{"id":1984,"date":"2024-05-09T16:55:26","date_gmt":"2024-05-09T16:55:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1984"},"modified":"2024-08-21T16:02:02","modified_gmt":"2024-08-21T16:02:02","slug":"review-of-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/review-of-functions-learn-it-1\/","title":{"raw":"Review of Functions: Learn It 1","rendered":"Review of Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Determine the set of all possible inputs (domain) and outputs (range) for a function from its graph or equation<\/li>\r\n\t<li>Find where functions cross the x-axis and y-axis by looking at equations, graphs, and data tables<\/li>\r\n\t<li>Interpret graphs and tables to describe function behaviors, including symmetry<\/li>\r\n\t<li>Combine two or more functions to create a new function<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Functions<\/h2>\r\n<p id=\"fs-id1170572203905\">Given two sets [latex]A[\/latex] and [latex]B[\/latex], a set with elements that are ordered pairs [latex](x,y)[\/latex], where [latex]x[\/latex] is an element of [latex]A[\/latex] and [latex]y[\/latex] is an element of [latex]B[\/latex], is a relation from [latex]A[\/latex] to [latex]B[\/latex]. A <strong>relation<\/strong> from [latex]A[\/latex] to [latex]B[\/latex] defines a relationship between those two sets.<\/p>\r\n<p>A <strong>function<\/strong> is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the<em> <span class=\"no-emphasis\">input<\/span><\/em>; the element of the second set is called the <em><span class=\"no-emphasis\">output<\/span><\/em>. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input.<\/p>\r\n<section class=\"textbox example\">\r\n<p>The area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>functions<\/h3>\r\n<p>A\u00a0<strong>function\u00a0<\/strong>[latex]f[\/latex] consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output.<\/p>\r\n<\/section>\r\n<p>For a general function [latex]f[\/latex] with domain [latex]D[\/latex], we often use [latex]x[\/latex] to denote the input and [latex]y[\/latex] to denote the output associated with [latex]x[\/latex]. When doing so, we refer to [latex]x[\/latex] as the <strong>independent variable<\/strong> and [latex]y[\/latex] as the <strong>dependent variable<\/strong>, because it depends on [latex]x[\/latex]. Using function notation, we write [latex]y=f(x)[\/latex], and we read this equation as \"[latex]y[\/latex] equals [latex]f[\/latex] of [latex]x[\/latex].\"\u00a0<\/p>\r\n<p id=\"fs-id1170572151421\">The concept of a function can be visualized using Figure 1.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202106\/CNX_Calc_Figure_01_01_001.jpg\" alt=\"An image with three items. The first item is text that reads \u201cInput, x\u201d. An arrow points from the first item to the second item, which is a box with the label \u201cfunction\u201d. An arrow points from the second item to the third item, which is text that reads \u201cOutput, f(x)\u201d.\" width=\"325\" height=\"110\" \/> Figure 1. A function can be visualized as an input\/output device.[\/caption]\r\n\r\n<h3>Evaluating a Function<\/h3>\r\n<p>Evaluating a function is like finding out what the function does when you give it a specific input. Think of a function as a machine in a factory: you put something in, the machine works on it, and then it gives you something back. In the case of a function, you give it a number, and it gives you another number according to a specific rule.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<h4>How to: Evaluate a Function:<\/h4>\r\n<ol>\r\n\t<li><strong>Identify the input:<\/strong> This is the value that you will put into the function, often represented as '[latex]x[\/latex]'.<\/li>\r\n\t<li><strong>Plug the input into the function:<\/strong> Replace the '[latex]x[\/latex]' in the function's formula with the value of your input.<\/li>\r\n\t<li><strong>Follow the operations:<\/strong> Perform the mathematical operations in the formula with your input value. This means you'll do any addition, subtraction, multiplication, division, exponentiation, etc., that the function tells you to do with that input.<\/li>\r\n\t<li><strong>Simplify:<\/strong> If the function's rule has more than one operation, follow the order of operations (parentheses, exponents, multiplication and division, addition and subtraction) to simplify the expression down to a single number.<\/li>\r\n\t<li><strong>Find the output:<\/strong> The number you end up with after doing all the operations is the output of the function, often represented as '[latex]f(x)[\/latex]' or '[latex]y[\/latex]'.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">For the function [latex]f(x)=3x^2+2x-1[\/latex], evaluate,\r\n\r\n<ol id=\"fs-id1170572177936\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>[latex]f(-2)[\/latex]<\/li>\r\n\t<li>[latex]f(\\sqrt{2})[\/latex]<\/li>\r\n\t<li>[latex]f(a+h)[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170572114772\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572114772\"]<\/p>\r\n<p id=\"fs-id1170572114772\">Substitute the given value for [latex]x[\/latex] in the formula for [latex]f(x)[\/latex].<\/p>\r\n<ol id=\"fs-id1170572223895\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>\r\n<div>\u00a0[latex]f(-2)=3(-2)^2+2(-2)-1=12-4-1=7[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li>\r\n<div>\u00a0[latex]f(\\sqrt{2})=3(\\sqrt{2})^2+2\\sqrt{2}-1=6+2\\sqrt{2}-1=5+2\\sqrt{2}[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li>\r\n<div>\u00a0[latex]\\begin{array}{cc}\\hfill f(a+h)=3(a+h)^2+2(a+h)-1&amp; =3(a^2+2ah+h^2)+2a+2h-1\\hfill \\\\ &amp; =3a^2+6ah+3h^2+2a+2h-1\\hfill \\end{array}[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n<\/ol>\r\n\r\nWatch the following video to see the worked solution to this example.For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\n<p>You can view the transcript for this segmented clip of \"1.1 Review of Functions\" <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions34to179_transcript.txt\" target=\"_blank\" rel=\"noopener\">using this link<\/a> (opens in new window). [\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>Representing Functions<\/h3>\r\n<p id=\"fs-id1170572171352\">Typically, a function is represented using one or more of the following tools:<\/p>\r\n<ul id=\"fs-id1170572295375\">\r\n\t<li>A table<\/li>\r\n\t<li>A graph<\/li>\r\n\t<li>A formula<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170572455608\">We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.<\/p>\r\n<div id=\"fs-id1170572163667\" class=\"bc-section section\">\r\n<h4>Tables<\/h4>\r\n<p id=\"fs-id1170572243966\">Functions described using a <strong>table of values<\/strong> arise frequently in real-world applications. Consider the following simple example.<\/p>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572243966\">We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable [latex]x[\/latex] be the time after midnight, measured in hours, and the output variable [latex]y[\/latex] be the temperature [latex]x[\/latex] hours after midnight, measured in degrees Fahrenheit. We record our data in Table 1.<\/p>\r\n<table id=\"fs-id1170572114646\" summary=\"A table with 12 rows and 4 columns is shown. The first column is labeled \u201chours after midnight\u201d and has the values \u201c0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11\u201d. The second column is labeled \u201cTemperature in Fahrenheit\u201d and has the values are \u201c58; 54; 53; 52; 52; 55; 60; 64; 72; 75; 78; 80.\u201d The third column is labeled \u201chours after midnight\u201d and continues counting where column 1 left off with \u201c12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23\u201d. The fourth column is labeled \u201cTemperature in Fahrenheit\u201d and continues counting where column 2 left off with \u201c84; 85; 85; 83; 82; 80; 77; 74; 69; 65; 60; 58\u201d.\">\r\n<caption>Table 1. Temperature as a Function of Time of Day<\/caption>\r\n<thead>\r\n<tr style=\"height: 30px;\" valign=\"top\">\r\n<th style=\"height: 30px;\">Hours after Midnight<\/th>\r\n<th style=\"height: 30px;\">Temperature [latex](\\text{\u00b0}F)[\/latex]<\/th>\r\n<th style=\"height: 30px;\">Hours after Midnight<\/th>\r\n<th style=\"height: 30px;\">Temperature [latex](\\text{\u00b0}F)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]58[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]84[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]54[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]13[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]85[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]53[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]14[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]85[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.75px;\" valign=\"top\">\r\n<td style=\"height: 15.75px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"height: 15.75px;\">[latex]52[\/latex]<\/td>\r\n<td style=\"height: 15.75px;\">[latex]15[\/latex]<\/td>\r\n<td style=\"height: 15.75px;\">[latex]83[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]52[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]82[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]55[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]17[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]80[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]60[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]18[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]77[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]7[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]64[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]19[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]74[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]72[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]20[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]69[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]9[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]75[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]21[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]65[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]78[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]22[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]60[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"top\">\r\n<td style=\"height: 15px;\">[latex]11[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]80[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]23[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]58[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572106936\">We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]218414[\/ohm_question]<\/p>\r\n<\/section>\r\n<div id=\"fs-id1170572226958\" class=\"bc-section section\">\r\n<h4>Graphs<\/h4>\r\n<p>Given a function [latex]f[\/latex] described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table 1 can give us a better idea of their fluctuation throughout the day. Figure 5 shows the plot of the temperature function.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202116\/CNX_Calc_Figure_01_01_005.jpg\" alt=\"An image of a graph. The y axis runs from 0 to 90 and has the label \u201cTemperature in Fahrenheit\u201d. The x axis runs from 0 to 24 and has the label \u201chours after midnight\u201d. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58).\" width=\"325\" height=\"420\" \/> Figure 5. The graph of the data from Table 1 shows temperature as a function of time.[\/caption]\r\n\r\n<div class=\"wp-caption-text\">\u00a0<\/div>\r\n<p>From the points plotted on the graph in Figure 5, we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table.\u00a0<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202119\/CNX_Calc_Figure_01_01_014.jpg\" alt=\"An image of a graph. The y axis runs from 0 to 90 and has the label \u201cTemperature in Fahrenheit\u201d. The x axis runs from 0 to 24 and has the label \u201chours after midnight\u201d. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58). A line connects all the points on the graph.\" width=\"325\" height=\"421\" \/> Figure 6. Connecting the dots in Figure 5 shows the general pattern of the data.[\/caption]\r\n\r\n<p>In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern.<\/p>\r\n<\/div>\r\n<div id=\"fs-id1170572450089\" class=\"bc-section section\">\r\n<h4>Algebraic Formulas<\/h4>\r\n<p id=\"fs-id1170572295705\">Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications.\u00a0<\/p>\r\n<section class=\"textbox example\">\r\n<p>The area of a circle of radius [latex]r[\/latex] is given by the formula [latex]A(r)=\\pi r^2[\/latex]. When an object is thrown upward from the ground with an initial velocity [latex]v_{0}[\/latex] ft\/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula [latex]s(t)=-16t^2+v_{0}t[\/latex]. When [latex]P[\/latex] dollars are invested in an account at an annual interest rate [latex]r[\/latex] compounded continuously, the amount of money after [latex]t[\/latex] years is given by the formula [latex]A(t)=Pe^{rt}[\/latex]. Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.<\/p>\r\n<\/section>\r\n<p id=\"fs-id1170572216050\">Given an algebraic formula for a function [latex]f[\/latex], the graph of [latex]f[\/latex] is the set of points [latex](x,f(x))[\/latex], where [latex]x[\/latex] is in the domain of [latex]f[\/latex] and [latex]f(x)[\/latex] is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of [latex]f[\/latex] consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.<\/p>\r\n<\/div>\r\n<\/div>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Determine the set of all possible inputs (domain) and outputs (range) for a function from its graph or equation<\/li>\n<li>Find where functions cross the x-axis and y-axis by looking at equations, graphs, and data tables<\/li>\n<li>Interpret graphs and tables to describe function behaviors, including symmetry<\/li>\n<li>Combine two or more functions to create a new function<\/li>\n<\/ul>\n<\/section>\n<h2>Functions<\/h2>\n<p id=\"fs-id1170572203905\">Given two sets [latex]A[\/latex] and [latex]B[\/latex], a set with elements that are ordered pairs [latex](x,y)[\/latex], where [latex]x[\/latex] is an element of [latex]A[\/latex] and [latex]y[\/latex] is an element of [latex]B[\/latex], is a relation from [latex]A[\/latex] to [latex]B[\/latex]. A <strong>relation<\/strong> from [latex]A[\/latex] to [latex]B[\/latex] defines a relationship between those two sets.<\/p>\n<p>A <strong>function<\/strong> is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the<em> <span class=\"no-emphasis\">input<\/span><\/em>; the element of the second set is called the <em><span class=\"no-emphasis\">output<\/span><\/em>. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input.<\/p>\n<section class=\"textbox example\">\n<p>The area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>functions<\/h3>\n<p>A\u00a0<strong>function\u00a0<\/strong>[latex]f[\/latex] consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output.<\/p>\n<\/section>\n<p>For a general function [latex]f[\/latex] with domain [latex]D[\/latex], we often use [latex]x[\/latex] to denote the input and [latex]y[\/latex] to denote the output associated with [latex]x[\/latex]. When doing so, we refer to [latex]x[\/latex] as the <strong>independent variable<\/strong> and [latex]y[\/latex] as the <strong>dependent variable<\/strong>, because it depends on [latex]x[\/latex]. Using function notation, we write [latex]y=f(x)[\/latex], and we read this equation as &#8220;[latex]y[\/latex] equals [latex]f[\/latex] of [latex]x[\/latex].&#8221;\u00a0<\/p>\n<p id=\"fs-id1170572151421\">The concept of a function can be visualized using Figure 1.<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202106\/CNX_Calc_Figure_01_01_001.jpg\" alt=\"An image with three items. The first item is text that reads \u201cInput, x\u201d. An arrow points from the first item to the second item, which is a box with the label \u201cfunction\u201d. An arrow points from the second item to the third item, which is text that reads \u201cOutput, f(x)\u201d.\" width=\"325\" height=\"110\" \/><figcaption class=\"wp-caption-text\">Figure 1. A function can be visualized as an input\/output device.<\/figcaption><\/figure>\n<h3>Evaluating a Function<\/h3>\n<p>Evaluating a function is like finding out what the function does when you give it a specific input. Think of a function as a machine in a factory: you put something in, the machine works on it, and then it gives you something back. In the case of a function, you give it a number, and it gives you another number according to a specific rule.<\/p>\n<section class=\"textbox questionHelp\">\n<h4>How to: Evaluate a Function:<\/h4>\n<ol>\n<li><strong>Identify the input:<\/strong> This is the value that you will put into the function, often represented as &#8216;[latex]x[\/latex]&#8216;.<\/li>\n<li><strong>Plug the input into the function:<\/strong> Replace the &#8216;[latex]x[\/latex]&#8216; in the function&#8217;s formula with the value of your input.<\/li>\n<li><strong>Follow the operations:<\/strong> Perform the mathematical operations in the formula with your input value. This means you&#8217;ll do any addition, subtraction, multiplication, division, exponentiation, etc., that the function tells you to do with that input.<\/li>\n<li><strong>Simplify:<\/strong> If the function&#8217;s rule has more than one operation, follow the order of operations (parentheses, exponents, multiplication and division, addition and subtraction) to simplify the expression down to a single number.<\/li>\n<li><strong>Find the output:<\/strong> The number you end up with after doing all the operations is the output of the function, often represented as &#8216;[latex]f(x)[\/latex]&#8216; or &#8216;[latex]y[\/latex]&#8216;.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">For the function [latex]f(x)=3x^2+2x-1[\/latex], evaluate,<\/p>\n<ol id=\"fs-id1170572177936\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(-2)[\/latex]<\/li>\n<li>[latex]f(\\sqrt{2})[\/latex]<\/li>\n<li>[latex]f(a+h)[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572114772\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572114772\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572114772\">Substitute the given value for [latex]x[\/latex] in the formula for [latex]f(x)[\/latex].<\/p>\n<ol id=\"fs-id1170572223895\" style=\"list-style-type: lower-alpha;\">\n<li>\n<div>\u00a0[latex]f(-2)=3(-2)^2+2(-2)-1=12-4-1=7[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/li>\n<li>\n<div>\u00a0[latex]f(\\sqrt{2})=3(\\sqrt{2})^2+2\\sqrt{2}-1=6+2\\sqrt{2}-1=5+2\\sqrt{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/li>\n<li>\n<div>\u00a0[latex]\\begin{array}{cc}\\hfill f(a+h)=3(a+h)^2+2(a+h)-1& =3(a^2+2ah+h^2)+2a+2h-1\\hfill \\\\ & =3a^2+6ah+3h^2+2a+2h-1\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution to this example.For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the transcript for this segmented clip of &#8220;1.1 Review of Functions&#8221; <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions34to179_transcript.txt\" target=\"_blank\" rel=\"noopener\">using this link<\/a> (opens in new window). <\/div>\n<\/div>\n<\/section>\n<h3>Representing Functions<\/h3>\n<p id=\"fs-id1170572171352\">Typically, a function is represented using one or more of the following tools:<\/p>\n<ul id=\"fs-id1170572295375\">\n<li>A table<\/li>\n<li>A graph<\/li>\n<li>A formula<\/li>\n<\/ul>\n<p id=\"fs-id1170572455608\">We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.<\/p>\n<div id=\"fs-id1170572163667\" class=\"bc-section section\">\n<h4>Tables<\/h4>\n<p id=\"fs-id1170572243966\">Functions described using a <strong>table of values<\/strong> arise frequently in real-world applications. Consider the following simple example.<\/p>\n<section class=\"textbox example\">\n<p>We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable [latex]x[\/latex] be the time after midnight, measured in hours, and the output variable [latex]y[\/latex] be the temperature [latex]x[\/latex] hours after midnight, measured in degrees Fahrenheit. We record our data in Table 1.<\/p>\n<table id=\"fs-id1170572114646\" summary=\"A table with 12 rows and 4 columns is shown. The first column is labeled \u201chours after midnight\u201d and has the values \u201c0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11\u201d. The second column is labeled \u201cTemperature in Fahrenheit\u201d and has the values are \u201c58; 54; 53; 52; 52; 55; 60; 64; 72; 75; 78; 80.\u201d The third column is labeled \u201chours after midnight\u201d and continues counting where column 1 left off with \u201c12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23\u201d. The fourth column is labeled \u201cTemperature in Fahrenheit\u201d and continues counting where column 2 left off with \u201c84; 85; 85; 83; 82; 80; 77; 74; 69; 65; 60; 58\u201d.\">\n<caption>Table 1. Temperature as a Function of Time of Day<\/caption>\n<thead>\n<tr style=\"height: 30px;\" valign=\"top\">\n<th style=\"height: 30px;\">Hours after Midnight<\/th>\n<th style=\"height: 30px;\">Temperature [latex](\\text{\u00b0}F)[\/latex]<\/th>\n<th style=\"height: 30px;\">Hours after Midnight<\/th>\n<th style=\"height: 30px;\">Temperature [latex](\\text{\u00b0}F)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]58[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]12[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]84[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]1[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]54[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]13[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]85[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]2[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]53[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]14[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]85[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.75px;\" valign=\"top\">\n<td style=\"height: 15.75px;\">[latex]3[\/latex]<\/td>\n<td style=\"height: 15.75px;\">[latex]52[\/latex]<\/td>\n<td style=\"height: 15.75px;\">[latex]15[\/latex]<\/td>\n<td style=\"height: 15.75px;\">[latex]83[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]4[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]52[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]16[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]82[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]5[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]55[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]17[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]80[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]60[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]18[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]77[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]7[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]64[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]19[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]74[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]8[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]72[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]20[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]69[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]9[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]75[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]21[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]65[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]10[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]78[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]22[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]60[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px;\">[latex]11[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]80[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]23[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]58[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572106936\">We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm218414\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218414&theme=lumen&iframe_resize_id=ohm218414&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<div id=\"fs-id1170572226958\" class=\"bc-section section\">\n<h4>Graphs<\/h4>\n<p>Given a function [latex]f[\/latex] described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in Table 1 can give us a better idea of their fluctuation throughout the day. Figure 5 shows the plot of the temperature function.<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202116\/CNX_Calc_Figure_01_01_005.jpg\" alt=\"An image of a graph. The y axis runs from 0 to 90 and has the label \u201cTemperature in Fahrenheit\u201d. The x axis runs from 0 to 24 and has the label \u201chours after midnight\u201d. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58).\" width=\"325\" height=\"420\" \/><figcaption class=\"wp-caption-text\">Figure 5. The graph of the data from Table 1 shows temperature as a function of time.<\/figcaption><\/figure>\n<div class=\"wp-caption-text\">\u00a0<\/div>\n<p>From the points plotted on the graph in Figure 5, we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table.\u00a0<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202119\/CNX_Calc_Figure_01_01_014.jpg\" alt=\"An image of a graph. The y axis runs from 0 to 90 and has the label \u201cTemperature in Fahrenheit\u201d. The x axis runs from 0 to 24 and has the label \u201chours after midnight\u201d. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58). A line connects all the points on the graph.\" width=\"325\" height=\"421\" \/><figcaption class=\"wp-caption-text\">Figure 6. Connecting the dots in Figure 5 shows the general pattern of the data.<\/figcaption><\/figure>\n<p>In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern.<\/p>\n<\/div>\n<div id=\"fs-id1170572450089\" class=\"bc-section section\">\n<h4>Algebraic Formulas<\/h4>\n<p id=\"fs-id1170572295705\">Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications.\u00a0<\/p>\n<section class=\"textbox example\">\n<p>The area of a circle of radius [latex]r[\/latex] is given by the formula [latex]A(r)=\\pi r^2[\/latex]. When an object is thrown upward from the ground with an initial velocity [latex]v_{0}[\/latex] ft\/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula [latex]s(t)=-16t^2+v_{0}t[\/latex]. When [latex]P[\/latex] dollars are invested in an account at an annual interest rate [latex]r[\/latex] compounded continuously, the amount of money after [latex]t[\/latex] years is given by the formula [latex]A(t)=Pe^{rt}[\/latex]. Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.<\/p>\n<\/section>\n<p id=\"fs-id1170572216050\">Given an algebraic formula for a function [latex]f[\/latex], the graph of [latex]f[\/latex] is the set of points [latex](x,f(x))[\/latex], where [latex]x[\/latex] is in the domain of [latex]f[\/latex] and [latex]f(x)[\/latex] is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of [latex]f[\/latex] consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":143,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1984"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1984\/revisions"}],"predecessor-version":[{"id":4621,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1984\/revisions\/4621"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1984\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1984"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1984"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1984"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1984"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}