{"id":16120,"date":"2019-07-30T15:45:34","date_gmt":"2019-07-30T15:45:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/exponential-functions\/"},"modified":"2021-11-08T02:00:39","modified_gmt":"2021-11-08T02:00:39","slug":"exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/exponential-functions\/","title":{"raw":"Exponential Functions","rendered":"Exponential Functions"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Evaluate exponential functions.<\/li>\r\n \t<li>Find the equation of an exponential function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3 data-type=\"title\">Defining Exponential Functions<\/h3>\r\nLet's take a look at the following situations:\r\n\r\n<span class=\"pullquote-left\">India is the second most populous country in the world with a population of about 1.25 billion people in 2013. The population is growing at a rate of about 1.2% each year<sup data-type=\"footnote-number\"><a href=\"#footnote1\" data-type=\"footnote-link\">1<\/a><\/sup>. If this rate continues, the population of India will exceed China\u2019s population by the year 2031.<\/span>\r\n\r\n<span class=\"pullquote-right\">A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products\u2014no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2021.<\/span>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p id=\"fs-id1165137640062\">When exploring linear growth, we observed a constant rate of change\u2014a constant number by which the output increased for each unit increase in input. For example, in the equation [latex]f(x)=3x+4[\/latex], the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a <em data-effect=\"italics\">percent<\/em> change per unit time (rather than a constant change) in the number of people.<\/p>\r\nWhen populations grow rapidly, we often say that the growth is \u201cexponential,\u201d meaning that something is growing very rapidly. To a mathematician, however, the term <strong><em data-effect=\"italics\">exponential growth <\/em><\/strong>has a very specific meaning. In this section, we will take a look at <em data-effect=\"italics\">exponential functions<\/em>, which model this kind of rapid growth.\r\n<div id=\"fs-id1165135159940\" class=\"bc-section section\" data-depth=\"1\">\r\n<div id=\"fs-id1165137477143\" class=\"bc-section section\" data-depth=\"2\">\r\n<p id=\"fs-id1165134069131\">What exactly does it mean to <em data-effect=\"italics\">grow exponentially<\/em>? What does the word <em data-effect=\"italics\">double <\/em>have in common with <em data-effect=\"italics\">percent increase<\/em>? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.<\/p>\r\n\r\n<ul id=\"fs-id1165134042783\">\r\n \t<li><strong>Percent change <\/strong>refers to a <em data-effect=\"italics\">change<\/em> based on a <em data-effect=\"italics\">percent<\/em> of the original amount.<\/li>\r\n \t<li><strong>Exponential growth <\/strong>refers to an <em data-effect=\"italics\">increase<\/em> based on a constant multiplicative rate of change over equal increments of time, that is, a <em data-effect=\"italics\">percent<\/em> increase of the original amount over time.<\/li>\r\n \t<li><strong>Exponential decay<\/strong> refers to a <em data-effect=\"italics\">decrease<\/em> based on a constant multiplicative rate of change over equal increments of time, that is, a <em data-effect=\"italics\">percent<\/em> decrease of the original amount over time.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137760753\">In order to gain a clear understanding of <span class=\"no-emphasis\" data-type=\"term\">exponential growth<\/span>, let us contrast exponential growth with <span class=\"no-emphasis\" data-type=\"term\">linear growth<\/span>.<\/p>\r\n<p id=\"fs-id1165137644244\">Consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex]. Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].<\/p>\r\n<p id=\"fs-id1165135512493\">A few years of growth for these companies are illustrated below.<\/p>\r\n\r\n<table id=\"Table_04_01_05\" summary=\"Six rows and three columns. The first column is labeled, \">\r\n<thead>\r\n<tr>\r\n<th>Year,\u00a0<em>x<\/em><\/th>\r\n<th>Stores, Company A<\/th>\r\n<th>Stores, Company B<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>100 + 50(0) = 100<\/td>\r\n<td>100(1 + 0.5)<sup>0<\/sup> = 100<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>100 + 50(1) = 150<\/td>\r\n<td>100(1 + 0.5)<sup>1<\/sup> = 150<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>100 + 50(2) = 200<\/td>\r\n<td>100(1 + 0.5)<sup>2<\/sup> = 225<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>100 + 50(3) = 250<\/td>\r\n<td>100(1 + 0.5)<sup>3<\/sup> =\u00a0337.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>x<\/em><\/td>\r\n<td><em>A<\/em>(<em>x<\/em>) = 100 + 50x<\/td>\r\n<td><em>B<\/em>(<em>x<\/em>) = 100(1 + 0.5)<sup><em>x<\/em><\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe graphs comparing the number of stores for each company over a five-year period are shown in below<strong>.<\/strong> We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010809\/CNX_Precalc_Figure_04_01_0012.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"487\" height=\"845\" data-media-type=\"image\/jpg\" \/> <b>Figure 2.<\/b> The graph shows the numbers of stores Companies A and B opened over a five-year period.[\/caption]\r\n<p id=\"fs-id1165135209682\">Notice that the domain for both functions is [latex]\\left[0,\\infty \\right)[\/latex], and the range for both functions is [latex]\\left[100,\\infty \\right)[\/latex]. After year 1, Company B always has more stores than Company A.<\/p>\r\n<p id=\"fs-id1165137836429\">Now we will turn our attention to the function representing the number of stores for Company B, [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex]. In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and [latex]1+0.5=1.5[\/latex] represents the growth factor. Generalizing further, we can write this function as [latex]B\\left(x\\right)=100{\\left(1.5\\right)}^{x}[\/latex], where 100 is the initial value, 1.5 is called the <em data-effect=\"italics\">base<\/em>, and <em>x<\/em>\u00a0is called the <em data-effect=\"italics\">exponent<\/em>.<\/p>\r\n\r\n<ul id=\"fs-id1165137725808\">\r\n \t<li><strong>Exponential growth <\/strong>refers to the original value from the range increases by the <em data-effect=\"italics\">same percentage<\/em> over equal increments found in the domain.<\/li>\r\n \t<li><strong>Linear growth<\/strong> refers to the original value from the range increases by the <em data-effect=\"italics\">same amount<\/em> over equal increments found in the domain.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137561507\">Clearly, the difference between \u201cthe same percentage\u201d and \u201cthe same amount\u201d is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in multiplying the output by 1.5 whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 50 to the output whenever the input was increased by one.<\/p>\r\nThe video below shows another example comparing linear and exponential growth.\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=t77tB6u27ik[\/embed]\r\n<p id=\"fs-id1165135445949\">The general form of the <span class=\"no-emphasis\" data-type=\"term\">exponential function<\/span> is [latex]f(x)=a{b}^{x} [\/latex], where [latex]a \\neq 0, \\; b&gt;0, \\; b \\neq 1[\/latex].<\/p>\r\n\r\n<ul id=\"fs-id1165137635065\">\r\n \t<li>If [latex]b&gt;1[\/latex], the function grows at a rate proportional to its size.<\/li>\r\n \t<li>If [latex]0&lt;b&lt;1[\/latex], the function decays at a rate proportional to its size.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137465225\">Let\u2019s look at the function [latex]f(x)={2}^{x} [\/latex]. We will create a table (below)\u00a0to determine the corresponding outputs over an interval in the domain from -3 to 3.<\/p>\r\n\r\n<table id=\"Table_04_01_02\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=2^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 2^(-3)=1\/8), (-2, 2^(-2)=1\/4), (-1, 2^(-1)=1\/2), (0, 2^(0)=1), (1, 2^(1)=2), (2, 2^(2)=4), and (3, 2^(3)=8).\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 81px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 118px;\">-3<\/td>\r\n<td style=\"width: 122px;\">-2<\/td>\r\n<td style=\"width: 123px;\">-1<\/td>\r\n<td style=\"width: 73px;\">0<\/td>\r\n<td style=\"width: 73px;\">1<\/td>\r\n<td style=\"width: 73px;\">2<\/td>\r\n<td style=\"width: 73px;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 81px;\">[latex]f(x)={2}^{x}[\/latex]<\/td>\r\n<td style=\"width: 118px;\">[latex]{2}^{-3}=\\frac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 122px;\">[latex]{2}^{-2}=\\frac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 123px;\">[latex]{2}^{-1}=f\\rac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 73px;\">[latex]{2}^{0}=1[\/latex]<\/td>\r\n<td style=\"width: 73px;\">[latex]{2}^{1}=2[\/latex]<\/td>\r\n<td style=\"width: 73px;\">[latex]{2}^{2}=4[\/latex]<\/td>\r\n<td style=\"width: 73px;\">[latex]{2}^{3}=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137640874\">Let us examine the graph of f(x) by plotting the ordered pairs from the table, and then make a few observations.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_01_006\" class=\"medium\"><span id=\"fs-id1165137571891\" data-type=\"media\" data-alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5742\/2021\/11\/07193633\/CNX_Precalc_Figure_04_01_006.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" data-media-type=\"image\/jpg\" \/><\/span><\/div>\r\n<p id=\"fs-id1165137408862\">Let\u2019s analyze the behavior of the graph of the exponential function[latex]f(x)={2}^{x} [\/latex] and highlight some of its key characteristics.<\/p>\r\n\r\n<ul id=\"fs-id1165137566018\">\r\n \t<li>the domain is [latex](-\\infty ,\\infty )[\/latex],<\/li>\r\n \t<li>the range is\u00a0[latex](0 ,\\infty )[\/latex],<\/li>\r\n \t<li>as [latex] x \\rightarrow \\infty , f(x)\\rightarrow \\infty [\/latex],<\/li>\r\n \t<li>as [latex] x \\rightarrow -\\infty , f(x)\\rightarrow 0 [\/latex],<\/li>\r\n \t<li>[latex] f(x) [\/latex] is always increasing,<\/li>\r\n \t<li>the graph of [latex] f(x) [\/latex] will never touch the <em data-effect=\"italics\">x<\/em>-axis because two raised to any exponent never has the result of zero,<\/li>\r\n \t<li>[latex] y=0 [\/latex] is the horizontal asymptote,<\/li>\r\n \t<li>the <em data-effect=\"italics\">y<\/em>-intercept is 1.<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1165137442472\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox\">\r\n<h3>A General Note: EXPONENTIAL FUNCTION<\/h3>\r\n<p id=\"fs-id1165137911387\">For any real number x an exponential function is a function of the form<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=a{b}^{x}[\/latex],<\/p>\r\nwhere\r\n<ul id=\"fs-id1165137401680\">\r\n \t<li>[latex]a [\/latex] is a non-zero real number called the initial or starting value of a function and<\/li>\r\n \t<li>[latex]b [\/latex] is any positive real number such that [latex]b \\neq 1[\/latex], called the growth factor or growth multiplier per unit <em>x.<\/em><\/li>\r\n \t<li>The domain of [latex] f(x)[\/latex] is all real numbers.<\/li>\r\n \t<li>The range of [latex] f(x)[\/latex] is all positive real numbers if [latex] a&gt;0[\/latex].<\/li>\r\n \t<li>The range of\u00a0[latex] f(x)[\/latex] is all negative real numbers if [latex] a&lt;0[\/latex].<\/li>\r\n \t<li>The <em data-effect=\"italics\">y<\/em>-intercept is [latex] (0,1) [\/latex] and the horizontal asymptote is\u00a0[latex]y=0 [\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: identifying exponential functions<\/h3>\r\n<div id=\"fs-id1165137659178\" data-type=\"problem\">\r\n<p id=\"fs-id1165137601478\">Which of the following equations are <em data-effect=\"italics\">not<\/em> exponential functions?<\/p>\r\n\r\n<ul id=\"fs-id1165135176602\">\r\n \t<li>[latex]f(x)={4}^{3(x-2)}[\/latex]<\/li>\r\n \t<li>[latex]g(x)={x}^{3}[\/latex]<\/li>\r\n \t<li>[latex]h(x)=(\\frac{1}{3})^{x}[\/latex]<\/li>\r\n \t<li>[latex]j(x)=(-2)^{x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165134108513\" data-type=\"solution\">\r\n<p id=\"fs-id1165137698136\">[reveal-answer q=\"751163\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"751163\"]<\/p>\r\nBy definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus,\r\n\r\n[latex]g(x)={x}^{3}[\/latex]\u00a0does not represent an exponential function because the base is an independent variable. In fact,[latex]g(x)={x}^{3}[\/latex] is a power function.\r\n\r\nRecall that the base <em data-effect=\"italics\">b <\/em>of an exponential function is always a positive constant, and [latex]b \\neq 1[\/latex]. Thus,\r\n\r\n[latex]j(x)=(-2)^{x}[\/latex]\u00a0does not represent an exponential function because the base (-2) is less than 0.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137436342\" class=\"precalculus try\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\r\n<div id=\"ti_04_01_01\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137862673\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<p id=\"fs-id1165137424249\">Which of the following equations represent exponential functions?<\/p>\r\n\r\n<ul id=\"fs-id1165135161022\">\r\n \t<li>[latex]f(x)=2{x}^{2}-3x+1[\/latex]<\/li>\r\n \t<li>[latex]g(x)={0.875}^{x}[\/latex]<\/li>\r\n \t<li>[latex]h(x)=1.75x+2[\/latex]<\/li>\r\n \t<li>[latex]j(x)={1095.6}^{-2x}[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"424165\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"424165\"]\r\n\r\n[latex]g(x)[\/latex] and [latex] j(x)[\/latex] represent exponential functions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137640042\" class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Evaluating Exponential Functions<\/h3>\r\n<p id=\"fs-id1165137784783\">Recall that the base of an exponential function must be a positive real number other than 1. Why do we limit the base <em>b<\/em>\u00a0to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\r\n\r\n<ul id=\"fs-id1165137754880\">\r\n \t<li>Let <em>b\u00a0<\/em>= \u20139 and [latex]x=\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137563360\">Why do we limit the base to positive values other than 1? Because base 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:<\/p>\r\n\r\n<ul id=\"fs-id1165137400268\">\r\n \t<li>Let <em>b\u00a0<\/em>= 1. Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137459694\">To evaluate an exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], we simply substitute <em>x<\/em>\u00a0with the given value, and calculate the resulting power. For example:<\/p>\r\n<p id=\"fs-id1165135403544\">Let [latex]f\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\r\n\r\n<div id=\"eip-id1165137643186\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}f\\left(x\\right)\\hfill &amp; ={2}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; ={2}^{3}\\text{ }\\hfill &amp; \\text{Substitute }x=3.\\hfill \\\\ \\hfill &amp; =8\\text{ }\\hfill &amp; \\text{Evaluate the power}\\text{.}\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137849020\">To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:<\/p>\r\n<p id=\"fs-id1165137849024\">Let [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\r\n\r\n<div id=\"eip-id1165134086025\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}f\\left(x\\right)\\hfill &amp; =30{\\left(2\\right)}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; =30{\\left(2\\right)}^{3}\\hfill &amp; \\text{Substitute }x=3.\\hfill \\\\ \\hfill &amp; =30\\left(8\\right)\\text{ }\\hfill &amp; \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill &amp; =240\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137841073\">Note that if the order of operations were not followed, the result would be incorrect:<\/p>\r\n\r\n<div id=\"eip-id1165135320147\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/div>\r\n<div data-type=\"equation\" data-label=\"\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Evaluating Exponential Functions<\/h3>\r\nLet [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"629847\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"629847\"]\r\n<p id=\"fs-id1165137598173\">Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\r\n\r\n<div id=\"eip-id1165135208555\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}f\\left(x\\right)\\hfill &amp; =5{\\left(3\\right)}^{x+1}\\hfill &amp; \\hfill \\\\ f\\left(2\\right)\\hfill &amp; =5{\\left(3\\right)}^{2+1}\\hfill &amp; \\text{Substitute }x=2.\\hfill \\\\ \\hfill &amp; =5{\\left(3\\right)}^{3}\\hfill &amp; \\text{Add the exponents}.\\hfill \\\\ \\hfill &amp; =5\\left(27\\right)\\hfill &amp; \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill &amp; =135\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{cases}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137762788\">Let [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.<\/p>\r\n[reveal-answer q=\"144970\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"144970\"]5.5556[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3 data-type=\"title\">Example: Evaluating a Real-World Exponential Model<\/h3>\r\n<p id=\"fs-id1165135541867\">At the beginning of this section, we learned that the population of India was about 1.25 billion in the year 2013, with an annual growth rate of about 1.2%. This situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex], where <em>t<\/em>\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?<\/p>\r\n[reveal-answer q=\"262079\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"262079\"]\r\n<p id=\"fs-id1165137786635\">To estimate the population in 2031, we evaluate the models for <em>t\u00a0<\/em>= 18, because 2031 is 18 years after 2013. Rounding to the nearest thousandth,<\/p>\r\n\r\n<div id=\"eip-id1165135657117\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/div>\r\n<p id=\"fs-id1165135394343\">There will be about 1.549 billion people in India in the year 2031.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137635314\">The population of China was about 1.39 billion in the year 2013, with an annual growth rate of about 0.6%. This situation is represented by the growth function [latex]P\\left(t\\right)=1.39{\\left(1.006\\right)}^{t}[\/latex], where <em>t<\/em>\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of China be for the year 2031? How does this compare to the population prediction we made for India in our example?<\/p>\r\n[reveal-answer q=\"595217\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"595217\"]\r\n\r\nAbout 1.548 billion people; by the year 2031, India\u2019s population will exceed China\u2019s by about 0.001 billion, or 1 million people.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"footnote-refs\">\r\n<h3 data-type=\"footnote-refs-title\">Footnotes<\/h3>\r\n<ul data-list-type=\"bulleted\" data-bullet-style=\"none\">\r\n \t<li data-type=\"footnote-ref\"><a href=\"#footnote-ref1\" data-type=\"footnote-ref-link\">1<\/a><span data-type=\"footnote-ref-content\">http:\/\/www.worldometers.info\/world-population\/. Accessed February 24, 2014.<\/span><\/li>\r\n \t<li id=\"footnote2\" data-type=\"footnote-ref\"><a href=\"#footnote-ref2\" data-type=\"footnote-ref-link\">2<\/a><span data-type=\"footnote-ref-content\">Oxford Dictionary. http:\/\/oxforddictionaries.com\/us\/definition\/american_english\/nomina.<\/span><\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Evaluate exponential functions.<\/li>\n<li>Find the equation of an exponential function.<\/li>\n<\/ul>\n<\/div>\n<h3 data-type=\"title\">Defining Exponential Functions<\/h3>\n<p>Let&#8217;s take a look at the following situations:<\/p>\n<p><span class=\"pullquote-left\">India is the second most populous country in the world with a population of about 1.25 billion people in 2013. The population is growing at a rate of about 1.2% each year<sup data-type=\"footnote-number\"><a href=\"#footnote1\" data-type=\"footnote-link\">1<\/a><\/sup>. If this rate continues, the population of India will exceed China\u2019s population by the year 2031.<\/span><\/p>\n<p><span class=\"pullquote-right\">A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products\u2014no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2021.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137640062\">When exploring linear growth, we observed a constant rate of change\u2014a constant number by which the output increased for each unit increase in input. For example, in the equation [latex]f(x)=3x+4[\/latex], the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a <em data-effect=\"italics\">percent<\/em> change per unit time (rather than a constant change) in the number of people.<\/p>\n<p>When populations grow rapidly, we often say that the growth is \u201cexponential,\u201d meaning that something is growing very rapidly. To a mathematician, however, the term <strong><em data-effect=\"italics\">exponential growth <\/em><\/strong>has a very specific meaning. In this section, we will take a look at <em data-effect=\"italics\">exponential functions<\/em>, which model this kind of rapid growth.<\/p>\n<div id=\"fs-id1165135159940\" class=\"bc-section section\" data-depth=\"1\">\n<div id=\"fs-id1165137477143\" class=\"bc-section section\" data-depth=\"2\">\n<p id=\"fs-id1165134069131\">What exactly does it mean to <em data-effect=\"italics\">grow exponentially<\/em>? What does the word <em data-effect=\"italics\">double <\/em>have in common with <em data-effect=\"italics\">percent increase<\/em>? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.<\/p>\n<ul id=\"fs-id1165134042783\">\n<li><strong>Percent change <\/strong>refers to a <em data-effect=\"italics\">change<\/em> based on a <em data-effect=\"italics\">percent<\/em> of the original amount.<\/li>\n<li><strong>Exponential growth <\/strong>refers to an <em data-effect=\"italics\">increase<\/em> based on a constant multiplicative rate of change over equal increments of time, that is, a <em data-effect=\"italics\">percent<\/em> increase of the original amount over time.<\/li>\n<li><strong>Exponential decay<\/strong> refers to a <em data-effect=\"italics\">decrease<\/em> based on a constant multiplicative rate of change over equal increments of time, that is, a <em data-effect=\"italics\">percent<\/em> decrease of the original amount over time.<\/li>\n<\/ul>\n<p id=\"fs-id1165137760753\">In order to gain a clear understanding of <span class=\"no-emphasis\" data-type=\"term\">exponential growth<\/span>, let us contrast exponential growth with <span class=\"no-emphasis\" data-type=\"term\">linear growth<\/span>.<\/p>\n<p id=\"fs-id1165137644244\">Consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex]. Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].<\/p>\n<p id=\"fs-id1165135512493\">A few years of growth for these companies are illustrated below.<\/p>\n<table id=\"Table_04_01_05\" summary=\"Six rows and three columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Year,\u00a0<em>x<\/em><\/th>\n<th>Stores, Company A<\/th>\n<th>Stores, Company B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>100 + 50(0) = 100<\/td>\n<td>100(1 + 0.5)<sup>0<\/sup> = 100<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>100 + 50(1) = 150<\/td>\n<td>100(1 + 0.5)<sup>1<\/sup> = 150<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>100 + 50(2) = 200<\/td>\n<td>100(1 + 0.5)<sup>2<\/sup> = 225<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>100 + 50(3) = 250<\/td>\n<td>100(1 + 0.5)<sup>3<\/sup> =\u00a0337.5<\/td>\n<\/tr>\n<tr>\n<td><em>x<\/em><\/td>\n<td><em>A<\/em>(<em>x<\/em>) = 100 + 50x<\/td>\n<td><em>B<\/em>(<em>x<\/em>) = 100(1 + 0.5)<sup><em>x<\/em><\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The graphs comparing the number of stores for each company over a five-year period are shown in below<strong>.<\/strong> We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010809\/CNX_Precalc_Figure_04_01_0012.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"487\" height=\"845\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> The graph shows the numbers of stores Companies A and B opened over a five-year period.<\/p>\n<\/div>\n<p id=\"fs-id1165135209682\">Notice that the domain for both functions is [latex]\\left[0,\\infty \\right)[\/latex], and the range for both functions is [latex]\\left[100,\\infty \\right)[\/latex]. After year 1, Company B always has more stores than Company A.<\/p>\n<p id=\"fs-id1165137836429\">Now we will turn our attention to the function representing the number of stores for Company B, [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex]. In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and [latex]1+0.5=1.5[\/latex] represents the growth factor. Generalizing further, we can write this function as [latex]B\\left(x\\right)=100{\\left(1.5\\right)}^{x}[\/latex], where 100 is the initial value, 1.5 is called the <em data-effect=\"italics\">base<\/em>, and <em>x<\/em>\u00a0is called the <em data-effect=\"italics\">exponent<\/em>.<\/p>\n<ul id=\"fs-id1165137725808\">\n<li><strong>Exponential growth <\/strong>refers to the original value from the range increases by the <em data-effect=\"italics\">same percentage<\/em> over equal increments found in the domain.<\/li>\n<li><strong>Linear growth<\/strong> refers to the original value from the range increases by the <em data-effect=\"italics\">same amount<\/em> over equal increments found in the domain.<\/li>\n<\/ul>\n<p id=\"fs-id1165137561507\">Clearly, the difference between \u201cthe same percentage\u201d and \u201cthe same amount\u201d is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in multiplying the output by 1.5 whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 50 to the output whenever the input was increased by one.<\/p>\n<p>The video below shows another example comparing linear and exponential growth.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Linear and Exponential Growth: Complete a Salary Table\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/t77tB6u27ik?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p id=\"fs-id1165135445949\">The general form of the <span class=\"no-emphasis\" data-type=\"term\">exponential function<\/span> is [latex]f(x)=a{b}^{x}[\/latex], where [latex]a \\neq 0, \\; b>0, \\; b \\neq 1[\/latex].<\/p>\n<ul id=\"fs-id1165137635065\">\n<li>If [latex]b>1[\/latex], the function grows at a rate proportional to its size.<\/li>\n<li>If [latex]0<b<1[\/latex], the function decays at a rate proportional to its size.<\/li>\n<\/ul>\n<p id=\"fs-id1165137465225\">Let\u2019s look at the function [latex]f(x)={2}^{x}[\/latex]. We will create a table (below)\u00a0to determine the corresponding outputs over an interval in the domain from -3 to 3.<\/p>\n<table id=\"Table_04_01_02\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=2^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 2^(-3)=1\/8), (-2, 2^(-2)=1\/4), (-1, 2^(-1)=1\/2), (0, 2^(0)=1), (1, 2^(1)=2), (2, 2^(2)=4), and (3, 2^(3)=8).\">\n<tbody>\n<tr>\n<td style=\"width: 81px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 118px;\">-3<\/td>\n<td style=\"width: 122px;\">-2<\/td>\n<td style=\"width: 123px;\">-1<\/td>\n<td style=\"width: 73px;\">0<\/td>\n<td style=\"width: 73px;\">1<\/td>\n<td style=\"width: 73px;\">2<\/td>\n<td style=\"width: 73px;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 81px;\">[latex]f(x)={2}^{x}[\/latex]<\/td>\n<td style=\"width: 118px;\">[latex]{2}^{-3}=\\frac{1}{8}[\/latex]<\/td>\n<td style=\"width: 122px;\">[latex]{2}^{-2}=\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 123px;\">[latex]{2}^{-1}=f\\rac{1}{2}[\/latex]<\/td>\n<td style=\"width: 73px;\">[latex]{2}^{0}=1[\/latex]<\/td>\n<td style=\"width: 73px;\">[latex]{2}^{1}=2[\/latex]<\/td>\n<td style=\"width: 73px;\">[latex]{2}^{2}=4[\/latex]<\/td>\n<td style=\"width: 73px;\">[latex]{2}^{3}=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137640874\">Let us examine the graph of f(x) by plotting the ordered pairs from the table, and then make a few observations.<\/p>\n<div id=\"CNX_Precalc_Figure_04_01_006\" class=\"medium\"><span id=\"fs-id1165137571891\" data-type=\"media\" data-alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5742\/2021\/11\/07193633\/CNX_Precalc_Figure_04_01_006.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" data-media-type=\"image\/jpg\" \/><\/span><\/div>\n<p id=\"fs-id1165137408862\">Let\u2019s analyze the behavior of the graph of the exponential function[latex]f(x)={2}^{x}[\/latex] and highlight some of its key characteristics.<\/p>\n<ul id=\"fs-id1165137566018\">\n<li>the domain is [latex](-\\infty ,\\infty )[\/latex],<\/li>\n<li>the range is\u00a0[latex](0 ,\\infty )[\/latex],<\/li>\n<li>as [latex]x \\rightarrow \\infty , f(x)\\rightarrow \\infty[\/latex],<\/li>\n<li>as [latex]x \\rightarrow -\\infty , f(x)\\rightarrow 0[\/latex],<\/li>\n<li>[latex]f(x)[\/latex] is always increasing,<\/li>\n<li>the graph of [latex]f(x)[\/latex] will never touch the <em data-effect=\"italics\">x<\/em>-axis because two raised to any exponent never has the result of zero,<\/li>\n<li>[latex]y=0[\/latex] is the horizontal asymptote,<\/li>\n<li>the <em data-effect=\"italics\">y<\/em>-intercept is 1.<\/li>\n<\/ul>\n<div id=\"fs-id1165137442472\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<div data-type=\"title\">\n<div class=\"textbox\">\n<h3>A General Note: EXPONENTIAL FUNCTION<\/h3>\n<p id=\"fs-id1165137911387\">For any real number x an exponential function is a function of the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a{b}^{x}[\/latex],<\/p>\n<p>where<\/p>\n<ul id=\"fs-id1165137401680\">\n<li>[latex]a[\/latex] is a non-zero real number called the initial or starting value of a function and<\/li>\n<li>[latex]b[\/latex] is any positive real number such that [latex]b \\neq 1[\/latex], called the growth factor or growth multiplier per unit <em>x.<\/em><\/li>\n<li>The domain of [latex]f(x)[\/latex] is all real numbers.<\/li>\n<li>The range of [latex]f(x)[\/latex] is all positive real numbers if [latex]a>0[\/latex].<\/li>\n<li>The range of\u00a0[latex]f(x)[\/latex] is all negative real numbers if [latex]a<0[\/latex].<\/li>\n<li>The <em data-effect=\"italics\">y<\/em>-intercept is [latex](0,1)[\/latex] and the horizontal asymptote is\u00a0[latex]y=0[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: identifying exponential functions<\/h3>\n<div id=\"fs-id1165137659178\" data-type=\"problem\">\n<p id=\"fs-id1165137601478\">Which of the following equations are <em data-effect=\"italics\">not<\/em> exponential functions?<\/p>\n<ul id=\"fs-id1165135176602\">\n<li>[latex]f(x)={4}^{3(x-2)}[\/latex]<\/li>\n<li>[latex]g(x)={x}^{3}[\/latex]<\/li>\n<li>[latex]h(x)=(\\frac{1}{3})^{x}[\/latex]<\/li>\n<li>[latex]j(x)=(-2)^{x}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134108513\" data-type=\"solution\">\n<p id=\"fs-id1165137698136\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q751163\">Show Answer<\/span><\/p>\n<div id=\"q751163\" class=\"hidden-answer\" style=\"display: none\">\n<p>By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus,<\/p>\n<p>[latex]g(x)={x}^{3}[\/latex]\u00a0does not represent an exponential function because the base is an independent variable. In fact,[latex]g(x)={x}^{3}[\/latex] is a power function.<\/p>\n<p>Recall that the base <em data-effect=\"italics\">b <\/em>of an exponential function is always a positive constant, and [latex]b \\neq 1[\/latex]. Thus,<\/p>\n<p>[latex]j(x)=(-2)^{x}[\/latex]\u00a0does not represent an exponential function because the base (-2) is less than 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436342\" class=\"precalculus try\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<div id=\"ti_04_01_01\" data-type=\"exercise\">\n<div id=\"fs-id1165137862673\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p id=\"fs-id1165137424249\">Which of the following equations represent exponential functions?<\/p>\n<ul id=\"fs-id1165135161022\">\n<li>[latex]f(x)=2{x}^{2}-3x+1[\/latex]<\/li>\n<li>[latex]g(x)={0.875}^{x}[\/latex]<\/li>\n<li>[latex]h(x)=1.75x+2[\/latex]<\/li>\n<li>[latex]j(x)={1095.6}^{-2x}[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q424165\">Show Answer<\/span><\/p>\n<div id=\"q424165\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g(x)[\/latex] and [latex]j(x)[\/latex] represent exponential functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137640042\" class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Evaluating Exponential Functions<\/h3>\n<p id=\"fs-id1165137784783\">Recall that the base of an exponential function must be a positive real number other than 1. Why do we limit the base <em>b<\/em>\u00a0to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\n<ul id=\"fs-id1165137754880\">\n<li>Let <em>b\u00a0<\/em>= \u20139 and [latex]x=\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\n<\/ul>\n<p id=\"fs-id1165137563360\">Why do we limit the base to positive values other than 1? Because base 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:<\/p>\n<ul id=\"fs-id1165137400268\">\n<li>Let <em>b\u00a0<\/em>= 1. Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137459694\">To evaluate an exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], we simply substitute <em>x<\/em>\u00a0with the given value, and calculate the resulting power. For example:<\/p>\n<p id=\"fs-id1165135403544\">Let [latex]f\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<div id=\"eip-id1165137643186\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}f\\left(x\\right)\\hfill & ={2}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & ={2}^{3}\\text{ }\\hfill & \\text{Substitute }x=3.\\hfill \\\\ \\hfill & =8\\text{ }\\hfill & \\text{Evaluate the power}\\text{.}\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137849020\">To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:<\/p>\n<p id=\"fs-id1165137849024\">Let [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<div id=\"eip-id1165134086025\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}f\\left(x\\right)\\hfill & =30{\\left(2\\right)}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & =30{\\left(2\\right)}^{3}\\hfill & \\text{Substitute }x=3.\\hfill \\\\ \\hfill & =30\\left(8\\right)\\text{ }\\hfill & \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill & =240\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137841073\">Note that if the order of operations were not followed, the result would be incorrect:<\/p>\n<div id=\"eip-id1165135320147\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/div>\n<div data-type=\"equation\" data-label=\"\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Evaluating Exponential Functions<\/h3>\n<p>Let [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q629847\">Solution<\/span><\/p>\n<div id=\"q629847\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137598173\">Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\n<div id=\"eip-id1165135208555\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}f\\left(x\\right)\\hfill & =5{\\left(3\\right)}^{x+1}\\hfill & \\hfill \\\\ f\\left(2\\right)\\hfill & =5{\\left(3\\right)}^{2+1}\\hfill & \\text{Substitute }x=2.\\hfill \\\\ \\hfill & =5{\\left(3\\right)}^{3}\\hfill & \\text{Add the exponents}.\\hfill \\\\ \\hfill & =5\\left(27\\right)\\hfill & \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill & =135\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{cases}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137762788\">Let [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q144970\">Show Answer<\/span><\/p>\n<div id=\"q144970\" class=\"hidden-answer\" style=\"display: none\">5.5556<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3 data-type=\"title\">Example: Evaluating a Real-World Exponential Model<\/h3>\n<p id=\"fs-id1165135541867\">At the beginning of this section, we learned that the population of India was about 1.25 billion in the year 2013, with an annual growth rate of about 1.2%. This situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex], where <em>t<\/em>\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q262079\">Show Answer<\/span><\/p>\n<div id=\"q262079\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137786635\">To estimate the population in 2031, we evaluate the models for <em>t\u00a0<\/em>= 18, because 2031 is 18 years after 2013. Rounding to the nearest thousandth,<\/p>\n<div id=\"eip-id1165135657117\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/div>\n<p id=\"fs-id1165135394343\">There will be about 1.549 billion people in India in the year 2031.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137635314\">The population of China was about 1.39 billion in the year 2013, with an annual growth rate of about 0.6%. This situation is represented by the growth function [latex]P\\left(t\\right)=1.39{\\left(1.006\\right)}^{t}[\/latex], where <em>t<\/em>\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of China be for the year 2031? How does this compare to the population prediction we made for India in our example?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q595217\">Show Answer<\/span><\/p>\n<div id=\"q595217\" class=\"hidden-answer\" style=\"display: none\">\n<p>About 1.548 billion people; by the year 2031, India\u2019s population will exceed China\u2019s by about 0.001 billion, or 1 million people.<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h3 data-type=\"footnote-refs-title\">Footnotes<\/h3>\n<ul data-list-type=\"bulleted\" data-bullet-style=\"none\">\n<li data-type=\"footnote-ref\"><a href=\"#footnote-ref1\" data-type=\"footnote-ref-link\">1<\/a><span data-type=\"footnote-ref-content\">http:\/\/www.worldometers.info\/world-population\/. Accessed February 24, 2014.<\/span><\/li>\n<li id=\"footnote2\" data-type=\"footnote-ref\"><a href=\"#footnote-ref2\" data-type=\"footnote-ref-link\">2<\/a><span data-type=\"footnote-ref-content\">Oxford Dictionary. http:\/\/oxforddictionaries.com\/us\/definition\/american_english\/nomina.<\/span><\/li>\n<\/ul>\n<\/div>\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-16120\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/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":473810,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["jay-abramson"],"pb_section_license":"cc-by"},"chapter-type":[],"contributor":[75],"license":[57],"class_list":["post-16120","chapter","type-chapter","status-publish","hentry","contributor-jay-abramson","license-cc-by"],"part":11277,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/16120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/473810"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/16120\/revisions"}],"predecessor-version":[{"id":16137,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/16120\/revisions\/16137"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/11277"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/16120\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=16120"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=16120"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=16120"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=16120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}