation","rendered":"Functions and Their Notation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Define a function using tables<\/li>\r\n \t<li>Define a function from a set of ordered pairs<\/li>\r\n \t<li>Define the domain and range of a function given as a table or a set of ordered pairs<\/li>\r\n \t<li>Write functions using algebraic notation<\/li>\r\n \t<li>Use the vertical line test to determine whether a graph represents a function<\/li>\r\n \t<li>Given a function described by an equation, find function values (outputs) for\u00a0numerical inputs<\/li>\r\n \t<li>Given a function described by an equation, find function values (outputs) for variable inputs<\/li>\r\n<\/ul>\r\n<\/div>\r\nAlgebra gives us a way to explore and describe relationships. Imagine tossing a ball straight up in the air and watching it rise to reach its highest point before dropping back down into your hands. As time passes, the height of the ball changes. There is a relationship between the amount of time that has elapsed since the toss and the height of the ball. In mathematics, a correspondence between variables that change together (such as time and height) is called a relation. Some, but not all, relations can also be described as functions.\r\n\r\nThere are many kinds of relations. Relations are simply correspondences between sets of values or information. Think about members of your family and their ages. The pairing of each member of your family and their age is a relation. Each family member can be paired with an age in the set of ages of your family members. Another example of a relation is the pairing of a state with its United States\u2019 senators. Each state can be matched with two individuals who have been elected to serve as senator. In turn, each senator can be matched with one specific state that he or she represents. Both of these are real-life examples of relations.\r\n\r\nThe first value of a relation is an input value and the second value is the output value. A <strong>function<\/strong> is a specific type of relation in which each input value has one and only one output value. An input is the <i>independent<\/i> value, and the output value is the <i>dependent <\/i>value, as it depends on the value of the input.\r\n\r\nNotice in the first table below, where the input is \u201cname\u201d and the output is \u201cage,\u201d each input matches with exactly one output. This is an example of a function.\r\n<table style=\"width: 50%;\">\r\n<thead>\r\n<tr>\r\n<th scope=\"row\">Family Member's Name (Input)<\/th>\r\n<th>Family Member's Age<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td scope=\"row\">Nellie<\/td>\r\n<td>13<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Marcos<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Esther<\/td>\r\n<td>46<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Samuel<\/td>\r\n<td>47<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Nina<\/td>\r\n<td>47<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Paul<\/td>\r\n<td>47<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Katrina<\/td>\r\n<td>21<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Andrew<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Maria<\/td>\r\n<td>13<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Ana<\/td>\r\n<td>81<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCompare this with the next table, where the input is \u201cage\u201d and the output is \u201cname.\u201d Some of the inputs result in more than one output. This is an example of a correspondence that is <i>not <\/i>a function.\r\n<table style=\"width: 50%;\">\r\n<thead>\r\n<tr>\r\n<th scope=\"row\">Starting Information (Input)\r\n\r\nFamily Member\u2019s Age<\/th>\r\n<th>Related Information (Output)\r\n\r\nFamily Member\u2019s Name<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td scope=\"row\">11<\/td>\r\n<td>Marcos<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>13<\/td>\r\n<td>\n\nNellie\r\n\r\nMaria<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">16<\/td>\r\n<td>Andrew<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">21<\/td>\r\n<td>Katrina<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">46<\/td>\r\n<td>Esther<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">47<\/td>\r\n<td>\n\nSamuel\r\n\r\nNina\r\n\r\nPaul<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">81<\/td>\r\n<td>Ana<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet\u2019s look back at our examples to determine whether the relations are functions or not and under what circumstances. Remember that a relation is a function if there is only <em><strong>one <\/strong><\/em>output for each input.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFill in the table.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th scope=\"row\">Input<\/th>\r\n<th>Output<\/th>\r\n<th>Function?<\/th>\r\n<th>Why or why not?<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td scope=\"row\">Name of senator<\/td>\r\n<td>Name of state<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Name of state<\/td>\r\n<td>Name of senator<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Time elapsed<\/td>\r\n<td>Height of a tossed ball<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Height of a tossed ball<\/td>\r\n<td>Time elapsed<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Number of cars<\/td>\r\n<td>Number of tires<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Number of tires<\/td>\r\n<td>Number of cars<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"842346\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"842346\"]\r\n<table>\r\n<thead>\r\n<tr>\r\n<th scope=\"row\">Input<\/th>\r\n<th>Output<\/th>\r\n<th>Function?<\/th>\r\n<th>Why or why not?<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td scope=\"row\">Name of senator<\/td>\r\n<td>Name of state<\/td>\r\n<td>Yes<\/td>\r\n<td>For each input, there will only be one output because a senator only represents one state.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Name of state<\/td>\r\n<td>Name of senator<\/td>\r\n<td>No<\/td>\r\n<td>For each state that is an input, 2 names of senators would result because each state has two senators.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Time elapsed<\/td>\r\n<td>Height of a tossed ball<\/td>\r\n<td>Yes<\/td>\r\n<td>At a specific time, the ball has one specific height.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Height of a tossed ball<\/td>\r\n<td>Time elapsed<\/td>\r\n<td>No<\/td>\r\n<td>Remember that the ball was tossed up and fell down. So for a given height, there could be two different times when the ball was at that height. The input height can result in more than one output.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Number of cars<\/td>\r\n<td>Number of tires<\/td>\r\n<td>Yes<\/td>\r\n<td>For any input of a specific number of cars, there is one specific output representing the number of tires.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Number of tires<\/td>\r\n<td>Number of cars<\/td>\r\n<td>Yes<\/td>\r\n<td>For any input of a specific number of tires, there is one specific output representing the number of cars.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRelations can be written as ordered pairs of numbers or as numbers in a table of values. By examining the inputs (<i>x<\/i>-coordinates) and outputs (<i>y<\/i>-coordinates), you can determine whether or not the relation is a function. Remember, in a function each input has only one output.\r\n\r\nThere 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\nIf you have a set of ordered pairs, you can find the domain by listing all of the input values, which are the <i>x<\/i>-coordinates. And to find the range, list all of the output values, which are the <i>y<\/i>-coordinates.\r\n\r\nSo for the following set of ordered pairs,\r\n\r\n[latex]\\{(\u22122,0),(0,6),(2,12),(4,18)\\}[\/latex]\r\n\r\nYou have the following:\r\n\r\n[latex]\\begin{array}{l}\\text{Domain}:\\{\u22122,0,2,4\\}\\\\\\text{Range}:\\{0,6,12,18\\}\\end{array}\\\\[\/latex]\r\n\r\nYou try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nList the domain and range for the following table of values where <em>x<\/em> is the input and <em>y<\/em> is the output.\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<th scope=\"row\"><i>x<\/i><\/th>\r\n<th><i>y<\/i><\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22123[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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.\r\n\r\n[latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}\\\\[\/latex]\r\n\r\nThe range describes all the outputs.\r\n\r\n[latex]\\text{Range}:\\{4\\}\\\\[\/latex]\r\n\r\nWe only listed 4 once because it is not necessary to list it every time it appears in the range.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we provide another example of identifying whether a table of values represents a function, as well as determining the domain and range of the sets.\r\nhttps:\/\/youtu.be\/y2TqnP_6M1s\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDefine the domain and range for the following set of ordered pairs, and determine whether the relation given is a function.\r\n<p style=\"text-align: center;\">[latex]\\{(\u22123,\u22126),(\u22122,\u22121),(1,0),(1,5),(2,0)\\}[\/latex]<\/p>\r\n[reveal-answer q=\"507050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507050\"]\r\n\r\nWe list all of the input values as the domain. \u00a0The input values are represented first in the ordered pair as a matter of convention.\r\n\r\nDomain: {-3,-2,1,2}\r\n\r\nNote how we didn't enter repeated values more than once, it is not necessary.\r\n\r\nThe range is the list of outputs for the relation, they are entered second in the ordered pair.\r\n\r\nRange: {-6, -1, 0, 5}\r\n\r\nOrganizing the ordered pairs in a table can help you tell whether this relation is a function. \u00a0By definition, the inputs in a function have only one output.\r\n<table style=\"width: 50%;\">\r\n<tbody>\r\n<tr>\r\n<th scope=\"row\"><i>x<\/i><\/th>\r\n<th><i>y<\/i><\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22123[\/latex]<\/td>\r\n<td>[latex]\u22126[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]1[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]2[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\nDomain: {-3,-2,1,2}\r\n\r\nRange: {-6, -1, 0, 5}\r\n\r\nThe relation is not a function because the input 1 has two outputs: 0 and 5.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show how to determine whether a relation is a function, and define the domain and range.\r\nhttps:\/\/youtu.be\/kzgLfwgxE8g\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDefine the domain and range of this relation and determine whether it is\u00a0a function.\r\n<p style=\"text-align: center;\">[latex]\\{(\u22123, 4),(\u22122, 4),( \u22121, 4),(2, 4),(3, 4)\\}[\/latex]<\/p>\r\n[reveal-answer q=\"587362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587362\"]\r\n\r\nDomain: {-3, -2, -1, 2, 3}\r\n\r\nRange: {4}\r\n\r\nTo help you determine whether this is a function, you could reorganize the information by creating a table.\r\n<table style=\"width: 50%;\">\r\n<tbody>\r\n<tr>\r\n<th scope=\"row\"><i>x<\/i><\/th>\r\n<th><i>y<\/i><\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22123[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEach input has only one output, and the fact that it is the same output (4) does not matter.\r\n<h4>Answer<\/h4>\r\nDomain: {-3, -2, -1, 2, 3}\r\n\r\nRange: {4}\r\n\r\nThis relation is a function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSome 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 3 and add 2\u201d or \u201cdivide by 5, add 25, and multiply by [latex]\u22121[\/latex].\u201d If you put a number into the machine, a new number will pop 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<i>f\u201d <\/i>for function. If you put <i>x <\/i>into the box, <i>f<\/i>(<i>x<\/i>)<i>, <\/i>comes out. Mathematically speaking, <i>x<\/i> is the input, or the \u201cindependent variable,\u201d and <i>f<\/i>(<i>x<\/i>) is the output, or the \u201cdependent variable,\u201d since it depends on the value of <i>x<\/i>.\r\n\r\n[latex]f(x)=4x+1[\/latex] is written in function notation and is read \u201c<i>f <\/i>of <i>x<\/i> equals 4<i>x<\/i> plus 1.\u201d It represents the following situation: A function named <i>f <\/i>acts upon an input, <i>x, <\/i>and produces <i>f<\/i>(<i>x<\/i>) which is equal to [latex]4x+1[\/latex]. This is the same as the equation as [latex]y=4x+1[\/latex].\r\n\r\nFunction notation gives you more flexibility because you don\u2019t have to use <i>y<\/i> for every equation. Instead, you could use <i>f<\/i>(<i>x<\/i>) or <i>g<\/i>(<i>x<\/i>) or <i>c<\/i>(<i>x<\/i>). This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.\r\n<h2>Using Function Notation<\/h2>\r\nOnce we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <strong>function notation<\/strong> is one representation that facilitates working with functions.\r\n\r\nNow you try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRepresent height as a function of age using function notation.\r\n[reveal-answer q=\"221008\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"221008\"]\r\n\r\nTo represent \"height is a function of age,\" we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age.\r\n\r\n[latex]\\begin{array}{ccc}h\\text{ is }f\\text{ of }a\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We name the function }f;\\text{ height is a function of age}.\\hfill \\\\ h=f\\left(a\\right)\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We use parentheses to indicate the function input}\\text{. }\\hfill \\\\ f\\left(a\\right)\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{We name the function }f;\\text{ the expression is read as ''}f\\text{ of }a\\text{.''}\\hfill \\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Analysis of the solution<\/h3>\r\nWe can use any letter to name the function; the notation [latex]h\\left(a\\right)[\/latex] shows us that [latex]h[\/latex] depends on [latex]a[\/latex]. The value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.\r\n\r\nLet's try another.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<ol>\r\n \t<li>Write the formula for perimeter of a square, [latex]P=4s[\/latex], as a function.<\/li>\r\n \t<li>Write the formula for area of a square, [latex]A=l^{2}[\/latex], as a function.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"136183\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"136183\"]\r\n<ol>\r\n \t<li>Name the function P. P is a function of the length of the sides, s. Perimeter as a function of side length is equal to 4 times side length.\u00a0[latex]P(s)=4s[\/latex]<\/li>\r\n \t<li>Name the function A. \u00a0Area as a function of the length of the sides is equal to the length squared.[latex]A(l)=l^{2}[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis would make it easy to graph both functions on the same graph without confusion about the variables.\r\n\r\nWe can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means \"first add <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, and the result is the input for the function <em data-effect=\"italics\">f<\/em>.\" The operations must be performed in this order to obtain the correct result.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Function Notation<\/h3>\r\nThe notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as \"[latex]y[\/latex] is a function of [latex]x[\/latex].\" The letter [latex]x[\/latex] represents the input value, or independent variable. The letter <em>y\u00a0<\/em> or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse function notation to represent a function whose input is the name of a month and output is the number of days in that month.\r\n\r\n[reveal-answer q=\"5489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5489\"]The number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a \"rule\" that associates a specific number (the output) with each input.\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\/25200459\/CNX_Precalc_Figure_01_01_0052.jpg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" data-media-type=\"image\/jpg\" \/>\r\n\r\nFor example, [latex]f\\left(\\text{March}\\right)=31[\/latex], because March has 31 days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?\r\n\r\n[reveal-answer q=\"226737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"226737\"]\r\n\r\nWhen we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers [latex]\\left(N\\right)[\/latex], is 300. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were 300 police officers in the town.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following videos we show two more\u00a0examples of how to express a relationship\u00a0using function notation.\r\n\r\nhttps:\/\/youtu.be\/lF0fzdaxU_8\r\n\r\nhttps:\/\/youtu.be\/nAF_GZFwU1g\r\n<h2>Graphs of functions<\/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 <i>x<\/i>-axis and the dependent value is plotted on the <i>y<\/i>-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 y = f(x), or y is a function of x, we would write ordered pairs (x, f(x)) using function notation instead of (x,y) as you may have seen previously.\r\n\r\n<img class=\"wp-image-2679 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16195741\/Screen-Shot-2016-07-16-at-12.56.58-PM-300x281.png\" alt=\"Coordinate axes with y-axis labeled f(x) and the x axis labeled x.\" width=\"408\" height=\"382\" \/>\r\n\r\nWe can identify whether the graph of a relation represents a function\u00a0because for each <i>x<\/i>-coordinate there will be exactly one <i>y<\/i>-coordinate.\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\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/><\/p>\r\nWhen a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of <i>x<\/i>.\r\n\r\nIf, on the other hand, a graph shows two or more intersections with a vertical line, then an input (<i>x<\/i>-coordinate) can have more than one output (<i>y<\/i>-coordinate), and <i>y<\/i> is not a function of <i>x<\/i>. 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 \u201cvertical line test.\u201d\r\n\r\nYou try it.\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 Answer[\/reveal-answer]\r\n[hidden-answer a=\"28965\"]\r\n\r\nThis relationship cannot be a function, because some of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-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\nThe vertical line method can also be applied to a set of ordered pairs plotted on a coordinate plane to determine if the relation is a function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the ordered pairs\r\n\r\n[latex]\\{(\u22121,3),(\u22122,5),(\u22123,3),(\u22125,\u22123)\\}[\/latex], plotted on the graph below. Use the vertical line test to determine whether the set of ordered pairs represents a function.\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\/18232420\/image003.jpg\" alt=\"The points (\u22121,3); (\u22122,5); (\u22123,3); and (\u22125,\u22123).\" width=\"329\" height=\"328\" \/><\/p>\r\n<p align=\"center\">[reveal-answer q=\"114452\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"114452\"]<\/p>\r\n<p align=\"center\">Drawing vertical lines through each point results in each line only touching one point. This means that none of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates, so this is a function.<\/p>\r\n<p align=\"center\"><img class=\"size-full wp-image-2514 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/14213709\/Edit_Page_%E2%80%B9_Intermediate_Algebra_%E2%80%94_Pressbooks.png\" alt=\"Edit_Page_\u2039_Intermediate_Algebra_\u2014_Pressbooks\" width=\"298\" height=\"294\" \/><\/p>\r\n<p align=\"center\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn another set of ordered pairs, [latex]\\{(3,\u22121),(5,\u22122),(3,\u22123),(\u22123,5)\\}[\/latex], one of the inputs, 3, can produce two different outputs, [latex]\u22121[\/latex] and [latex]\u22123[\/latex]. You know what that means\u2014this set of ordered pairs is not a function. A plot confirms this.\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\/18232421\/image004.jpg\" alt=\"The point negative 3, 5; the point 5, negative 2. A line through the points negative 1, 3; and the point negative 3, negative 3.\" width=\"337\" height=\"336\" \/><\/p>\r\nNotice that a vertical line passes through two plotted points. One <i>x<\/i>-coordinate has multiple <i>y<\/i>-coordinates. This relation is not a function.\r\n\r\nIn the following video we show another example of determining whether a graph represents a function using the vertical line test.\r\n\r\nhttps:\/\/youtu.be\/5Z8DaZPJLKY\r\n<h2>Evaluate Functions<\/h2>\r\nThroughout this course, you have been working with algebraic equations. Many of these equations are functions. For example, [latex]y=4x+1[\/latex] is an equation that can also represent a function. When you input values for <em>x<\/em>, you can determine a single output for <em>y<\/em>. In this case, if you substitute [latex]x=10[\/latex] into the equation you will find that y must be 41; there is no other value of y that would make the equation true.\r\n\r\nRather than using the variable y, the equations of functions can be written using <strong>function notation<\/strong>. Function notation is very useful when you are working with more than one function at a time, and substituting more than one variable in for <em>x<\/em>.\r\n\r\nEquations written using function notation can also be evaluated. With function notation, you might see a problem like this.\r\n\r\n<i>Given <\/i>[latex]f(x)=4x+1[\/latex]<i>, find f<\/i>(<i>2<\/i>)<i>.<\/i>\r\n\r\nYou read this problem like this: \u201cgiven <i>f<\/i> of <i>x<\/i> equals 4<i>x<\/i> plus one, find <i>f<\/i> of 2.\u201d While the notation and wording is different, the process of evaluating a function is the same as evaluating an equation: in both cases, you substitute 2 for <i>x<\/i>, multiply it by 4 and add 1, simplifying to get 9. In both a function and an equation, an input of 2 results in an output of 9.\r\n<p style=\"text-align: center;\">[latex]f(x)=4x+1\\\\f(2)=4(2)+1=8+1=9[\/latex]<\/p>\r\nYou can simply apply what you already know about evaluating expressions to evaluate a function. It\u2019s important to note that the parentheses that are part of function notation do not mean multiply. The notation <i>f<\/i>(<i>x<\/i>) does not mean <i>f<\/i> multiplied by <i>x<\/i>. Instead the notation means \u201c<i>f<\/i> of <i>x<\/i>\u201d or \u201cthe function of <i>x<\/i>\u201d To evaluate the function, take the value given for <i>x<\/i>, and substitute that value in for <i>x<\/i> in the expression. Let\u2019s look at a couple of examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGiven [latex]f(x)=3x\u20134[\/latex],\u00a0find f(5).\r\n\r\n[reveal-answer q=\"42679\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42679\"]Substitute 5 in for <i>x <\/i>in the function.\r\n<p style=\"text-align: center;\">[latex]f(5)=3(5)-4[\/latex]<\/p>\r\nSimplify the expression on the right side of the equation.\r\n<p style=\"text-align: center;\">[latex]f(5)=15-4\\\\f(5)=11[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nGiven [latex]f(x)=3x\u20134[\/latex], [latex]f(5)=11[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nFunctions can be evaluated for negative values of <i>x<\/i>, too. Keep in mind the rules for integer operations.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGiven [latex]p(x)=2x^{2}+5[\/latex], find [latex]p(\u22123)[\/latex].\r\n\r\n[reveal-answer q=\"489384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489384\"]Substitute [latex]-3[\/latex] in for <i>x <\/i>in the function.\r\n<p style=\"text-align: center;\">[latex]p(\u22123)=2(\u22123)^{2}+5[\/latex]<\/p>\r\nSimplify the expression on the right side of the equation.\r\n<p style=\"text-align: center;\">[latex]p(\u22123)=2(9)+5\\\\p(\u22123)=18+5\\\\p(\u22123)=23[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nGiven [latex]p(x)=2x^{2}+5[\/latex], [latex]p(\u22123)=23[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may also be asked to evaluate a function for more than one value as shown in the example that follows.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGiven [latex]f(x)=|4x-3|[\/latex], find [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\"]Treat each of these like three separate problems. In each case, you substitute the value in for <em>x<\/em> and simplify. Start with [latex]x=0[\/latex].\r\n<p style=\"text-align: center;\">[latex]f(0)=|4(0)-3|\\\\=|-3|\\\\f(0)=3[\/latex]<\/p>\r\nEvaluate for [latex]x=2[\/latex].\r\n<p style=\"text-align: center;\">[latex]f(2)=|4(2)-3|\\\\=|5|\\\\f(2)=5[\/latex]<\/p>\r\nEvaluate for [latex]x=\u22121[\/latex].\r\n<p style=\"text-align: center;\">[latex]f(\u22121)=|4(-1)-3|\\\\=|-7|\\\\f(-1)=7[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nGiven\u00a0[latex]f(x)=|4x-3|[\/latex], [latex]f(0)=3[\/latex], [latex]f(2)=5[\/latex], and [latex]f(\u20121)=7[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Variable Inputs<\/h2>\r\nSo far, you have evaluated functions for inputs that have been constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following examples show how to evaluate a function for a variable input.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGiven [latex]f(x)=3x^{2}+2x+1[\/latex], find <i>f<\/i>(<i>b<\/i>).\r\n[reveal-answer q=\"213691\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"213691\"]This problem is asking you to evaluate the function for <i>b<\/i>. This means substitute <i>b<\/i> in the equation for <i>x.<\/i>\r\n\r\n[latex]f(b)=3b^{2}+2b+1[\/latex]\r\n\r\n(That\u2019s it\u2014you\u2019re done.)\r\n<h4>Answer<\/h4>\r\nGiven [latex]f(x)=3x^{2}+2x+1[\/latex], [latex]f(b)=3b^{2}+2b+1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following example, you evaluate a function for an expression. So here you will substitute the entire expression in for <i>x<\/i> and simplify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGiven [latex]f(x)=4x+1[\/latex], find [latex]f(h+1)[\/latex].\r\n\r\n[reveal-answer q=\"943471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"943471\"]This time, you substitute [latex](h+1)[\/latex] into the equation for <i>x.<\/i>\r\n\r\n[latex]f(h+1)=4(h+1)+1[\/latex]<i>\u00a0<\/i>\r\n\r\nUse the distributive property on the right side, and then combine like terms to simplify.\r\n\r\n[latex]f(h+1)=4h+4+1=4h+5[\/latex]\r\n<h4>Answer<\/h4>\r\nGiven [latex]f(x)=4x+1[\/latex], [latex]f(h+1)=4h+5[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show more examples of evaluating functions for both integer and variable inputs.\r\n\r\nhttps:\/\/youtu.be\/_bi0B2zibOg\r\n<h2>Summary<\/h2>\r\n<div class=\"textbox\">\r\n\r\nGiven a relationship between two quantities, determine whether the relationship is a function.\r\n<ol>\r\n \t<li>Identify the input values - this is your domain.<\/li>\r\n \t<li>Identify the output values - this is your range.<\/li>\r\n \t<li>If each value in the domain leads to only one value in the range, classify the relationship as a function. If any value in the domain leads to two or more values in the range, do not classify the relationship as a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\nFunction notation takes the form such as [latex]f(x)=18x\u201310[\/latex] and is read \u201c<i>f <\/i>of <i>x <\/i>equals eighteen times <i>x <\/i>minus 10.\u201d Function notation can use letters other than <i>f, <\/i>such as <i>c<\/i>(<i>x<\/i>)<i>,<\/i> <i>g<\/i>(<i>x<\/i>), or <i>h<\/i>(<i>x<\/i>). As you go further in your study of functions, this notation will provide you more flexibility, allowing you to examine and compare different functions more easily. Just as an algebraic equation written in <i>x <\/i>and <i>y<\/i> can be evaluated for different values of the input <i>x, <\/i>an equation written in function notation can also be evaluated for different values of <i>x<\/i>. To evaluate a function, substitute in values for <i>x <\/i>and simplify to find the related output.\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Define a function using tables<\/li>\n<li>Define a function from a set of ordered pairs<\/li>\n<li>Define the domain and range of a function given as a table or a set of ordered pairs<\/li>\n<li>Write functions using algebraic notation<\/li>\n<li>Use the vertical line test to determine whether a graph represents a function<\/li>\n<li>Given a function described by an equation, find function values (outputs) for\u00a0numerical inputs<\/li>\n<li>Given a function described by an equation, find function values (outputs) for variable inputs<\/li>\n<\/ul>\n<\/div>\n<p>Algebra gives us a way to explore and describe relationships. Imagine tossing a ball straight up in the air and watching it rise to reach its highest point before dropping back down into your hands. As time passes, the height of the ball changes. There is a relationship between the amount of time that has elapsed since the toss and the height of the ball. In mathematics, a correspondence between variables that change together (such as time and height) is called a relation. Some, but not all, relations can also be described as functions.<\/p>\n<p>There are many kinds of relations. Relations are simply correspondences between sets of values or information. Think about members of your family and their ages. The pairing of each member of your family and their age is a relation. Each family member can be paired with an age in the set of ages of your family members. Another example of a relation is the pairing of a state with its United States\u2019 senators. Each state can be matched with two individuals who have been elected to serve as senator. In turn, each senator can be matched with one specific state that he or she represents. Both of these are real-life examples of relations.<\/p>\n<p>The first value of a relation is an input value and the second value is the output value. A <strong>function<\/strong> is a specific type of relation in which each input value has one and only one output value. An input is the <i>independent<\/i> value, and the output value is the <i>dependent <\/i>value, as it depends on the value of the input.<\/p>\n<p>Notice in the first table below, where the input is \u201cname\u201d and the output is \u201cage,\u201d each input matches with exactly one output. This is an example of a function.<\/p>\n<table style=\"width: 50%;\">\n<thead>\n<tr>\n<th scope=\"row\">Family Member&#8217;s Name (Input)<\/th>\n<th>Family Member&#8217;s Age<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td scope=\"row\">Nellie<\/td>\n<td>13<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Marcos<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Esther<\/td>\n<td>46<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Samuel<\/td>\n<td>47<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Nina<\/td>\n<td>47<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Paul<\/td>\n<td>47<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Katrina<\/td>\n<td>21<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Andrew<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Maria<\/td>\n<td>13<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Ana<\/td>\n<td>81<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Compare this with the next table, where the input is \u201cage\u201d and the output is \u201cname.\u201d Some of the inputs result in more than one output. This is an example of a correspondence that is <i>not <\/i>a function.<\/p>\n<table style=\"width: 50%;\">\n<thead>\n<tr>\n<th scope=\"row\">Starting Information (Input)<\/p>\n<p>Family Member\u2019s Age<\/th>\n<th>Related Information (Output)<\/p>\n<p>Family Member\u2019s Name<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td scope=\"row\">11<\/td>\n<td>Marcos<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>\n<p>Nellie<\/p>\n<p>Maria<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">16<\/td>\n<td>Andrew<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">21<\/td>\n<td>Katrina<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">46<\/td>\n<td>Esther<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">47<\/td>\n<td>\n<p>Samuel<\/p>\n<p>Nina<\/p>\n<p>Paul<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">81<\/td>\n<td>Ana<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let\u2019s look back at our examples to determine whether the relations are functions or not and under what circumstances. Remember that a relation is a function if there is only <em><strong>one <\/strong><\/em>output for each input.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Fill in the table.<\/p>\n<table>\n<thead>\n<tr>\n<th scope=\"row\">Input<\/th>\n<th>Output<\/th>\n<th>Function?<\/th>\n<th>Why or why not?<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td scope=\"row\">Name of senator<\/td>\n<td>Name of state<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Name of state<\/td>\n<td>Name of senator<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Time elapsed<\/td>\n<td>Height of a tossed ball<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Height of a tossed ball<\/td>\n<td>Time elapsed<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Number of cars<\/td>\n<td>Number of tires<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Number of tires<\/td>\n<td>Number of cars<\/td>\n<td><\/td>\n<td><\/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=\"q842346\">Show Answer<\/span><\/p>\n<div id=\"q842346\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<thead>\n<tr>\n<th scope=\"row\">Input<\/th>\n<th>Output<\/th>\n<th>Function?<\/th>\n<th>Why or why not?<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td scope=\"row\">Name of senator<\/td>\n<td>Name of state<\/td>\n<td>Yes<\/td>\n<td>For each input, there will only be one output because a senator only represents one state.<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Name of state<\/td>\n<td>Name of senator<\/td>\n<td>No<\/td>\n<td>For each state that is an input, 2 names of senators would result because each state has two senators.<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Time elapsed<\/td>\n<td>Height of a tossed ball<\/td>\n<td>Yes<\/td>\n<td>At a specific time, the ball has one specific height.<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Height of a tossed ball<\/td>\n<td>Time elapsed<\/td>\n<td>No<\/td>\n<td>Remember that the ball was tossed up and fell down. So for a given height, there could be two different times when the ball was at that height. The input height can result in more than one output.<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Number of cars<\/td>\n<td>Number of tires<\/td>\n<td>Yes<\/td>\n<td>For any input of a specific number of cars, there is one specific output representing the number of tires.<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Number of tires<\/td>\n<td>Number of cars<\/td>\n<td>Yes<\/td>\n<td>For any input of a specific number of tires, there is one specific output representing the number of cars.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Relations can be written as ordered pairs of numbers or as numbers in a table of values. By examining the inputs (<i>x<\/i>-coordinates) and outputs (<i>y<\/i>-coordinates), you can determine whether or not the relation is a function. Remember, in a function each input has only one output.<\/p>\n<p>There 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>If you have a set of ordered pairs, you can find the domain by listing all of the input values, which are the <i>x<\/i>-coordinates. And to find the range, list all of the output values, which are the <i>y<\/i>-coordinates.<\/p>\n<p>So for the following set of ordered pairs,<\/p>\n<p>[latex]\\{(\u22122,0),(0,6),(2,12),(4,18)\\}[\/latex]<\/p>\n<p>You have the following:<\/p>\n<p>[latex]\\begin{array}{l}\\text{Domain}:\\{\u22122,0,2,4\\}\\\\\\text{Range}:\\{0,6,12,18\\}\\end{array}\\\\[\/latex]<\/p>\n<p>You try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>List the domain and range for the following table of values where <em>x<\/em> is the input and <em>y<\/em> is the output.<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<th scope=\"row\"><i>x<\/i><\/th>\n<th><i>y<\/i><\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22123[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22122[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22121[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/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=\"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.<\/p>\n<p>[latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}\\\\[\/latex]<\/p>\n<p>The range describes all the outputs.<\/p>\n<p>[latex]\\text{Range}:\\{4\\}\\\\[\/latex]<\/p>\n<p>We only listed 4 once because it is not necessary to list it every time it appears in the range.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we provide another example of identifying whether a table of values represents a function, as well as determining the domain and range of the sets.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Determine if a Table of Values Represents a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/y2TqnP_6M1s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Define the domain and range for the following set of ordered pairs, and determine whether the relation given is a function.<\/p>\n<p style=\"text-align: center;\">[latex]\\{(\u22123,\u22126),(\u22122,\u22121),(1,0),(1,5),(2,0)\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q507050\">Show Solution<\/span><\/p>\n<div id=\"q507050\" class=\"hidden-answer\" style=\"display: none\">\n<p>We list all of the input values as the domain. \u00a0The input values are represented first in the ordered pair as a matter of convention.<\/p>\n<p>Domain: {-3,-2,1,2}<\/p>\n<p>Note how we didn&#8217;t enter repeated values more than once, it is not necessary.<\/p>\n<p>The range is the list of outputs for the relation, they are entered second in the ordered pair.<\/p>\n<p>Range: {-6, -1, 0, 5}<\/p>\n<p>Organizing the ordered pairs in a table can help you tell whether this relation is a function. \u00a0By definition, the inputs in a function have only one output.<\/p>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<th scope=\"row\"><i>x<\/i><\/th>\n<th><i>y<\/i><\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22123[\/latex]<\/td>\n<td>[latex]\u22126[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22122[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]1[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]2[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>Domain: {-3,-2,1,2}<\/p>\n<p>Range: {-6, -1, 0, 5}<\/p>\n<p>The relation is not a function because the input 1 has two outputs: 0 and 5.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show how to determine whether a relation is a function, and define the domain and range.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1: Find Domain and Range of Ordered Pairs, Function or Not\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/kzgLfwgxE8g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Define the domain and range of this relation and determine whether it is\u00a0a function.<\/p>\n<p style=\"text-align: center;\">[latex]\\{(\u22123, 4),(\u22122, 4),( \u22121, 4),(2, 4),(3, 4)\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587362\">Show Solution<\/span><\/p>\n<div id=\"q587362\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain: {-3, -2, -1, 2, 3}<\/p>\n<p>Range: {4}<\/p>\n<p>To help you determine whether this is a function, you could reorganize the information by creating a table.<\/p>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<th scope=\"row\"><i>x<\/i><\/th>\n<th><i>y<\/i><\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22123[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22122[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]\u22121[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Each input has only one output, and the fact that it is the same output (4) does not matter.<\/p>\n<h4>Answer<\/h4>\n<p>Domain: {-3, -2, -1, 2, 3}<\/p>\n<p>Range: {4}<\/p>\n<p>This relation is a function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>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 3 and add 2\u201d or \u201cdivide by 5, add 25, and multiply by [latex]\u22121[\/latex].\u201d If you put a number into the machine, a new number will pop 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<i>f\u201d <\/i>for function. If you put <i>x <\/i>into the box, <i>f<\/i>(<i>x<\/i>)<i>, <\/i>comes out. Mathematically speaking, <i>x<\/i> is the input, or the \u201cindependent variable,\u201d and <i>f<\/i>(<i>x<\/i>) is the output, or the \u201cdependent variable,\u201d since it depends on the value of <i>x<\/i>.<\/p>\n<p>[latex]f(x)=4x+1[\/latex] is written in function notation and is read \u201c<i>f <\/i>of <i>x<\/i> equals 4<i>x<\/i> plus 1.\u201d It represents the following situation: A function named <i>f <\/i>acts upon an input, <i>x, <\/i>and produces <i>f<\/i>(<i>x<\/i>) which is equal to [latex]4x+1[\/latex]. This is the same as the equation as [latex]y=4x+1[\/latex].<\/p>\n<p>Function notation gives you more flexibility because you don\u2019t have to use <i>y<\/i> for every equation. Instead, you could use <i>f<\/i>(<i>x<\/i>) or <i>g<\/i>(<i>x<\/i>) or <i>c<\/i>(<i>x<\/i>). This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.<\/p>\n<h2>Using Function Notation<\/h2>\n<p>Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <strong>function notation<\/strong> is one representation that facilitates working with functions.<\/p>\n<p>Now you try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Represent height as a function of age using function notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q221008\">Show Answer<\/span><\/p>\n<div id=\"q221008\" class=\"hidden-answer\" style=\"display: none\">\n<p>To represent &#8220;height is a function of age,&#8221; we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age.<\/p>\n<p>[latex]\\begin{array}{ccc}h\\text{ is }f\\text{ of }a\\hfill & \\hfill & \\hfill & \\hfill & \\text{We name the function }f;\\text{ height is a function of age}.\\hfill \\\\ h=f\\left(a\\right)\\hfill & \\hfill & \\hfill & \\hfill & \\text{We use parentheses to indicate the function input}\\text{. }\\hfill \\\\ f\\left(a\\right)\\hfill & \\hfill & \\hfill & \\hfill & \\text{We name the function }f;\\text{ the expression is read as ''}f\\text{ of }a\\text{.''}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the solution<\/h3>\n<p>We can use any letter to name the function; the notation [latex]h\\left(a\\right)[\/latex] shows us that [latex]h[\/latex] depends on [latex]a[\/latex]. The value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\n<p>Let&#8217;s try another.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<ol>\n<li>Write the formula for perimeter of a square, [latex]P=4s[\/latex], as a function.<\/li>\n<li>Write the formula for area of a square, [latex]A=l^{2}[\/latex], as a function.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q136183\">Show Answer<\/span><\/p>\n<div id=\"q136183\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Name the function P. P is a function of the length of the sides, s. Perimeter as a function of side length is equal to 4 times side length.\u00a0[latex]P(s)=4s[\/latex]<\/li>\n<li>Name the function A. \u00a0Area as a function of the length of the sides is equal to the length squared.[latex]A(l)=l^{2}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>This would make it easy to graph both functions on the same graph without confusion about the variables.<\/p>\n<p>We can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means &#8220;first add <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, and the result is the input for the function <em data-effect=\"italics\">f<\/em>.&#8221; The operations must be performed in this order to obtain the correct result.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Function Notation<\/h3>\n<p>The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as &#8220;[latex]y[\/latex] is a function of [latex]x[\/latex].&#8221; The letter [latex]x[\/latex] represents the input value, or independent variable. The letter <em>y\u00a0<\/em> or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q5489\">Show Solution<\/span><\/p>\n<div id=\"q5489\" class=\"hidden-answer\" style=\"display: none\">The number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a &#8220;rule&#8221; that associates a specific number (the output) with each input.<\/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\/25200459\/CNX_Precalc_Figure_01_01_0052.jpg\" alt=\"The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.\" width=\"487\" height=\"107\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>For example, [latex]f\\left(\\text{March}\\right)=31[\/latex], because March has 31 days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q226737\">Show Solution<\/span><\/p>\n<div id=\"q226737\" class=\"hidden-answer\" style=\"display: none\">\n<p>When we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers [latex]\\left(N\\right)[\/latex], is 300. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were 300 police officers in the town.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following videos we show two more\u00a0examples of how to express a relationship\u00a0using function notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Function Notation Application Problem\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lF0fzdaxU_8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Function Notation Application\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/nAF_GZFwU1g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Graphs of functions<\/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 <i>x<\/i>-axis and the dependent value is plotted on the <i>y<\/i>-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 y = f(x), or y is a function of x, we would write ordered pairs (x, f(x)) using function notation instead of (x,y) as you may have seen previously.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2679 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/16195741\/Screen-Shot-2016-07-16-at-12.56.58-PM-300x281.png\" alt=\"Coordinate axes with y-axis labeled f(x) and the x axis labeled x.\" width=\"408\" height=\"382\" \/><\/p>\n<p>We can identify whether the graph of a relation represents a function\u00a0because for each <i>x<\/i>-coordinate there will be exactly one <i>y<\/i>-coordinate.<\/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\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/><\/p>\n<p>When a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of <i>x<\/i>.<\/p>\n<p>If, on the other hand, a graph shows two or more intersections with a vertical line, then an input (<i>x<\/i>-coordinate) can have more than one output (<i>y<\/i>-coordinate), and <i>y<\/i> is not a function of <i>x<\/i>. 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 \u201cvertical line test.\u201d<\/p>\n<p>You try it.<\/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 Answer<\/span><\/p>\n<div id=\"q28965\" class=\"hidden-answer\" style=\"display: none\">\n<p>This relationship cannot be a function, because some of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-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<p>The vertical line method can also be applied to a set of ordered pairs plotted on a coordinate plane to determine if the relation is a function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the ordered pairs<\/p>\n<p>[latex]\\{(\u22121,3),(\u22122,5),(\u22123,3),(\u22125,\u22123)\\}[\/latex], plotted on the graph below. Use the vertical line test to determine whether the set of ordered pairs represents a function.<\/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\/18232420\/image003.jpg\" alt=\"The points (\u22121,3); (\u22122,5); (\u22123,3); and (\u22125,\u22123).\" width=\"329\" height=\"328\" \/><\/p>\n<p style=\"text-align: center;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q114452\">Show Answer<\/span><\/p>\n<div id=\"q114452\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">Drawing vertical lines through each point results in each line only touching one point. This means that none of the <i>x<\/i>-coordinates have two corresponding <i>y<\/i>-coordinates, so this is a function.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2514 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/14213709\/Edit_Page_%E2%80%B9_Intermediate_Algebra_%E2%80%94_Pressbooks.png\" alt=\"Edit_Page_\u2039_Intermediate_Algebra_\u2014_Pressbooks\" width=\"298\" height=\"294\" \/><\/p>\n<p style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<p>In another set of ordered pairs, [latex]\\{(3,\u22121),(5,\u22122),(3,\u22123),(\u22123,5)\\}[\/latex], one of the inputs, 3, can produce two different outputs, [latex]\u22121[\/latex] and [latex]\u22123[\/latex]. You know what that means\u2014this set of ordered pairs is not a function. A plot confirms this.<\/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\/18232421\/image004.jpg\" alt=\"The point negative 3, 5; the point 5, negative 2. A line through the points negative 1, 3; and the point negative 3, negative 3.\" width=\"337\" height=\"336\" \/><\/p>\n<p>Notice that a vertical line passes through two plotted points. One <i>x<\/i>-coordinate has multiple <i>y<\/i>-coordinates. This relation is not a function.<\/p>\n<p>In the following video we show another example of determining whether a graph represents a function using the vertical line test.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5Z8DaZPJLKY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Evaluate Functions<\/h2>\n<p>Throughout this course, you have been working with algebraic equations. Many of these equations are functions. For example, [latex]y=4x+1[\/latex] is an equation that can also represent a function. When you input values for <em>x<\/em>, you can determine a single output for <em>y<\/em>. In this case, if you substitute [latex]x=10[\/latex] into the equation you will find that y must be 41; there is no other value of y that would make the equation true.<\/p>\n<p>Rather than using the variable y, the equations of functions can be written using <strong>function notation<\/strong>. Function notation is very useful when you are working with more than one function at a time, and substituting more than one variable in for <em>x<\/em>.<\/p>\n<p>Equations written using function notation can also be evaluated. With function notation, you might see a problem like this.<\/p>\n<p><i>Given <\/i>[latex]f(x)=4x+1[\/latex]<i>, find f<\/i>(<i>2<\/i>)<i>.<\/i><\/p>\n<p>You read this problem like this: \u201cgiven <i>f<\/i> of <i>x<\/i> equals 4<i>x<\/i> plus one, find <i>f<\/i> of 2.\u201d While the notation and wording is different, the process of evaluating a function is the same as evaluating an equation: in both cases, you substitute 2 for <i>x<\/i>, multiply it by 4 and add 1, simplifying to get 9. In both a function and an equation, an input of 2 results in an output of 9.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=4x+1\\\\f(2)=4(2)+1=8+1=9[\/latex]<\/p>\n<p>You can simply apply what you already know about evaluating expressions to evaluate a function. It\u2019s important to note that the parentheses that are part of function notation do not mean multiply. The notation <i>f<\/i>(<i>x<\/i>) does not mean <i>f<\/i> multiplied by <i>x<\/i>. Instead the notation means \u201c<i>f<\/i> of <i>x<\/i>\u201d or \u201cthe function of <i>x<\/i>\u201d To evaluate the function, take the value given for <i>x<\/i>, and substitute that value in for <i>x<\/i> in the expression. Let\u2019s look at a couple of examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Given [latex]f(x)=3x\u20134[\/latex],\u00a0find f(5).<\/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\">Substitute 5 in for <i>x <\/i>in the function.<\/p>\n<p style=\"text-align: center;\">[latex]f(5)=3(5)-4[\/latex]<\/p>\n<p>Simplify the expression on the right side of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]f(5)=15-4\\\\f(5)=11[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Given [latex]f(x)=3x\u20134[\/latex], [latex]f(5)=11[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Functions can be evaluated for negative values of <i>x<\/i>, too. Keep in mind the rules for integer operations.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Given [latex]p(x)=2x^{2}+5[\/latex], find [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\">Substitute [latex]-3[\/latex] in for <i>x <\/i>in the function.<\/p>\n<p style=\"text-align: center;\">[latex]p(\u22123)=2(\u22123)^{2}+5[\/latex]<\/p>\n<p>Simplify the expression on the right side of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]p(\u22123)=2(9)+5\\\\p(\u22123)=18+5\\\\p(\u22123)=23[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Given [latex]p(x)=2x^{2}+5[\/latex], [latex]p(\u22123)=23[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may also be asked to evaluate a function for more than one value as shown in the example that follows.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Given [latex]f(x)=|4x-3|[\/latex], find [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\">Treat each of these like three separate problems. In each case, you substitute the value in for <em>x<\/em> and simplify. Start with [latex]x=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(0)=|4(0)-3|\\\\=|-3|\\\\f(0)=3[\/latex]<\/p>\n<p>Evaluate for [latex]x=2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(2)=|4(2)-3|\\\\=|5|\\\\f(2)=5[\/latex]<\/p>\n<p>Evaluate for [latex]x=\u22121[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(\u22121)=|4(-1)-3|\\\\=|-7|\\\\f(-1)=7[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Given\u00a0[latex]f(x)=|4x-3|[\/latex], [latex]f(0)=3[\/latex], [latex]f(2)=5[\/latex], and [latex]f(\u20121)=7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Variable Inputs<\/h2>\n<p>So far, you have evaluated functions for inputs that have been constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following examples show how to evaluate a function for a variable input.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Given [latex]f(x)=3x^{2}+2x+1[\/latex], find <i>f<\/i>(<i>b<\/i>).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q213691\">Show Solution<\/span><\/p>\n<div id=\"q213691\" class=\"hidden-answer\" style=\"display: none\">This problem is asking you to evaluate the function for <i>b<\/i>. This means substitute <i>b<\/i> in the equation for <i>x.<\/i><\/p>\n<p>[latex]f(b)=3b^{2}+2b+1[\/latex]<\/p>\n<p>(That\u2019s it\u2014you\u2019re done.)<\/p>\n<h4>Answer<\/h4>\n<p>Given [latex]f(x)=3x^{2}+2x+1[\/latex], [latex]f(b)=3b^{2}+2b+1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following example, you evaluate a function for an expression. So here you will substitute the entire expression in for <i>x<\/i> and simplify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Given [latex]f(x)=4x+1[\/latex], find [latex]f(h+1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q943471\">Show Solution<\/span><\/p>\n<div id=\"q943471\" class=\"hidden-answer\" style=\"display: none\">This time, you substitute [latex](h+1)[\/latex] into the equation for <i>x.<\/i><\/p>\n<p>[latex]f(h+1)=4(h+1)+1[\/latex]<i>\u00a0<\/i><\/p>\n<p>Use the distributive property on the right side, and then combine like terms to simplify.<\/p>\n<p>[latex]f(h+1)=4h+4+1=4h+5[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Given [latex]f(x)=4x+1[\/latex], [latex]f(h+1)=4h+5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show more examples of evaluating functions for both integer and variable inputs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex: Determine Various Function Outputs for a Quadratic Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_bi0B2zibOg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<div class=\"textbox\">\n<p>Given a relationship between two quantities, determine whether the relationship is a function.<\/p>\n<ol>\n<li>Identify the input values &#8211; this is your domain.<\/li>\n<li>Identify the output values &#8211; this is your range.<\/li>\n<li>If each value in the domain leads to only one value in the range, classify the relationship as a function. If any value in the domain leads to two or more values in the range, do not classify the relationship as a function.<\/li>\n<\/ol>\n<\/div>\n<p>Function notation takes the form such as [latex]f(x)=18x\u201310[\/latex] and is read \u201c<i>f <\/i>of <i>x <\/i>equals eighteen times <i>x <\/i>minus 10.\u201d Function notation can use letters other than <i>f, <\/i>such as <i>c<\/i>(<i>x<\/i>)<i>,<\/i> <i>g<\/i>(<i>x<\/i>), or <i>h<\/i>(<i>x<\/i>). As you go further in your study of functions, this notation will provide you more flexibility, allowing you to examine and compare different functions more easily. Just as an algebraic equation written in <i>x <\/i>and <i>y<\/i> can be evaluated for different values of the input <i>x, <\/i>an equation written in function notation can also be evaluated for different values of <i>x<\/i>. To evaluate a function, substitute in values for <i>x <\/i>and simplify to find the related output.<\/p>\n<h2><\/h2>\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-71\">\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><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><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><li>Ex: Function Notation Application Problem. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lF0fzdaxU_8\">https:\/\/youtu.be\/lF0fzdaxU_8<\/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>Function Notation Application. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nAF_GZFwU1g\">https:\/\/youtu.be\/nAF_GZFwU1g<\/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\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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><li>Ex: Determine Various Function Outputs for a Quadratic Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/_bi0B2zibOg\">https:\/\/youtu.be\/_bi0B2zibOg<\/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":21,"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) 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