{"id":17722,"date":"2021-08-20T19:05:08","date_gmt":"2021-08-20T19:05:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=17722"},"modified":"2021-08-20T19:39:35","modified_gmt":"2021-08-20T19:39:35","slug":"section-2-3-properties-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/section-2-3-properties-of-functions\/","title":{"raw":"Section 2.3: Properties of Functions","rendered":"Section 2.3: Properties of Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\r\n \t<li>Use a graph to locate local and absolute maxima and local minima.<\/li>\r\n \t<li>Find the average rate of change of a function.<\/li>\r\n \t<li>Determine whether a function is even, odd, or neither from its graph and equation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Increasing and Decreasing Functions<\/h2>\r\n<p id=\"fs-id1165137784644\">As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 1\u00a0shows examples of increasing and decreasing intervals on a function.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0042.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/>\r\n<p style=\"text-align: center;\"><strong>Figure 1.<\/strong> The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].<\/p>\r\nThis video further explains how to find where a function is increasing or decreasing.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=78b4HOMVcKM\r\n<p id=\"fs-id1165134272749\">While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a <strong>local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a <strong>local minimum<\/strong>. The plural form is \"local minima.\" Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is \"extremum.\") Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>local<\/em>.<\/p>\r\n<p id=\"fs-id1165134547216\">Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em>local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/p>\r\nFor the function in Figure 2, the local maximum is 16, and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0142.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum. The local maximum is 16 and occurs at x = negative 2. This is the point negative 2, 16. The local minimum is negative 16 and occurs at x = 2. This is the point 2, negative 16.\" width=\"731\" height=\"467\" \/> <strong>Figure 2<\/strong>[\/caption]\r\n<p id=\"fs-id1165133316450\">To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 3\u00a0illustrates these ideas for a local maximum.<\/p>\r\n\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\/03010554\/CNX_Precalc_Figure_01_03_0052.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/> <strong>Figure 3.<\/strong> Definition of a local maximum.[\/caption]\r\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\r\n\r\n<div id=\"fs-id1165134169419\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Local Minima and Local Maxima<\/h3>\r\n<p id=\"fs-id1165134169426\">A function [latex]f[\/latex] is an <strong>increasing function<\/strong> on an open interval if [latex]f\\left(b\\right)&gt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex].<\/p>\r\n<p id=\"fs-id1165137668624\">A function [latex]f[\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right)&lt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex].<\/p>\r\n<p id=\"fs-id1165135389881\">A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a&lt;b&lt;c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex]. Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a&lt;b&lt;c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_07\" class=\"example\">\r\n<div id=\"fs-id1165134266716\" class=\"exercise\">\r\n<div id=\"fs-id1165134266718\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Finding Increasing and Decreasing Intervals on a Graph<\/h3>\r\nGiven the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.\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\/03010554\/CNX_Precalc_Figure_01_03_0062.jpg\" alt=\"Graph of a polynomial. As x gets large in the negative direction, the outputs of the function get large in the positive direction. As inputs approach 1, then the function value approaches a minimum of negative one. As x approaches 3, the values increase again and between 3 and 4 decrease one last time. As x gets large in the positive direction, the function values increase without bound.\" width=\"487\" height=\"295\" \/> <strong>Figure 4<\/strong>[\/caption]\r\n\r\n[reveal-answer q=\"331055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"331055\"]\r\n<p id=\"fs-id1165133067197\">We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on.<\/p>\r\n<p id=\"fs-id1165135369127\">In <strong>interval notation<\/strong>, we would say the function appears to be increasing on the interval (1,3) and the interval [latex]\\left(4,\\infty \\right)[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165134104021\">Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[\/latex] , [latex]t=3[\/latex] , and [latex]t=4[\/latex] . These points are the local extrema (two minima and a maximum).<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_08\" class=\"example\">\r\n<div id=\"fs-id1165135261521\" class=\"exercise\">\r\n<div id=\"fs-id1165135261523\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Finding Local Extrema from a Graph<\/h3>\r\n<p id=\"fs-id1165135261528\">Graph the function [latex]f\\left(x\\right)=\\frac{2}{x}+\\frac{x}{3}[\/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\r\n[reveal-answer q=\"453777\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"453777\"]\r\n\r\nUsing technology, we find that the graph of the function looks like that in Figure 5. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3[\/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2[\/latex].\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\/03010554\/CNX_Precalc_Figure_01_03_0072.jpg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/> <b>Figure 5<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nMost graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 6\u00a0provides screen images from two different technologies, showing the estimate for the local maximum and minimum.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_008ab2.jpg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1165134075625\">Based on these estimates, the function is increasing on the interval [latex]\\left(-\\infty ,-{2.449}\\right)[\/latex]\r\nand [latex]\\left(2.449\\text{,}\\infty \\right)[\/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at [latex]\\pm \\sqrt{6}[\/latex], but determining this requires calculus.)<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\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-id1165135640967\">Graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.<\/p>\r\n[reveal-answer q=\"48622\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"48622\"]\r\n\r\nThe local maximum is 28 at\u00a0<em>x\u00a0<\/em>= -1\u00a0and the local minimum is -80 at\u00a0<em>x<\/em> = 5. The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right)[\/latex].\r\n\r\n<span id=\"fs-id1165134043615\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_0102.jpg\" alt=\"Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).\" width=\"487\" height=\"328\" \/><\/span>\r\n\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[ohm_question hide_question_numbers=1]165724[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_09\" class=\"example\">\r\n<div id=\"fs-id1165135367558\" class=\"exercise\">\r\n<div id=\"fs-id1165137896103\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Finding Local Maxima and Minima from a Graph<\/h3>\r\nFor the function [latex]f[\/latex] whose graph is shown in Figure 7, find all local maxima and minima.\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\/03010555\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"209462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"209462\"]\r\n<p id=\"fs-id1165135527085\">Observe the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].<\/p>\r\n<p id=\"fs-id1165134485672\">The graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134544960\">\r\n<h2>Use\u00a0A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h2>\r\nThere is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.\r\n\r\nTo locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.\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\/03010557\/CNX_Precalc_Figure_01_03_0152.jpg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/> <b>Figure 10<\/b>[\/caption]\r\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\r\n\r\n<div id=\"fs-id1165135251290\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Absolute Maxima and Minima<\/h3>\r\n<p id=\"fs-id1165132939786\">The <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\r\n<p id=\"fs-id1165137932685\">The <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_10\" class=\"example\">\r\n<div id=\"fs-id1165134047533\" class=\"exercise\">\r\n<div id=\"fs-id1165134047535\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Finding Absolute Maxima and Minima from a Graph<\/h3>\r\nFor the function [latex]f[\/latex] shown in Figure 11, find all absolute maxima and minima.\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\/03010557\/CNX_Precalc_Figure_01_03_0132.jpg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/> <b>Figure 11<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"502428\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"502428\"]\r\n<p id=\"fs-id1165135532371\">Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].<\/p>\r\n<p id=\"fs-id1165137863670\">The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<p id=\"fs-id1165135194500\">Gasoline costs have experienced some wild fluctuations over the last several decades. The table below[footnote]http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.[\/footnote]\u00a0lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.<\/p>\r\n\r\n<h2>Finding the Average Rate of Change of a Function<\/h2>\r\n<p id=\"fs-id1165137834011\">The price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\r\n\r\n<div id=\"fs-id1165135452482\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{\\text{Change in output}}{\\text{Change in input}} \\\\[1mm] &amp;=\\frac{\\Delta y}{\\Delta x} \\\\[1mm] &amp;= \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &amp;= \\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165135471272\">The Greek letter [latex]\\Delta [\/latex] (delta) signifies the change in a quantity; we read the ratio as \"delta-<em>y<\/em> over delta-<em>x<\/em>\" or \"the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].\" Occasionally we write [latex]\\Delta f[\/latex] instead of [latex]\\Delta y[\/latex], which still represents the change in the function\u2019s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.<\/p>\r\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\r\n\r\n<div id=\"fs-id1165137526960\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/div>\r\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\r\n<p id=\"fs-id1165135397217\">Other examples of rates of change include:<\/p>\r\n\r\n<ul id=\"fs-id1165137424067\">\r\n \t<li>A population of rats increasing by 40 rats per week<\/li>\r\n \t<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\r\n \t<li>A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)<\/li>\r\n \t<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\r\n \t<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3 class=\"title\">A General Note: Rate of Change<\/h3>\r\n<p id=\"fs-id1165137780744\">A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are \"output units per input units.\"<\/p>\r\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\r\n\r\n<div id=\"fs-id1165134060431\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137762240\">How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/h3>\r\n<ol id=\"fs-id1165137442714\">\r\n \t<li>Calculate the difference [latex]{y}_{2}-{y}_{1}=\\Delta y[\/latex].<\/li>\r\n \t<li>Calculate the difference [latex]{x}_{2}-{x}_{1}=\\Delta x[\/latex].<\/li>\r\n \t<li>Find the ratio [latex]\\frac{\\Delta y}{\\Delta x}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_03_01\" class=\"example\">\r\n<div id=\"fs-id1165135485962\" class=\"exercise\">\r\n<div id=\"fs-id1165137464225\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Computing an Average Rate of Change<\/h3>\r\n<p id=\"fs-id1165137603118\">Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"562005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"562005\"]\r\n<p id=\"fs-id1165135209401\">In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&amp;=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &amp;=\\frac{2.41-2.84}{2009 - 2007} \\\\[1mm] &amp;=\\frac{-0.43}{2\\text{ years}} \\\\[1mm] &amp;={-0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137784092\">Note that a decrease is expressed by a negative change or \"negative increase.\" A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of how to find the average rate of change between two points from a table of values.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=iJ_0nPUUlOg&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135160759\">Using the data in the table below,\u00a0find the average rate of change between 2005 and 2010.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"175600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"175600\"]\r\n\r\n[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_02\" class=\"example\">\r\n<div id=\"fs-id1165137851963\" class=\"exercise\">\r\n<div id=\"fs-id1165137437853\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Computing Average Rate of Change from a Graph<\/h3>\r\nGiven the function [latex]g\\left(t\\right)[\/latex] shown in Figure 5, find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].\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\/03010553\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"806109\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"806109\"]\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\/03010553\/CNX_Precalc_Figure_01_03_0022.jpg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/> <b>Figure 6<\/b>[\/caption]\r\n\r\nAt [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].<span id=\"fs-id1165137387448\">\r\n<\/span>\r\n<p id=\"fs-id1165137591169\">The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{1 - 4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135538482\">Note that the order we choose is very important. If, for example, we use [latex]\\frac{{y}_{2}-{y}_{1}}{{x}_{1}-{x}_{2}}[\/latex], we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_03\" class=\"example\">\r\n<div id=\"fs-id1165135536188\" class=\"exercise\">\r\n<div id=\"fs-id1165137835656\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Computing Average Rate of Change from a Table<\/h3>\r\n<p id=\"fs-id1165135515898\">After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in the table below.\u00a0Find her average speed over the first 6 hours.<\/p>\r\n\r\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong><em>t<\/em> (hours)<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>55<\/td>\r\n<td>90<\/td>\r\n<td>153<\/td>\r\n<td>214<\/td>\r\n<td>240<\/td>\r\n<td>282<\/td>\r\n<td>300<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"566859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"566859\"]\r\n<p id=\"fs-id1165137891478\">Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{292 - 10}{6 - 0} =\\frac{282}{6} =47[\/latex]<\/p>\r\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137731074\">Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_04\" class=\"example\">\r\n<div id=\"fs-id1165135353057\" class=\"exercise\">\r\n<div id=\"fs-id1165135383644\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Computing Average Rate of Change for a Function Expressed as a Formula<\/h3>\r\n<p id=\"fs-id1165131958324\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{4].}[\/latex]<\/p>\r\n[reveal-answer q=\"222718\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222718\"]\r\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&amp;={2}^{2}-\\frac{1}{2} &amp;&amp;&amp; f\\left(4\\right)&amp;={4}^{2}-\\frac{1}{4} \\\\ &amp;=4-\\frac{1}{2} &amp;&amp;&amp;&amp; =16-{1}{4} \\\\ &amp;=\\frac{7}{2} &amp;&amp;&amp;&amp; =\\frac{63}{4} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[1mm] &amp;=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[1mm] &amp;=\\frac{\\frac{49}{4}}{2} \\\\[1mm] &amp;=\\frac{49}{8} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video provides another example of finding the average rate of change of a function given a formula and an interval.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=g93QEKJXeu4&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].<\/p>\r\n[reveal-answer q=\"191250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"191250\"]\r\n\r\n[latex]\\frac{1}{2}[\/latex]\r\n\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[ohm_question hide_question_numbers=1]165703[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_05\" class=\"example\">\r\n<div id=\"fs-id1165137772170\" class=\"exercise\">\r\n<div id=\"fs-id1165137772173\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Finding the Average Rate of Change of a Force<\/h3>\r\n<p id=\"fs-id1165135443718\">The <strong>electrostatic force<\/strong> [latex]F[\/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[\/latex], in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex]. Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.<\/p>\r\n[reveal-answer q=\"271117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"271117\"]\r\n<p id=\"fs-id1165137770364\">We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right][\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change }&amp;=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2} \\\\[1mm] &amp;=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2} &amp;&amp; \\text{Simplify}. \\\\[1mm] &amp;=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4} \\\\[1mm] &amp;=\\frac{-\\frac{16}{36}}{4} &amp;&amp;\\text{Combine numerator terms}. \\\\[1mm] &amp;=-\\frac{1}{9}&amp;&amp;\\text{Simplify}\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135543242\">The average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_06\" class=\"example\">\r\n<div id=\"fs-id1165135174952\" class=\"exercise\">\r\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\r\n<h3>Example 10: Finding an Average Rate of Change as an Expression<\/h3>\r\n<p id=\"fs-id1165135155397\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].<\/p>\r\n[reveal-answer q=\"138559\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138559\"]\r\n<p id=\"fs-id1165137418913\">We use the average rate of change formula.<\/p>\r\n<p style=\"text-align: center;\">\u200b[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&amp;&amp;\\text{Evaluate} \\\\[1mm] &amp;\u200b=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&amp;&amp;\\text{Simplify}\u200b \\\\[1mm] &amp;=\\frac{{a}^{2}+3a+1 - 1}{a}&amp;&amp;\\text{Simplify and factor}\u200b \\\\[1mm] &amp;=\\frac{a\\left(a+3\\right)}{a}&amp;&amp;\\text{Divide by the common factor }a\u200b \\\\[1mm] &amp;=a+3 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165133316469\">This result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a[\/latex]. For example, on the interval [latex]\\left[0,5\\right][\/latex], the average rate of change would be [latex]5+3=8[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\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-id1165134149846\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x - 8[\/latex]. on the interval [latex]\\left[5,a\\right][\/latex].<\/p>\r\n[reveal-answer q=\"944513\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"944513\"]\r\n\r\n[latex]a+7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Determining Even and Odd Functions<\/h2>\r\n<p id=\"fs-id1165135532474\">Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\\left(x\\right)={x}^{2}[\/latex] or [latex]f\\left(x\\right)=|x|[\/latex] will result in the original graph. We say that these types of graphs are symmetric about the <em>y<\/em>-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong><\/p>\r\nIf the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010520\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/> <b>Figure 7.<\/b> (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.[\/caption]\r\n<p id=\"fs-id1165137406881\">We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.<\/p>\r\nWe use the expressions, \"odd\" and \"even\" because of polynomials. A polynomial function with only odd degree terms (odd powers of\u00a0<em>x<\/em>) will be an odd function. A polynomial function with only even degree terms (even powers of\u00a0<em>x<\/em>) will be an even function.\r\n<p id=\"fs-id1165134573214\">Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0[\/latex].<\/p>\r\nhttps:\/\/www.youtube.com\/watch?v=VvUI6E78cN4\r\n<div id=\"fs-id1165137619398\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Even and Odd Functions<\/h3>\r\n<p id=\"fs-id1165137407995\">A function is called an even function if for every input [latex]x[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135424702\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165135552902\">The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.<\/p>\r\n<p id=\"fs-id1165137501973\">A function is called an odd function if for every input [latex]x[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137762060\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165135503845\">The graph of an odd function is symmetric about the origin.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135503849\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165133353947\">How To: Given the formula for a function, determine if the function is even, odd, or neither.<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137552979\">\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]. If it does, it is even.<\/li>\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]. If it does, it is odd.<\/li>\r\n \t<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_12\" class=\"example\">\r\n<div id=\"fs-id1165137415536\" class=\"exercise\">\r\n<div id=\"fs-id1165137415539\" class=\"problem textbox shaded\">\r\n<h3>Example 11: Determining whether a Function Is Even, Odd, or Neither<\/h3>\r\n<p id=\"fs-id1165135252115\">Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\r\n[reveal-answer q=\"920669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"920669\"]\r\n<p id=\"fs-id1165137784968\">Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\r\n<p id=\"fs-id1165137771042\">This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\r\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\r\n<p id=\"fs-id1165135667851\">Because [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nConsider the graph of [latex]f[\/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[\/latex], and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010520\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\" \/> <b>Figure 8<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\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-id1165137897941\">Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\r\n[reveal-answer q=\"990690\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"990690\"]\r\n\r\nEven.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165135481945\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135481952\">\r\n \t<li>A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.<\/li>\r\n \t<li>Identifying points that mark the interval on a graph can be used to find the average rate of change.<\/li>\r\n \t<li>Comparing pairs of input and output values in a table can also be used to find the average rate of change.<\/li>\r\n \t<li>An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.<\/li>\r\n \t<li>The average rate of change can sometimes be determined as an expression.<\/li>\r\n \t<li>A graph can be symmetric about the x-axis, y-axis, origin, or none at all.<\/li>\r\n \t<li>Even functions are symmetric about the [latex]y\\text{-}[\/latex] axis, whereas odd functions are symmetric about the origin.<\/li>\r\n \t<li>Even functions satisfy the condition [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].<\/li>\r\n \t<li>Odd functions satisfy the condition [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex].<\/li>\r\n \t<li>A function can be odd, even, or neither.<\/li>\r\n \t<li>A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.<\/li>\r\n \t<li>A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.<\/li>\r\n \t<li>A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.<\/li>\r\n \t<li>Minima and maxima are also called extrema.<\/li>\r\n \t<li>We can find local extrema from a graph.<\/li>\r\n \t<li>The highest and lowest points on a graph indicate the maxima and minima.<\/li>\r\n<\/ul>\r\n<div style=\"line-height: 1.5;\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165133052903\" class=\"definition\">\r\n \t<dt><\/dt>\r\n \t<dt><strong>absolute maximum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133052908\">the greatest value of a function over an interval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133052911\" class=\"definition\">\r\n \t<dt><strong>absolute minimum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133052916\">the lowest value of a function over an interval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133052921\" class=\"definition\">\r\n \t<dt><strong>average rate of change<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133052926\">the difference in the output values of a function found for two values of the input divided by the difference between the inputs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135264639\" class=\"definition\">\r\n \t<dt><strong>decreasing function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135264645\">a function is decreasing in some open interval if [latex]f\\left(b\\right)&lt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137448239\" class=\"definition\">\r\n \t<dt>even function<\/dt>\r\n \t<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex], and is symmetric about the [latex]y\\text{-}[\/latex] axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135639824\" class=\"definition\">\r\n \t<dt><strong>increasing function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135639829\">a function is increasing in some open interval if [latex]f\\left(b\\right)&gt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135536408\" class=\"definition\">\r\n \t<dt><strong>local extrema<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135536413\">collectively, all of a function's local maxima and minima<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135536416\" class=\"definition\">\r\n \t<dt><strong>local maximum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135412035\">a value of the input where a function changes from increasing to decreasing as the input value increases.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135412040\" class=\"definition\">\r\n \t<dt><strong>local minimum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135412046\">a value of the input where a function changes from decreasing to increasing as the input value increases.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134259240\" class=\"definition\">\r\n \t<dt>odd function<\/dt>\r\n \t<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex], and is symmetric about the origin<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135412050\" class=\"definition\">\r\n \t<dt><strong>rate of change<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135412054\">the change of an output quantity relative to the change of the input quantity<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>\r\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 2.3 Homework Exercises<\/span><\/h2>\r\n1. Can the average rate of change of a function be constant?\r\n\r\n2. If a function [latex]f[\/latex] is increasing on [latex]\\left(a,b\\right)[\/latex] and decreasing on [latex]\\left(b,c\\right)[\/latex], then what can be said about the local extremum of [latex]f[\/latex] on [latex]\\left(a,c\\right)?[\/latex]\r\n\r\n3. How are the absolute maximum and minimum similar to and different from the local extrema?\r\n\r\n4. How does the graph of the absolute value function compare to the graph of the quadratic function, [latex]y={x}^{2}[\/latex], in terms of increasing and decreasing intervals?\r\n\r\n5. What is the meaning if the average rate of change is zero?\r\n\r\nFor exercises 6\u201310, find the average rate of change of each function on the interval specified for real numbers [latex]b[\/latex] or [latex]h[\/latex].\r\n\r\n6. [latex]f\\left(x\\right)=4{x}^{2}-7[\/latex] on [latex]\\left[1,\\text{ }b\\right][\/latex]\r\n\r\n7. [latex]g\\left(x\\right)=2{x}^{2}-9[\/latex] on [latex]\\left[4,\\text{ }b\\right][\/latex]\r\n\r\n8. [latex]p\\left(x\\right)=3x+4[\/latex] on [latex]\\left[2,\\text{ }2+h\\right][\/latex]\r\n\r\n9. [latex]g\\left(x\\right)=3{x}^{2}-2[\/latex] on [latex]\\left[x,x+h\\right][\/latex]\r\n\r\nFor exercises 10\u201312, consider the graph of [latex]f[\/latex].<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_2012.jpg\" alt=\"Graph of a polynomial. As y increases, the line increases to x = 5, decreases to x =3, increases to x = 7, decreases to x = 3, and then increases infinitely.\" width=\"731\" height=\"364\" \/>\r\n\r\n10. Estimate the average rate of change from [latex]x=1[\/latex] to [latex]x=4[\/latex].\r\n\r\n11. Estimate the average rate of change from [latex]x=2[\/latex] to [latex]x=5[\/latex].\r\n\r\n12. Estimate the average rate of change from [latex]x=5[\/latex] to [latex]x=7[\/latex].\r\n<p id=\"fs-id1165132924966\">For the following exercises, determine whether the function is odd, even, or neither.<\/p>\r\n\r\n<div id=\"fs-id1165132924969\" class=\"exercise\">\r\n<div id=\"fs-id1165132924971\" class=\"problem\">\r\n<p id=\"fs-id1165137812602\">13. [latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135671506\" class=\"solution\">\r\n<p id=\"fs-id1165135671508\">14.\u00a0[latex]g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137828008\" class=\"exercise\">\r\n<div id=\"fs-id1165137828010\" class=\"problem\">\r\n<p id=\"fs-id1165133408839\">15. [latex]h\\left(x\\right)=\\frac{1}{x}+3x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134187165\" class=\"exercise\">\r\n<div id=\"fs-id1165134187167\" class=\"problem\">\r\n<p id=\"fs-id1165134271332\">16. [latex]f\\left(x\\right)={\\left(x - 2\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134039317\" class=\"exercise\">\r\n<div id=\"fs-id1165137679220\" class=\"solution\">\r\n<p id=\"fs-id1165137679222\">17.\u00a0[latex]h\\left(x\\right)=2x-{x}^{3}[\/latex]<\/p>\r\n18. [latex]f(x)=\\dfrac{x^2}{x^4-5}[\/latex]\r\n\r\n19. [latex]f(x)=\\dfrac{x^4}{x^2+6}[\/latex]\r\n\r\n20. [latex]g(x)=\\dfrac{4}{x^3-2x}[\/latex]\r\n\r\n21. [latex]g(x)=\\dfrac{8}{x-4x^3}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\nFor exercises 22\u201325, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.\r\n\r\n22.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_2022.jpg\" alt=\"Graph of an absolute value function with minimum at (1, -3), and f(x) approaching positive infinity as x approaches negative infinity, f(x) approaches positive infinity as x approaches infinity. \" width=\"487\" height=\"251\" \/>\r\n\r\n23.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010558\/CNX_Precalc_Figure_01_03_2032.jpg\" alt=\"Graph of a cubic function with f(x) decreasing to negative infinity as x approaches negative infinity, a local maximum at (-2.5, 5.5), passing through the origin, a local min. at (-1.5, 1) and f(x) increasing to positive infinity as x approaches positive infinity.\" width=\"487\" height=\"345\" \/>\r\n\r\n24.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010558\/CNX_Precalc_Figure_01_03_2042.jpg\" alt=\"Graph of a cubic function with f(x) increasing to positive infinity as x approaches negative infinity, a local minimum at (-1.5, -2.5), passing through the origin, a local max. at (2, 4.5) and f(x) decreasing to negative infinity as x approaches positive infinity.\" width=\"487\" height=\"344\" \/>\r\n\r\n25.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010600\/CNX_Precalc_Figure_01_03_2052.jpg\" alt=\"Graph of a reciprocal function with an asymptote between 2 and 3, f(x) decreases to negative infinity as x approaches negative infinity, f(x) has a local max at (1,0), and approaches negative infinity as x approaches 3 from the left. f(x) approaches negative infinity as x approaches positive infinity, with f(x) approaching negative infinity as x approaches 3 from the right side.\" width=\"487\" height=\"284\" \/>\r\n\r\nFor exercises 26\u201327, consider the graph shown\u00a0below.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010600\/CNX_Precalc_Figure_01_03_2062.jpg\" alt=\"Graph of a cubic function passing through the origin, with local max at approximately (-3, 50) and decreasing to negative infinity as x approaches negative infinity. f(x) has a local minimum at (3, -50) and approaches infinity as x approaches positive infinity.\" width=\"731\" height=\"288\" \/>\r\n\r\n26. Estimate the intervals where the function is increasing or decreasing.\r\n\r\n27. Estimate the point(s) at which the graph of [latex]f[\/latex] has a local maximum or a local minimum.\r\n\r\nFor exercises 28\u201329, consider the graph\u00a0below.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010600\/CNX_Precalc_Figure_01_03_2072.jpg\" alt=\"Graph of a cubic function.\" \/>\r\n\r\n28.\u00a0If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.\r\n\r\n29. If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.\r\n\r\n30. The table below gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td><strong>Sales (millions of dollars)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1998<\/td>\r\n<td>201<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1999<\/td>\r\n<td>219<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2000<\/td>\r\n<td>233<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2001<\/td>\r\n<td>243<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2002<\/td>\r\n<td>249<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2003<\/td>\r\n<td>251<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2004<\/td>\r\n<td>249<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2005<\/td>\r\n<td>243<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2006<\/td>\r\n<td>233<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n31. The table below\u00a0gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td><strong>Population (thousands)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2000<\/td>\r\n<td>87<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2001<\/td>\r\n<td>84<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2002<\/td>\r\n<td>83<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2003<\/td>\r\n<td>80<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2004<\/td>\r\n<td>77<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2005<\/td>\r\n<td>76<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2006<\/td>\r\n<td>78<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2007<\/td>\r\n<td>81<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2008<\/td>\r\n<td>85<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor the following exercises, find the average rate of change of each function on the interval specified.\r\n\r\n32. [latex]f\\left(x\\right)={x}^{2}[\/latex] on [latex]\\left[1,\\text{ }5\\right][\/latex]\r\n\r\n33. [latex]h\\left(x\\right)=5 - 2{x}^{2}[\/latex] on [latex]\\left[-2,\\text{4}\\right][\/latex]\r\n\r\n34.\u00a0[latex]q\\left(x\\right)={x}^{3}[\/latex] on [latex]\\left[-4,\\text{2}\\right][\/latex]\r\n\r\n35. [latex]g\\left(x\\right)=3{x}^{3}-1[\/latex] on [latex]\\left[-3,\\text{3}\\right][\/latex]\r\n\r\n36. [latex]y=\\frac{1}{x}[\/latex] on [latex]\\left[1,\\text{ 3}\\right][\/latex]\r\n\r\n37. [latex]p\\left(t\\right)=\\frac{\\left({t}^{2}-4\\right)\\left(t+1\\right)}{{t}^{2}+3}[\/latex] on [latex]\\left[-3,\\text{1}\\right][\/latex]\r\n\r\n38.\u00a0[latex]k\\left(t\\right)=6{t}^{2}+\\frac{4}{{t}^{3}}[\/latex] on [latex]\\left[-1,3\\right][\/latex]\r\n\r\nFor the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.\r\n\r\n39. [latex]f\\left(x\\right)={x}^{4}-4{x}^{3}+5[\/latex]\r\n\r\n40. [latex]h\\left(x\\right)={x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}-1[\/latex]\r\n\r\n41. [latex]g\\left(t\\right)=t\\sqrt{t+3}[\/latex]\r\n\r\n42.\u00a0[latex]k\\left(t\\right)=3{t}^{\\frac{2}{3}}-t[\/latex]\r\n\r\n43. [latex]m\\left(x\\right)={x}^{4}+2{x}^{3}-12{x}^{2}-10x+4[\/latex]\r\n\r\n44. [latex]n\\left(x\\right)={x}^{4}-8{x}^{3}+18{x}^{2}-6x+2[\/latex]\r\n\r\n45. The graph of the function [latex]f[\/latex] is shown below.\r\n<img style=\"line-height: 1.5;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010601\/CNX_Precalc_Figure_01_03_213n2.jpg\" alt=\"Graph of f(x) on a graphing calculator.\" \/>\r\n\r\nBased on the calculator screenshot,\u00a0the point [latex]\\left(1.333,\\text{ }5.185\\right)[\/latex]\u00a0is which of the following?\r\nA) a relative (local) maximum of the function\r\nB) the vertex of the function\r\nC) the absolute maximum of the function\r\nD) a zero of the function.\r\n\r\n46. Let [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]. Find a number [latex]c[\/latex] such that the average rate of change of the function [latex]f[\/latex] on the interval [latex]\\left(1,c\\right)[\/latex] is [latex]-\\frac{1}{4}[\/latex].\r\n\r\n47. Let [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] . Find the number [latex]b[\/latex] such that the average rate of change of [latex]f[\/latex] on the interval [latex]\\left(2,b\\right)[\/latex] is [latex]-\\frac{1}{10}[\/latex].\r\n\r\n48.\u00a0At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?\r\n\r\n49.\u00a0A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?\r\n\r\n50.\u00a0Near the surface of the moon, the distance that an object falls is a function of time. It is given by [latex]d\\left(t\\right)=2.6667{t}^{2}[\/latex], where [latex]t[\/latex] is in seconds and [latex]d\\left(t\\right)[\/latex] is in feet. If an object is dropped from a certain height, find the average velocity of the object from [latex]t=1[\/latex] to [latex]t=2[\/latex].\r\n\r\n51.\u00a0The graph below\u00a0illustrates the decay of a radioactive substance over [latex]t[\/latex] days.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005048\/CNX_Precalc_Figure_01_03_214n.jpg\" alt=\"Graph of an exponential function.\" \/>\r\nUse the graph to estimate the average decay rate from [latex]t=5[\/latex] to [latex]t=15[\/latex].","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\n<li>Use a graph to locate local and absolute maxima and local minima.<\/li>\n<li>Find the average rate of change of a function.<\/li>\n<li>Determine whether a function is even, odd, or neither from its graph and equation.<\/li>\n<\/ul>\n<\/div>\n<h2>Increasing and Decreasing Functions<\/h2>\n<p id=\"fs-id1165137784644\">As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 1\u00a0shows examples of increasing and decreasing intervals on a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0042.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 1.<\/strong> The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].<\/p>\n<p>This video further explains how to find where a function is increasing or decreasing.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine Where a Function is Increasing and Decreasing\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/78b4HOMVcKM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p id=\"fs-id1165134272749\">While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a <strong>local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a <strong>local minimum<\/strong>. The plural form is &#8220;local minima.&#8221; Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is &#8220;extremum.&#8221;) Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>local<\/em>.<\/p>\n<p id=\"fs-id1165134547216\">Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em>local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/p>\n<p>For the function in Figure 2, the local maximum is 16, and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].<\/p>\n<div style=\"width: 741px\" 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\/03010554\/CNX_Precalc_Figure_01_03_0142.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum. The local maximum is 16 and occurs at x = negative 2. This is the point negative 2, 16. The local minimum is negative 16 and occurs at x = 2. This is the point 2, negative 16.\" width=\"731\" height=\"467\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2<\/strong><\/p>\n<\/div>\n<p id=\"fs-id1165133316450\">To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 3\u00a0illustrates these ideas for a local maximum.<\/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\/03010554\/CNX_Precalc_Figure_01_03_0052.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong> Definition of a local maximum.<\/p>\n<\/div>\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\n<div id=\"fs-id1165134169419\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Local Minima and Local Maxima<\/h3>\n<p id=\"fs-id1165134169426\">A function [latex]f[\/latex] is an <strong>increasing function<\/strong> on an open interval if [latex]f\\left(b\\right)>f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex].<\/p>\n<p id=\"fs-id1165137668624\">A function [latex]f[\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right)<f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex].<\/p>\n<p id=\"fs-id1165135389881\">A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a<b<c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex]. Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a<b<c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].<\/p>\n<\/div>\n<div id=\"Example_01_03_07\" class=\"example\">\n<div id=\"fs-id1165134266716\" class=\"exercise\">\n<div id=\"fs-id1165134266718\" class=\"problem textbox shaded\">\n<h3>Example 1: Finding Increasing and Decreasing Intervals on a Graph<\/h3>\n<p>Given the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.<\/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\/03010554\/CNX_Precalc_Figure_01_03_0062.jpg\" alt=\"Graph of a polynomial. As x gets large in the negative direction, the outputs of the function get large in the positive direction. As inputs approach 1, then the function value approaches a minimum of negative one. As x approaches 3, the values increase again and between 3 and 4 decrease one last time. As x gets large in the positive direction, the function values increase without bound.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4<\/strong><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q331055\">Show Solution<\/span><\/p>\n<div id=\"q331055\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133067197\">We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on.<\/p>\n<p id=\"fs-id1165135369127\">In <strong>interval notation<\/strong>, we would say the function appears to be increasing on the interval (1,3) and the interval [latex]\\left(4,\\infty \\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165134104021\">Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[\/latex] , [latex]t=3[\/latex] , and [latex]t=4[\/latex] . These points are the local extrema (two minima and a maximum).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_08\" class=\"example\">\n<div id=\"fs-id1165135261521\" class=\"exercise\">\n<div id=\"fs-id1165135261523\" class=\"problem textbox shaded\">\n<h3>Example 2: Finding Local Extrema from a Graph<\/h3>\n<p id=\"fs-id1165135261528\">Graph the function [latex]f\\left(x\\right)=\\frac{2}{x}+\\frac{x}{3}[\/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453777\">Show Solution<\/span><\/p>\n<div id=\"q453777\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using technology, we find that the graph of the function looks like that in Figure 5. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3[\/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2[\/latex].<\/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\/03010554\/CNX_Precalc_Figure_01_03_0072.jpg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>Most graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 6\u00a0provides screen images from two different technologies, showing the estimate for the local maximum and minimum.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_008ab2.jpg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134075625\">Based on these estimates, the function is increasing on the interval [latex]\\left(-\\infty ,-{2.449}\\right)[\/latex]<br \/>\nand [latex]\\left(2.449\\text{,}\\infty \\right)[\/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at [latex]\\pm \\sqrt{6}[\/latex], but determining this requires calculus.)<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135640967\">Graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q48622\">Show Solution<\/span><\/p>\n<div id=\"q48622\" class=\"hidden-answer\" style=\"display: none\">\n<p>The local maximum is 28 at\u00a0<em>x\u00a0<\/em>= -1\u00a0and the local minimum is -80 at\u00a0<em>x<\/em> = 5. The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right)[\/latex].<\/p>\n<p><span id=\"fs-id1165134043615\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_0102.jpg\" alt=\"Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).\" width=\"487\" height=\"328\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm165724\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=165724&theme=oea&iframe_resize_id=ohm165724\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"Example_01_03_09\" class=\"example\">\n<div id=\"fs-id1165135367558\" class=\"exercise\">\n<div id=\"fs-id1165137896103\" class=\"problem textbox shaded\">\n<h3>Example 3: Finding Local Maxima and Minima from a Graph<\/h3>\n<p>For the function [latex]f[\/latex] whose graph is shown in Figure 7, find all local maxima and minima.<\/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\/03010555\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q209462\">Show Solution<\/span><\/p>\n<div id=\"q209462\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135527085\">Observe the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].<\/p>\n<p id=\"fs-id1165134485672\">The graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134544960\">\n<h2>Use\u00a0A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h2>\n<p>There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.<\/p>\n<p>To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.<\/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\/03010557\/CNX_Precalc_Figure_01_03_0152.jpg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\n<div id=\"fs-id1165135251290\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Absolute Maxima and Minima<\/h3>\n<p id=\"fs-id1165132939786\">The <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<p id=\"fs-id1165137932685\">The <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<\/div>\n<div id=\"Example_01_03_10\" class=\"example\">\n<div id=\"fs-id1165134047533\" class=\"exercise\">\n<div id=\"fs-id1165134047535\" class=\"problem textbox shaded\">\n<h3>Example 4: Finding Absolute Maxima and Minima from a Graph<\/h3>\n<p>For the function [latex]f[\/latex] shown in Figure 11, find all absolute maxima and minima.<\/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\/03010557\/CNX_Precalc_Figure_01_03_0132.jpg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q502428\">Show Solution<\/span><\/p>\n<div id=\"q502428\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135532371\">Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].<\/p>\n<p id=\"fs-id1165137863670\">The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p id=\"fs-id1165135194500\">Gasoline costs have experienced some wild fluctuations over the last several decades. The table below<a class=\"footnote\" title=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.\" id=\"return-footnote-17722-1\" href=\"#footnote-17722-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.<\/p>\n<h2>Finding the Average Rate of Change of a Function<\/h2>\n<p id=\"fs-id1165137834011\">The price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\n<div id=\"fs-id1165135452482\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\frac{\\text{Change in output}}{\\text{Change in input}} \\\\[1mm] &=\\frac{\\Delta y}{\\Delta x} \\\\[1mm] &= \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &= \\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165135471272\">The Greek letter [latex]\\Delta[\/latex] (delta) signifies the change in a quantity; we read the ratio as &#8220;delta-<em>y<\/em> over delta-<em>x<\/em>&#8221; or &#8220;the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].&#8221; Occasionally we write [latex]\\Delta f[\/latex] instead of [latex]\\Delta y[\/latex], which still represents the change in the function\u2019s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.<\/p>\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\n<div id=\"fs-id1165137526960\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/div>\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\n<p id=\"fs-id1165135397217\">Other examples of rates of change include:<\/p>\n<ul id=\"fs-id1165137424067\">\n<li>A population of rats increasing by 40 rats per week<\/li>\n<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\n<li>A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)<\/li>\n<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\n<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3 class=\"title\">A General Note: Rate of Change<\/h3>\n<p id=\"fs-id1165137780744\">A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are &#8220;output units per input units.&#8221;<\/p>\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\n<div id=\"fs-id1165134060431\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137762240\">How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/h3>\n<ol id=\"fs-id1165137442714\">\n<li>Calculate the difference [latex]{y}_{2}-{y}_{1}=\\Delta y[\/latex].<\/li>\n<li>Calculate the difference [latex]{x}_{2}-{x}_{1}=\\Delta x[\/latex].<\/li>\n<li>Find the ratio [latex]\\frac{\\Delta y}{\\Delta x}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_03_01\" class=\"example\">\n<div id=\"fs-id1165135485962\" class=\"exercise\">\n<div id=\"fs-id1165137464225\" class=\"problem textbox shaded\">\n<h3>Example 5: Computing an Average Rate of Change<\/h3>\n<p id=\"fs-id1165137603118\">Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/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=\"q562005\">Show Solution<\/span><\/p>\n<div id=\"q562005\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135209401\">In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &=\\frac{2.41-2.84}{2009 - 2007} \\\\[1mm] &=\\frac{-0.43}{2\\text{ years}} \\\\[1mm] &={-0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137784092\">Note that a decrease is expressed by a negative change or &#8220;negative increase.&#8221; A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of how to find the average rate of change between two points from a table of values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Find the Average Rate of Change From a Table - Temperatures\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iJ_0nPUUlOg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135160759\">Using the data in the table below,\u00a0find the average rate of change between 2005 and 2010.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/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=\"q175600\">Show Solution<\/span><\/p>\n<div id=\"q175600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_02\" class=\"example\">\n<div id=\"fs-id1165137851963\" class=\"exercise\">\n<div id=\"fs-id1165137437853\" class=\"problem textbox shaded\">\n<h3>Example 6: Computing Average Rate of Change from a Graph<\/h3>\n<p>Given the function [latex]g\\left(t\\right)[\/latex] shown in Figure 5, find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].<\/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\/03010553\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q806109\">Show Solution<\/span><\/p>\n<div id=\"q806109\" class=\"hidden-answer\" style=\"display: none\">\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\/03010553\/CNX_Precalc_Figure_01_03_0022.jpg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p>At [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].<span id=\"fs-id1165137387448\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137591169\">The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1 - 4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135538482\">Note that the order we choose is very important. If, for example, we use [latex]\\frac{{y}_{2}-{y}_{1}}{{x}_{1}-{x}_{2}}[\/latex], we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_03\" class=\"example\">\n<div id=\"fs-id1165135536188\" class=\"exercise\">\n<div id=\"fs-id1165137835656\" class=\"problem textbox shaded\">\n<h3>Example 7: Computing Average Rate of Change from a Table<\/h3>\n<p id=\"fs-id1165135515898\">After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in the table below.\u00a0Find her average speed over the first 6 hours.<\/p>\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong><em>t<\/em> (hours)<\/strong><\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\n<td>10<\/td>\n<td>55<\/td>\n<td>90<\/td>\n<td>153<\/td>\n<td>214<\/td>\n<td>240<\/td>\n<td>282<\/td>\n<td>300<\/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=\"q566859\">Show Solution<\/span><\/p>\n<div id=\"q566859\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137891478\">Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{292 - 10}{6 - 0} =\\frac{282}{6} =47[\/latex]<\/p>\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137731074\">Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_04\" class=\"example\">\n<div id=\"fs-id1165135353057\" class=\"exercise\">\n<div id=\"fs-id1165135383644\" class=\"problem textbox shaded\">\n<h3>Example 8: Computing Average Rate of Change for a Function Expressed as a Formula<\/h3>\n<p id=\"fs-id1165131958324\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{4].}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q222718\">Show Solution<\/span><\/p>\n<div id=\"q222718\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&={2}^{2}-\\frac{1}{2} &&& f\\left(4\\right)&={4}^{2}-\\frac{1}{4} \\\\ &=4-\\frac{1}{2} &&&& =16-{1}{4} \\\\ &=\\frac{7}{2} &&&& =\\frac{63}{4} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\frac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[1mm] &=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[1mm] &=\\frac{\\frac{49}{4}}{2} \\\\[1mm] &=\\frac{49}{8} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of finding the average rate of change of a function given a formula and an interval.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Find the Average Rate of Change Given a Function Rule\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/g93QEKJXeu4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q191250\">Show Solution<\/span><\/p>\n<div id=\"q191250\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm165703\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=165703&theme=oea&iframe_resize_id=ohm165703\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"Example_01_03_05\" class=\"example\">\n<div id=\"fs-id1165137772170\" class=\"exercise\">\n<div id=\"fs-id1165137772173\" class=\"problem textbox shaded\">\n<h3>Example 9: Finding the Average Rate of Change of a Force<\/h3>\n<p id=\"fs-id1165135443718\">The <strong>electrostatic force<\/strong> [latex]F[\/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[\/latex], in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex]. Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q271117\">Show Solution<\/span><\/p>\n<div id=\"q271117\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137770364\">We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right][\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change }&=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2} \\\\[1mm] &=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2} && \\text{Simplify}. \\\\[1mm] &=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4} \\\\[1mm] &=\\frac{-\\frac{16}{36}}{4} &&\\text{Combine numerator terms}. \\\\[1mm] &=-\\frac{1}{9}&&\\text{Simplify}\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135543242\">The average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_06\" class=\"example\">\n<div id=\"fs-id1165135174952\" class=\"exercise\">\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\n<h3>Example 10: Finding an Average Rate of Change as an Expression<\/h3>\n<p id=\"fs-id1165135155397\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q138559\">Show Solution<\/span><\/p>\n<div id=\"q138559\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137418913\">We use the average rate of change formula.<\/p>\n<p style=\"text-align: center;\">\u200b[latex]\\begin{align}\\text{Average rate of change}&=\\frac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&&\\text{Evaluate} \\\\[1mm] &\u200b=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&&\\text{Simplify}\u200b \\\\[1mm] &=\\frac{{a}^{2}+3a+1 - 1}{a}&&\\text{Simplify and factor}\u200b \\\\[1mm] &=\\frac{a\\left(a+3\\right)}{a}&&\\text{Divide by the common factor }a\u200b \\\\[1mm] &=a+3 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165133316469\">This result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a[\/latex]. For example, on the interval [latex]\\left[0,5\\right][\/latex], the average rate of change would be [latex]5+3=8[\/latex].<\/p>\n<\/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-id1165134149846\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x - 8[\/latex]. on the interval [latex]\\left[5,a\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q944513\">Show Solution<\/span><\/p>\n<div id=\"q944513\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]a+7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Determining Even and Odd Functions<\/h2>\n<p id=\"fs-id1165135532474\">Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\\left(x\\right)={x}^{2}[\/latex] or [latex]f\\left(x\\right)=|x|[\/latex] will result in the original graph. We say that these types of graphs are symmetric about the <em>y<\/em>-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong><\/p>\n<p>If the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010520\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7.<\/b> (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.<\/p>\n<\/div>\n<p id=\"fs-id1165137406881\">We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.<\/p>\n<p>We use the expressions, &#8220;odd&#8221; and &#8220;even&#8221; because of polynomials. A polynomial function with only odd degree terms (odd powers of\u00a0<em>x<\/em>) will be an odd function. A polynomial function with only even degree terms (even powers of\u00a0<em>x<\/em>) will be an even function.<\/p>\n<p id=\"fs-id1165134573214\">Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Introduction to Odd and Even Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VvUI6E78cN4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"fs-id1165137619398\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Even and Odd Functions<\/h3>\n<p id=\"fs-id1165137407995\">A function is called an even function if for every input [latex]x[\/latex]<\/p>\n<div id=\"fs-id1165135424702\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135552902\">The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.<\/p>\n<p id=\"fs-id1165137501973\">A function is called an odd function if for every input [latex]x[\/latex]<\/p>\n<div id=\"fs-id1165137762060\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135503845\">The graph of an odd function is symmetric about the origin.<\/p>\n<\/div>\n<div id=\"fs-id1165135503849\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165133353947\">How To: Given the formula for a function, determine if the function is even, odd, or neither.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137552979\">\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]. If it does, it is even.<\/li>\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]. If it does, it is odd.<\/li>\n<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_12\" class=\"example\">\n<div id=\"fs-id1165137415536\" class=\"exercise\">\n<div id=\"fs-id1165137415539\" class=\"problem textbox shaded\">\n<h3>Example 11: Determining whether a Function Is Even, Odd, or Neither<\/h3>\n<p id=\"fs-id1165135252115\">Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q920669\">Show Solution<\/span><\/p>\n<div id=\"q920669\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137784968\">Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\n<p id=\"fs-id1165137771042\">This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\n<p id=\"fs-id1165135667851\">Because [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Consider the graph of [latex]f[\/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[\/latex], and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.<\/p>\n<div style=\"width: 741px\" 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\/03010520\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137897941\">Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q990690\">Show Solution<\/span><\/p>\n<div id=\"q990690\" class=\"hidden-answer\" style=\"display: none\">\n<p>Even.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135481945\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135481952\">\n<li>A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.<\/li>\n<li>Identifying points that mark the interval on a graph can be used to find the average rate of change.<\/li>\n<li>Comparing pairs of input and output values in a table can also be used to find the average rate of change.<\/li>\n<li>An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.<\/li>\n<li>The average rate of change can sometimes be determined as an expression.<\/li>\n<li>A graph can be symmetric about the x-axis, y-axis, origin, or none at all.<\/li>\n<li>Even functions are symmetric about the [latex]y\\text{-}[\/latex] axis, whereas odd functions are symmetric about the origin.<\/li>\n<li>Even functions satisfy the condition [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].<\/li>\n<li>Odd functions satisfy the condition [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex].<\/li>\n<li>A function can be odd, even, or neither.<\/li>\n<li>A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.<\/li>\n<li>A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.<\/li>\n<li>A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.<\/li>\n<li>Minima and maxima are also called extrema.<\/li>\n<li>We can find local extrema from a graph.<\/li>\n<li>The highest and lowest points on a graph indicate the maxima and minima.<\/li>\n<\/ul>\n<div style=\"line-height: 1.5;\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133052903\" class=\"definition\">\n<dt><\/dt>\n<dt><strong>absolute maximum<\/strong><\/dt>\n<dd id=\"fs-id1165133052908\">the greatest value of a function over an interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133052911\" class=\"definition\">\n<dt><strong>absolute minimum<\/strong><\/dt>\n<dd id=\"fs-id1165133052916\">the lowest value of a function over an interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133052921\" class=\"definition\">\n<dt><strong>average rate of change<\/strong><\/dt>\n<dd id=\"fs-id1165133052926\">the difference in the output values of a function found for two values of the input divided by the difference between the inputs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135264639\" class=\"definition\">\n<dt><strong>decreasing function<\/strong><\/dt>\n<dd id=\"fs-id1165135264645\">a function is decreasing in some open interval if [latex]f\\left(b\\right)<f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137448239\" class=\"definition\">\n<dt>even function<\/dt>\n<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex], and is symmetric about the [latex]y\\text{-}[\/latex] axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135639824\" class=\"definition\">\n<dt><strong>increasing function<\/strong><\/dt>\n<dd id=\"fs-id1165135639829\">a function is increasing in some open interval if [latex]f\\left(b\\right)>f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135536408\" class=\"definition\">\n<dt><strong>local extrema<\/strong><\/dt>\n<dd id=\"fs-id1165135536413\">collectively, all of a function&#8217;s local maxima and minima<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135536416\" class=\"definition\">\n<dt><strong>local maximum<\/strong><\/dt>\n<dd id=\"fs-id1165135412035\">a value of the input where a function changes from increasing to decreasing as the input value increases.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135412040\" class=\"definition\">\n<dt><strong>local minimum<\/strong><\/dt>\n<dd id=\"fs-id1165135412046\">a value of the input where a function changes from decreasing to increasing as the input value increases.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134259240\" class=\"definition\">\n<dt>odd function<\/dt>\n<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex], and is symmetric about the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135412050\" class=\"definition\">\n<dt><strong>rate of change<\/strong><\/dt>\n<dd id=\"fs-id1165135412054\">the change of an output quantity relative to the change of the input quantity<\/dd>\n<\/dl>\n<\/div>\n<\/section>\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 2.3 Homework Exercises<\/span><\/h2>\n<p>1. Can the average rate of change of a function be constant?<\/p>\n<p>2. If a function [latex]f[\/latex] is increasing on [latex]\\left(a,b\\right)[\/latex] and decreasing on [latex]\\left(b,c\\right)[\/latex], then what can be said about the local extremum of [latex]f[\/latex] on [latex]\\left(a,c\\right)?[\/latex]<\/p>\n<p>3. How are the absolute maximum and minimum similar to and different from the local extrema?<\/p>\n<p>4. How does the graph of the absolute value function compare to the graph of the quadratic function, [latex]y={x}^{2}[\/latex], in terms of increasing and decreasing intervals?<\/p>\n<p>5. What is the meaning if the average rate of change is zero?<\/p>\n<p>For exercises 6\u201310, find the average rate of change of each function on the interval specified for real numbers [latex]b[\/latex] or [latex]h[\/latex].<\/p>\n<p>6. [latex]f\\left(x\\right)=4{x}^{2}-7[\/latex] on [latex]\\left[1,\\text{ }b\\right][\/latex]<\/p>\n<p>7. [latex]g\\left(x\\right)=2{x}^{2}-9[\/latex] on [latex]\\left[4,\\text{ }b\\right][\/latex]<\/p>\n<p>8. [latex]p\\left(x\\right)=3x+4[\/latex] on [latex]\\left[2,\\text{ }2+h\\right][\/latex]<\/p>\n<p>9. [latex]g\\left(x\\right)=3{x}^{2}-2[\/latex] on [latex]\\left[x,x+h\\right][\/latex]<\/p>\n<p>For exercises 10\u201312, consider the graph of [latex]f[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_2012.jpg\" alt=\"Graph of a polynomial. As y increases, the line increases to x = 5, decreases to x =3, increases to x = 7, decreases to x = 3, and then increases infinitely.\" width=\"731\" height=\"364\" \/><\/p>\n<p>10. Estimate the average rate of change from [latex]x=1[\/latex] to [latex]x=4[\/latex].<\/p>\n<p>11. Estimate the average rate of change from [latex]x=2[\/latex] to [latex]x=5[\/latex].<\/p>\n<p>12. Estimate the average rate of change from [latex]x=5[\/latex] to [latex]x=7[\/latex].<\/p>\n<p id=\"fs-id1165132924966\">For the following exercises, determine whether the function is odd, even, or neither.<\/p>\n<div id=\"fs-id1165132924969\" class=\"exercise\">\n<div id=\"fs-id1165132924971\" class=\"problem\">\n<p id=\"fs-id1165137812602\">13. [latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135671506\" class=\"solution\">\n<p id=\"fs-id1165135671508\">14.\u00a0[latex]g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137828008\" class=\"exercise\">\n<div id=\"fs-id1165137828010\" class=\"problem\">\n<p id=\"fs-id1165133408839\">15. [latex]h\\left(x\\right)=\\frac{1}{x}+3x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134187165\" class=\"exercise\">\n<div id=\"fs-id1165134187167\" class=\"problem\">\n<p id=\"fs-id1165134271332\">16. [latex]f\\left(x\\right)={\\left(x - 2\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134039317\" class=\"exercise\">\n<div id=\"fs-id1165137679220\" class=\"solution\">\n<p id=\"fs-id1165137679222\">17.\u00a0[latex]h\\left(x\\right)=2x-{x}^{3}[\/latex]<\/p>\n<p>18. [latex]f(x)=\\dfrac{x^2}{x^4-5}[\/latex]<\/p>\n<p>19. [latex]f(x)=\\dfrac{x^4}{x^2+6}[\/latex]<\/p>\n<p>20. [latex]g(x)=\\dfrac{4}{x^3-2x}[\/latex]<\/p>\n<p>21. [latex]g(x)=\\dfrac{8}{x-4x^3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>For exercises 22\u201325, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.<\/p>\n<p>22.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_2022.jpg\" alt=\"Graph of an absolute value function with minimum at (1, -3), and f(x) approaching positive infinity as x approaches negative infinity, f(x) approaches positive infinity as x approaches infinity.\" width=\"487\" height=\"251\" \/><\/p>\n<p>23.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010558\/CNX_Precalc_Figure_01_03_2032.jpg\" alt=\"Graph of a cubic function with f(x) decreasing to negative infinity as x approaches negative infinity, a local maximum at (-2.5, 5.5), passing through the origin, a local min. at (-1.5, 1) and f(x) increasing to positive infinity as x approaches positive infinity.\" width=\"487\" height=\"345\" \/><\/p>\n<p>24.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010558\/CNX_Precalc_Figure_01_03_2042.jpg\" alt=\"Graph of a cubic function with f(x) increasing to positive infinity as x approaches negative infinity, a local minimum at (-1.5, -2.5), passing through the origin, a local max. at (2, 4.5) and f(x) decreasing to negative infinity as x approaches positive infinity.\" width=\"487\" height=\"344\" \/><\/p>\n<p>25.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010600\/CNX_Precalc_Figure_01_03_2052.jpg\" alt=\"Graph of a reciprocal function with an asymptote between 2 and 3, f(x) decreases to negative infinity as x approaches negative infinity, f(x) has a local max at (1,0), and approaches negative infinity as x approaches 3 from the left. f(x) approaches negative infinity as x approaches positive infinity, with f(x) approaching negative infinity as x approaches 3 from the right side.\" width=\"487\" height=\"284\" \/><\/p>\n<p>For exercises 26\u201327, consider the graph shown\u00a0below.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010600\/CNX_Precalc_Figure_01_03_2062.jpg\" alt=\"Graph of a cubic function passing through the origin, with local max at approximately (-3, 50) and decreasing to negative infinity as x approaches negative infinity. f(x) has a local minimum at (3, -50) and approaches infinity as x approaches positive infinity.\" width=\"731\" height=\"288\" \/><\/p>\n<p>26. Estimate the intervals where the function is increasing or decreasing.<\/p>\n<p>27. Estimate the point(s) at which the graph of [latex]f[\/latex] has a local maximum or a local minimum.<\/p>\n<p>For exercises 28\u201329, consider the graph\u00a0below.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010600\/CNX_Precalc_Figure_01_03_2072.jpg\" alt=\"Graph of a cubic function.\" \/><\/p>\n<p>28.\u00a0If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.<\/p>\n<p>29. If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.<\/p>\n<p>30. The table below gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td><strong>Sales (millions of dollars)<\/strong><\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>201<\/td>\n<\/tr>\n<tr>\n<td>1999<\/td>\n<td>219<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>233<\/td>\n<\/tr>\n<tr>\n<td>2001<\/td>\n<td>243<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>249<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>251<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>249<\/td>\n<\/tr>\n<tr>\n<td>2005<\/td>\n<td>243<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>233<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>31. The table below\u00a0gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td><strong>Population (thousands)<\/strong><\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>87<\/td>\n<\/tr>\n<tr>\n<td>2001<\/td>\n<td>84<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>83<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>80<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>77<\/td>\n<\/tr>\n<tr>\n<td>2005<\/td>\n<td>76<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>78<\/td>\n<\/tr>\n<tr>\n<td>2007<\/td>\n<td>81<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>85<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For the following exercises, find the average rate of change of each function on the interval specified.<\/p>\n<p>32. [latex]f\\left(x\\right)={x}^{2}[\/latex] on [latex]\\left[1,\\text{ }5\\right][\/latex]<\/p>\n<p>33. [latex]h\\left(x\\right)=5 - 2{x}^{2}[\/latex] on [latex]\\left[-2,\\text{4}\\right][\/latex]<\/p>\n<p>34.\u00a0[latex]q\\left(x\\right)={x}^{3}[\/latex] on [latex]\\left[-4,\\text{2}\\right][\/latex]<\/p>\n<p>35. [latex]g\\left(x\\right)=3{x}^{3}-1[\/latex] on [latex]\\left[-3,\\text{3}\\right][\/latex]<\/p>\n<p>36. [latex]y=\\frac{1}{x}[\/latex] on [latex]\\left[1,\\text{ 3}\\right][\/latex]<\/p>\n<p>37. [latex]p\\left(t\\right)=\\frac{\\left({t}^{2}-4\\right)\\left(t+1\\right)}{{t}^{2}+3}[\/latex] on [latex]\\left[-3,\\text{1}\\right][\/latex]<\/p>\n<p>38.\u00a0[latex]k\\left(t\\right)=6{t}^{2}+\\frac{4}{{t}^{3}}[\/latex] on [latex]\\left[-1,3\\right][\/latex]<\/p>\n<p>For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.<\/p>\n<p>39. [latex]f\\left(x\\right)={x}^{4}-4{x}^{3}+5[\/latex]<\/p>\n<p>40. [latex]h\\left(x\\right)={x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}-1[\/latex]<\/p>\n<p>41. [latex]g\\left(t\\right)=t\\sqrt{t+3}[\/latex]<\/p>\n<p>42.\u00a0[latex]k\\left(t\\right)=3{t}^{\\frac{2}{3}}-t[\/latex]<\/p>\n<p>43. [latex]m\\left(x\\right)={x}^{4}+2{x}^{3}-12{x}^{2}-10x+4[\/latex]<\/p>\n<p>44. [latex]n\\left(x\\right)={x}^{4}-8{x}^{3}+18{x}^{2}-6x+2[\/latex]<\/p>\n<p>45. The graph of the function [latex]f[\/latex] is shown below.<br \/>\n<img decoding=\"async\" style=\"line-height: 1.5;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010601\/CNX_Precalc_Figure_01_03_213n2.jpg\" alt=\"Graph of f(x) on a graphing calculator.\" \/><\/p>\n<p>Based on the calculator screenshot,\u00a0the point [latex]\\left(1.333,\\text{ }5.185\\right)[\/latex]\u00a0is which of the following?<br \/>\nA) a relative (local) maximum of the function<br \/>\nB) the vertex of the function<br \/>\nC) the absolute maximum of the function<br \/>\nD) a zero of the function.<\/p>\n<p>46. Let [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]. Find a number [latex]c[\/latex] such that the average rate of change of the function [latex]f[\/latex] on the interval [latex]\\left(1,c\\right)[\/latex] is [latex]-\\frac{1}{4}[\/latex].<\/p>\n<p>47. Let [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] . Find the number [latex]b[\/latex] such that the average rate of change of [latex]f[\/latex] on the interval [latex]\\left(2,b\\right)[\/latex] is [latex]-\\frac{1}{10}[\/latex].<\/p>\n<p>48.\u00a0At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?<\/p>\n<p>49.\u00a0A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?<\/p>\n<p>50.\u00a0Near the surface of the moon, the distance that an object falls is a function of time. It is given by [latex]d\\left(t\\right)=2.6667{t}^{2}[\/latex], where [latex]t[\/latex] is in seconds and [latex]d\\left(t\\right)[\/latex] is in feet. If an object is dropped from a certain height, find the average velocity of the object from [latex]t=1[\/latex] to [latex]t=2[\/latex].<\/p>\n<p>51.\u00a0The graph below\u00a0illustrates the decay of a radioactive substance over [latex]t[\/latex] days.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005048\/CNX_Precalc_Figure_01_03_214n.jpg\" alt=\"Graph of an exponential function.\" \/><br \/>\nUse the graph to estimate the average decay rate from [latex]t=5[\/latex] to [latex]t=15[\/latex].<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-17722-1\">http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014. <a href=\"#return-footnote-17722-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":264444,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17722","chapter","type-chapter","status-publish","hentry"],"part":17684,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17722","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17722\/revisions"}],"predecessor-version":[{"id":17732,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17722\/revisions\/17732"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/17684"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17722\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=17722"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=17722"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=17722"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=17722"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}