{"id":1990,"date":"2024-05-10T16:29:05","date_gmt":"2024-05-10T16:29:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1990"},"modified":"2024-08-04T10:49:09","modified_gmt":"2024-08-04T10:49:09","slug":"basic-functions-and-graphs-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-functions-and-graphs-get-stronger\/","title":{"raw":"Basic Functions and Graphs: Get Stronger","rendered":"Basic Functions and Graphs: Get Stronger"},"content":{"raw":"<h2 class=\"entry-title\">Review of Functions<\/h2>\r\n<p><strong>For the following exercises (1-3),<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>determine the domain and the range of each relation<\/strong><\/li>\r\n\t<li><strong>state whether the relation is a function.<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<table id=\"fs-id1170572554567\" class=\"unnumbered\" summary=\"A table with 7 rows and 2 columns. The first column is labeled \u201cx\u201d and has the values \u201c-3; -2; -1; 0; 1; 2; 3\u201d. The second column is labeled \u201cy\u201d and the values are \u201c9; 4; 1; 0; 1; 4; 9\u201d.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22123[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table id=\"fs-id1170572549270\" class=\"unnumbered\" summary=\"A table with 7 rows and 2 columns. The first column is labeled \u201cx\u201d and has the values \u201c1; 2; 3; 0; 1; 2; 3\u201d. The second column is labeled \u201cy\u201d and the values are \u201c-3; -2; -1; 0; 1; 2; 3\u201d.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]\u22123[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]\u22121[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table id=\"fs-id1170572203536\" class=\"unnumbered\" summary=\"A table with 7 rows and 2 columns. The first column is labeled \u201cx\u201d and has the values \u201c3; 5; 8; 10; 15; 21; 33\u201d. The second column is labeled \u201cy\u201d and the values are \u201c3; 2; 1; 0; 1; 2; 3\u201d.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]21[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]33[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572453203\"><strong>For the following exercises (4-7), find the values for each of the following functions (a-f), if they exist, then simplify.<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>[latex]f(0)[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f(1)[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f(3)[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f(\u2212x)[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f(a)[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f(a+h)[\/latex]<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\" start=\"4\">\r\n\t<li>[latex]f(x)=5x-2[\/latex]<\/li>\r\n\t<li>[latex]f(x)=\\dfrac{2}{x}[\/latex]<\/li>\r\n\t<li>[latex]f(x)=\\sqrt{6x+5}[\/latex]<\/li>\r\n\t<li>[latex]f(x)=9[\/latex]<\/li>\r\n<\/ol>\r\n<p><strong>For the following exercises (8-11), find the,<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>domain of the functions<\/strong><\/li>\r\n\t<li><strong>range of the functions<\/strong><\/li>\r\n\t<li><strong>all zeros\/intercepts, if any, of the functions<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\" start=\"8\">\r\n\t<li>[latex]g(x)=\\sqrt{8x-1}[\/latex]<\/li>\r\n\t<li>[latex]f(x)=-1+\\sqrt{x+2}[\/latex]<\/li>\r\n\t<li>[latex]g(x)=\\dfrac{3}{x-4}[\/latex]<\/li>\r\n\t<li>[latex]g(x)=\\sqrt{\\dfrac{7}{x-5}}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572451338\"><strong>For the following exercises (12-14), use a table of values to sketch the graph of each function using the following values:<\/strong><\/p>\r\n<p style=\"text-align: center;\"><strong>[latex]x=-3,-2,-1,0,1,2,3[\/latex]<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"12\">\r\n\t<li>[latex]f(x)=3x-6[\/latex]<\/li>\r\n\t<li>[latex]f(x)=2|x|[\/latex]<\/li>\r\n\t<li>[latex]f(x)=x^3[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572169071\"><strong>For the following exercises (15-18), use the vertical line test to determine whether each of the given graphs represents a function. <em>Assume that a graph continues at both ends if it extends beyond the given grid.<\/em> If the graph represents a function, then determine the following for each graph:<\/strong><\/p>\r\n<ol id=\"fs-id1170572169081\" style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>Domain and range<\/strong><\/li>\r\n\t<li><strong>[latex]x[\/latex]-intercept, if any (estimate where necessary)<\/strong><\/li>\r\n\t<li><strong>[latex]y[\/latex]-intercept, if any (estimate where necessary)<\/strong><\/li>\r\n\t<li><strong>The intervals for which the function is increasing<\/strong><\/li>\r\n\t<li><strong>The intervals for which the function is decreasing<\/strong><\/li>\r\n\t<li><strong>The intervals for which the function is constant<\/strong><\/li>\r\n\t<li><strong>Symmetry about any axis and\/or the origin<\/strong><\/li>\r\n\t<li><strong>Whether the function is even, odd, or neither<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\" start=\"15\">\r\n\t<li><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202154\/CNX_Calc_Figure_01_01_208.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The relation decreases until it hits the point (-1, 0), then increases until it hits the point (0, 1), then decreases until it hits the point (1, 0), then increases again.\" \/><\/li>\r\n\t<li><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202200\/CNX_Calc_Figure_01_01_217.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The curved relation increases the entire time. The x intercept and y intercept are both at the origin.\" \/><\/li>\r\n\t<li><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202206\/CNX_Calc_Figure_01_01_212.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a horizontal line until the point (-2, -2), then it begins increasing in a straight line until the point (2, 2). After these points, the relation becomes a horizontal line again. The x intercept and y intercept are both at the origin.\" \/><\/li>\r\n\t<li><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202212\/CNX_Calc_Figure_01_01_214.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that starts at the point (-4, 4) and is a horizontal line until the point (0, 4), then it begins decreasing in a curved line until it hits the point (4, -4), where the graph ends. The x intercept is approximately at the point (1.2, 0) and y intercept is at the point (0, 4).\" \/><\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572229185\"><strong>For the following exercises (19-21), for each pair of functions, find<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>[latex]f+g[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f-g[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f\u00b7g[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex]f\/g[\/latex]<\/strong><\/li>\r\n<\/ol>\r\n<p><strong>Determine the domain of each of these new functions.<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"19\">\r\n\t<li>[latex]f(x)=x-8,\\,\\,\\, g(x)=5x^2[\/latex]<\/li>\r\n\t<li>[latex]f(x)=9-x^2,\\,\\,\\, g(x)=x^2-2x-3[\/latex]<\/li>\r\n\t<li>[latex]f(x)=6+\\dfrac{1}{x},\\,\\,\\, g(x)=\\dfrac{1}{x}[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572174804\"><strong>For the following exercises (22-24), for each pair of functions, find<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>[latex](f\\circ g)(x)[\/latex]<\/strong><\/li>\r\n\t<li><strong>[latex](g\\circ f)(x)[\/latex]<\/strong><\/li>\r\n<\/ol>\r\n<p><strong>Simplify the results. Find the domain of each of the results.<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\" 22\">\r\n\t<li>[latex]f(x)=x+4,\\,\\,\\, g(x)=4x-1[\/latex]<\/li>\r\n\t<li>[latex]f(x)=x^2+7,\\,\\,\\, g(x)=x^2-3[\/latex]<\/li>\r\n\t<li>[latex]f(x)=\\dfrac{3}{2x+1},\\,\\,\\, g(x)=\\dfrac{2}{x}[\/latex]<\/li>\r\n<\/ol>\r\n<p><strong>For the following exercises (25-29), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\" 25\">\r\n\t<li>The table below lists the NBA championship winners for the years 2001 to 2012.\r\n\r\n<table id=\"fs-id1170572128778\" class=\"unnumbered\" summary=\"A table with 12 rows and 2 columns. The first column is labeled \u201cyear\u201d and has the values \u201c2001; 2002; 2003; 2004; 2005; 2006; 2007; 2008; 2009; 2010; 2011; 2012\u201d. The second column is labeled \u201cwinner\u201d and the values are \u201cLA Lakers; LA Lakers; San Antonio Spurs; Detroit Pistons; San Antonio Spurs; Miami Heat; San Antonio Spurs; Boston Celtics; LA Lakers; LA Lakers; Dallas Mavericks; Miami Heat\u201d.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Year<\/th>\r\n<th>Winner<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>2001<\/td>\r\n<td>LA Lakers<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2002<\/td>\r\n<td>LA Lakers<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2003<\/td>\r\n<td>San Antonio Spurs<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2004<\/td>\r\n<td>Detroit Pistons<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2005<\/td>\r\n<td>San Antonio Spurs<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2006<\/td>\r\n<td>Miami Heat<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2007<\/td>\r\n<td>San Antonio Spurs<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2008<\/td>\r\n<td>Boston Celtics<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2009<\/td>\r\n<td>LA Lakers<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2010<\/td>\r\n<td>LA Lakers<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2011<\/td>\r\n<td>Dallas Mavericks<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2012<\/td>\r\n<td>Miami Heat<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol id=\"fs-id1170572233892\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.<\/li>\r\n\t<li>Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>The volume of a cube depends on the length of the sides [latex]s[\/latex].\r\n\r\n<ol id=\"fs-id1170572478086\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Write a function [latex]V(s)[\/latex] for the area of a square.<\/li>\r\n\t<li>Find and interpret [latex]V(11.8)[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>A vehicle has a [latex]20[\/latex]-gal tank and gets [latex]15[\/latex] mpg. The number of miles [latex]N[\/latex] that can be driven depends on the amount of gas [latex]x[\/latex] in the tank.\r\n\r\n<ol id=\"fs-id1170572431541\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Write a formula that models this situation.<\/li>\r\n\t<li>Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) [latex]3\/4[\/latex] of a tank of gas.<\/li>\r\n\t<li>Determine the domain and range of the function.<\/li>\r\n\t<li>Determine how many times the driver had to stop for gas if she has driven a total of [latex]578[\/latex] mi.<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by [latex]r(t)=6-\\left[\\dfrac{5}{(t^2+1)}\\right][\/latex], where [latex]t[\/latex] is time measured in hours since a circle of a [latex]1[\/latex] cm radius of the bacterium was put into the culture.\r\n\r\n<ol id=\"fs-id1170572171809\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Express the area of the bacteria as a function of time.<\/li>\r\n\t<li>Find the exact and approximate area of the bacterial culture in [latex]3[\/latex] hours.<\/li>\r\n\t<li>Express the circumference of the bacteria as a function of time.<\/li>\r\n\t<li>Find the exact and approximate circumference of the bacteria in [latex]3[\/latex] hours.<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>The manager at a skateboard shop pays his workers a monthly salary [latex]S[\/latex] of [latex]$750[\/latex] plus a commission of [latex]$8.50[\/latex] for each skateboard they sell.\r\n\r\n<ol id=\"fs-id1170572425360\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Write a function [latex]y=S(x)[\/latex] that models a worker\u2019s monthly salary based on the number of skateboards [latex]x[\/latex] he or she sells.<\/li>\r\n\t<li>Find the approximate monthly salary when a worker sells [latex]25[\/latex], [latex]40[\/latex], or [latex]55[\/latex] skateboards.<\/li>\r\n\t<li>Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of [latex]$1400[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<h2>Basic Classes of Functions<\/h2>\r\n<p id=\"fs-id1170573587694\"><strong>For the following exercises (1-4), for each pair of points,<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>find the slope of the line passing through the points<\/strong><\/li>\r\n\t<li><strong>indicate whether the line is increasing, decreasing, horizontal, or vertical<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex](-2,4)[\/latex] and [latex](1,1)[\/latex]<\/li>\r\n\t<li>[latex](3,5)[\/latex] and [latex](-1,2)[\/latex]<\/li>\r\n\t<li>[latex](2,3)[\/latex] and [latex](5,7)[\/latex]<\/li>\r\n\t<li>[latex](2,4)[\/latex] and [latex](1,4)[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573588100\"><strong>For the following exercises (5-8), write the equation of the line satisfying the given conditions in slope-intercept form.<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"5\">\r\n\t<li>Slope [latex]=-6[\/latex], passes through [latex](1,3)[\/latex]<\/li>\r\n\t<li>Slope [latex]=\\dfrac{1}{3}[\/latex], passes through [latex](0,4)[\/latex]<\/li>\r\n\t<li>Passing through [latex](2,1)[\/latex] and [latex](-2,-1)[\/latex]<\/li>\r\n\t<li>[latex]x[\/latex]-intercept [latex]=5[\/latex] and [latex]y[\/latex]-intercept [latex]=-3[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573588583\"><strong>For the following exercises (9-12), for each linear equation,<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>give the slope [latex](m)[\/latex], and [latex]y[\/latex]-intercept [latex](b)[\/latex], if any<\/strong><\/li>\r\n\t<li><strong>graph the line<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\" start=\"9\">\r\n\t<li>[latex]y=2x-3[\/latex]<\/li>\r\n\t<li>[latex]f(x)=-6x[\/latex]<\/li>\r\n\t<li>[latex]4y+24=0[\/latex]<\/li>\r\n\t<li>[latex]2x+3y=6[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573589155\"><strong>For the following exercises (13-15), for each polynomial,<\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>find the degree<\/strong><\/li>\r\n\t<li><strong>find the zeros, if any<\/strong><\/li>\r\n\t<li><strong>find the [latex]y[\/latex]-intercept(s), if any<\/strong><\/li>\r\n\t<li><strong>use the leading coefficient to determine the graph\u2019s end behavior<\/strong><\/li>\r\n\t<li><strong>determine algebraically whether the polynomial is even, odd, or neither.<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\" start=\"13\">\r\n\t<li>[latex]f(x)=2x^2-3x-5[\/latex]<\/li>\r\n\t<li>[latex]f(x)=\\frac{1}{2}x^2-1[\/latex]<\/li>\r\n\t<li>[latex]f(x)=3x-x^3[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573589452\"><strong>For the following exercise (16), use the graph of [latex]f(x)=x^2[\/latex] to graph the transformed function [latex]g[\/latex].<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"16\">\r\n\t<li>[latex]g(x)=(x+3)^2+1[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573589595\"><strong>For the following exercise (17), use the graph of [latex]f(x)=\\sqrt{x}[\/latex] to graph the transformed function [latex]g[\/latex].<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"17\">\r\n\t<li>[latex]g(x)=\u2212\\sqrt{x}-1[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573589723\"><strong>For the following exercise (18), use the graph of [latex]y=f(x)[\/latex] to graph the transformed function [latex]g[\/latex].<\/strong><\/p>\r\n<p><span id=\"fs-id1170573589753\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202403\/CNX_Calc_Figure_01_02_213.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a function that starts at point (-3, 0), where it begins to increase until the point (-1, 2). After the point (-1, 2), the function becomes a horizontal line and stays that way until the point (1, 2). After the point (1, 2), the function begins to decrease until the point (3, 0), where the function ends.\" \/><\/span><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"18\">\r\n\t<li>[latex]g(x)=f(x-1)+2[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573589885\"><strong>For the following exercises (19-20), for each of the piecewise-defined functions, <\/strong><\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><strong>Evaluate at the given values of the independent variable <\/strong><\/li>\r\n\t<li><strong>Sketch the graph<\/strong><\/li>\r\n<\/ol>\r\n<ol style=\"list-style-type: decimal;\" start=\"19\">\r\n\t<li>[latex]f(x)=\\begin{cases}x^2-3, &amp; x &lt; 0 \\\\ 4x-3, &amp; x \\ge 0 \\end{cases}[\/latex];\u00a0 \u00a0[latex]f(-4); \\, f(0); \\, f(2)[\/latex]<\/li>\r\n\t<li>[latex]g(x)=\\begin{cases} \\left(\\dfrac{3}{x-2}\\right), &amp; x \\ne 2 \\\\ 4, &amp; x = 2 \\end{cases}[\/latex];\u00a0 \u00a0[latex]g(0); \\, g(-4); \\, g(2)[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573595284\"><strong>For the following exercises (21-22), determine whether the statement is <em>true or false<\/em>. Explain why.<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"21\">\r\n\t<li>[latex]g(x)=\\sqrt[3]{x}[\/latex] is an odd root function<\/li>\r\n\t<li>A function of the form [latex]f(x)=x^b[\/latex], where [latex]b[\/latex] is a real valued constant, is an exponential function.<\/li>\r\n<\/ol>\r\n<p><strong>For the following exercises (23-27), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.<\/strong><\/p>\r\n<ol style=\"list-style-type: decimal;\" start=\"23\">\r\n\t<li>A company purchases some computer equipment for [latex]$20,500[\/latex]. At the end of a [latex]3[\/latex]-year period, the value of the equipment has decreased linearly to [latex]$12,300[\/latex].\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li>Find a function [latex]y=V(t)[\/latex] that determines the value [latex]V[\/latex] of the equipment at the end of [latex]t[\/latex] years.<\/li>\r\n\t<li>Find and interpret the meaning of the [latex]x[\/latex]- and [latex]y[\/latex]-intercepts for this situation.<\/li>\r\n\t<li>What is the value of the equipment at the end of [latex]5[\/latex] years?<\/li>\r\n\t<li>When will the value of the equipment be [latex]$3000[\/latex]?<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of [latex]$125[\/latex] to set up a cupcake stand. The owner estimates that it costs [latex]$0.75[\/latex] to make each cupcake. The owner is interested in determining the total cost [latex]C[\/latex] as a function of number of cupcakes made.\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li>Find a linear function that relates cost [latex]C[\/latex] to [latex]x[\/latex], the number of cupcakes made.<\/li>\r\n\t<li>Find the cost to bake [latex]160[\/latex] cupcakes.<\/li>\r\n\t<li>If the owner sells the cupcakes for [latex]$1.50[\/latex] apiece, how many cupcakes does she need to sell to start making profit? (<em>Hint<\/em>: <em>Use the INTERSECTION function on a calculator to find this number.)<\/em><\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>A car was purchased for [latex]$26,000[\/latex]. The value of the car depreciates by [latex]$1500[\/latex] per year.\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li>Find a linear function that models the value [latex]V[\/latex] of the car after [latex]t[\/latex] years.<\/li>\r\n\t<li>Find and interpret [latex]V(4)[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t<li>The total cost [latex]C[\/latex] (in thousands of dollars) to produce a certain item is modeled by the function [latex]C(x)=10.50x+28,500[\/latex], where [latex]x[\/latex] is the number of items produced. Determine the cost to produce [latex]175[\/latex] items.<\/li>\r\n\t<li>The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function [latex]P(t)=1.8576t+68.052[\/latex], where [latex]t[\/latex] is time in years and [latex]t=0[\/latex] corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.<\/li>\r\n<\/ol>","rendered":"<h2 class=\"entry-title\">Review of Functions<\/h2>\n<p><strong>For the following exercises (1-3),<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>determine the domain and the range of each relation<\/strong><\/li>\n<li><strong>state whether the relation is a function.<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\">\n<li>\n<table id=\"fs-id1170572554567\" class=\"unnumbered\" summary=\"A table with 7 rows and 2 columns. The first column is labeled \u201cx\u201d and has the values \u201c-3; -2; -1; 0; 1; 2; 3\u201d. The second column is labeled \u201cy\u201d and the values are \u201c9; 4; 1; 0; 1; 4; 9\u201d.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\u22123[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table id=\"fs-id1170572549270\" class=\"unnumbered\" summary=\"A table with 7 rows and 2 columns. The first column is labeled \u201cx\u201d and has the values \u201c1; 2; 3; 0; 1; 2; 3\u201d. The second column is labeled \u201cy\u201d and the values are \u201c-3; -2; -1; 0; 1; 2; 3\u201d.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\u22122[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table id=\"fs-id1170572203536\" class=\"unnumbered\" summary=\"A table with 7 rows and 2 columns. The first column is labeled \u201cx\u201d and has the values \u201c3; 5; 8; 10; 15; 21; 33\u201d. The second column is labeled \u201cy\u201d and the values are \u201c3; 2; 1; 0; 1; 2; 3\u201d.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]21[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]33[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<p id=\"fs-id1170572453203\"><strong>For the following exercises (4-7), find the values for each of the following functions (a-f), if they exist, then simplify.<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>[latex]f(0)[\/latex]<\/strong><\/li>\n<li><strong>[latex]f(1)[\/latex]<\/strong><\/li>\n<li><strong>[latex]f(3)[\/latex]<\/strong><\/li>\n<li><strong>[latex]f(\u2212x)[\/latex]<\/strong><\/li>\n<li><strong>[latex]f(a)[\/latex]<\/strong><\/li>\n<li><strong>[latex]f(a+h)[\/latex]<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\" start=\"4\">\n<li>[latex]f(x)=5x-2[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{2}{x}[\/latex]<\/li>\n<li>[latex]f(x)=\\sqrt{6x+5}[\/latex]<\/li>\n<li>[latex]f(x)=9[\/latex]<\/li>\n<\/ol>\n<p><strong>For the following exercises (8-11), find the,<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>domain of the functions<\/strong><\/li>\n<li><strong>range of the functions<\/strong><\/li>\n<li><strong>all zeros\/intercepts, if any, of the functions<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\" start=\"8\">\n<li>[latex]g(x)=\\sqrt{8x-1}[\/latex]<\/li>\n<li>[latex]f(x)=-1+\\sqrt{x+2}[\/latex]<\/li>\n<li>[latex]g(x)=\\dfrac{3}{x-4}[\/latex]<\/li>\n<li>[latex]g(x)=\\sqrt{\\dfrac{7}{x-5}}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170572451338\"><strong>For the following exercises (12-14), use a table of values to sketch the graph of each function using the following values:<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>[latex]x=-3,-2,-1,0,1,2,3[\/latex]<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"12\">\n<li>[latex]f(x)=3x-6[\/latex]<\/li>\n<li>[latex]f(x)=2|x|[\/latex]<\/li>\n<li>[latex]f(x)=x^3[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170572169071\"><strong>For the following exercises (15-18), use the vertical line test to determine whether each of the given graphs represents a function. <em>Assume that a graph continues at both ends if it extends beyond the given grid.<\/em> If the graph represents a function, then determine the following for each graph:<\/strong><\/p>\n<ol id=\"fs-id1170572169081\" style=\"list-style-type: lower-alpha;\">\n<li><strong>Domain and range<\/strong><\/li>\n<li><strong>[latex]x[\/latex]-intercept, if any (estimate where necessary)<\/strong><\/li>\n<li><strong>[latex]y[\/latex]-intercept, if any (estimate where necessary)<\/strong><\/li>\n<li><strong>The intervals for which the function is increasing<\/strong><\/li>\n<li><strong>The intervals for which the function is decreasing<\/strong><\/li>\n<li><strong>The intervals for which the function is constant<\/strong><\/li>\n<li><strong>Symmetry about any axis and\/or the origin<\/strong><\/li>\n<li><strong>Whether the function is even, odd, or neither<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\" start=\"15\">\n<li><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202154\/CNX_Calc_Figure_01_01_208.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The relation decreases until it hits the point (-1, 0), then increases until it hits the point (0, 1), then decreases until it hits the point (1, 0), then increases again.\" \/><\/li>\n<li><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202200\/CNX_Calc_Figure_01_01_217.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The curved relation increases the entire time. The x intercept and y intercept are both at the origin.\" \/><\/li>\n<li><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202206\/CNX_Calc_Figure_01_01_212.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a horizontal line until the point (-2, -2), then it begins increasing in a straight line until the point (2, 2). After these points, the relation becomes a horizontal line again. The x intercept and y intercept are both at the origin.\" \/><\/li>\n<li><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202212\/CNX_Calc_Figure_01_01_214.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that starts at the point (-4, 4) and is a horizontal line until the point (0, 4), then it begins decreasing in a curved line until it hits the point (4, -4), where the graph ends. The x intercept is approximately at the point (1.2, 0) and y intercept is at the point (0, 4).\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1170572229185\"><strong>For the following exercises (19-21), for each pair of functions, find<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>[latex]f+g[\/latex]<\/strong><\/li>\n<li><strong>[latex]f-g[\/latex]<\/strong><\/li>\n<li><strong>[latex]f\u00b7g[\/latex]<\/strong><\/li>\n<li><strong>[latex]f\/g[\/latex]<\/strong><\/li>\n<\/ol>\n<p><strong>Determine the domain of each of these new functions.<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"19\">\n<li>[latex]f(x)=x-8,\\,\\,\\, g(x)=5x^2[\/latex]<\/li>\n<li>[latex]f(x)=9-x^2,\\,\\,\\, g(x)=x^2-2x-3[\/latex]<\/li>\n<li>[latex]f(x)=6+\\dfrac{1}{x},\\,\\,\\, g(x)=\\dfrac{1}{x}[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170572174804\"><strong>For the following exercises (22-24), for each pair of functions, find<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>[latex](f\\circ g)(x)[\/latex]<\/strong><\/li>\n<li><strong>[latex](g\\circ f)(x)[\/latex]<\/strong><\/li>\n<\/ol>\n<p><strong>Simplify the results. Find the domain of each of the results.<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"22\">\n<li>[latex]f(x)=x+4,\\,\\,\\, g(x)=4x-1[\/latex]<\/li>\n<li>[latex]f(x)=x^2+7,\\,\\,\\, g(x)=x^2-3[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{3}{2x+1},\\,\\,\\, g(x)=\\dfrac{2}{x}[\/latex]<\/li>\n<\/ol>\n<p><strong>For the following exercises (25-29), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"25\">\n<li>The table below lists the NBA championship winners for the years 2001 to 2012.<br \/>\n<table id=\"fs-id1170572128778\" class=\"unnumbered\" summary=\"A table with 12 rows and 2 columns. The first column is labeled \u201cyear\u201d and has the values \u201c2001; 2002; 2003; 2004; 2005; 2006; 2007; 2008; 2009; 2010; 2011; 2012\u201d. The second column is labeled \u201cwinner\u201d and the values are \u201cLA Lakers; LA Lakers; San Antonio Spurs; Detroit Pistons; San Antonio Spurs; Miami Heat; San Antonio Spurs; Boston Celtics; LA Lakers; LA Lakers; Dallas Mavericks; Miami Heat\u201d.\">\n<thead>\n<tr valign=\"top\">\n<th>Year<\/th>\n<th>Winner<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>2001<\/td>\n<td>LA Lakers<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2002<\/td>\n<td>LA Lakers<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2003<\/td>\n<td>San Antonio Spurs<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2004<\/td>\n<td>Detroit Pistons<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2005<\/td>\n<td>San Antonio Spurs<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2006<\/td>\n<td>Miami Heat<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2007<\/td>\n<td>San Antonio Spurs<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2008<\/td>\n<td>Boston Celtics<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2009<\/td>\n<td>LA Lakers<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2010<\/td>\n<td>LA Lakers<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2011<\/td>\n<td>Dallas Mavericks<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2012<\/td>\n<td>Miami Heat<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1170572233892\" style=\"list-style-type: lower-alpha;\">\n<li>Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.<\/li>\n<li>Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.<\/li>\n<\/ol>\n<\/li>\n<li>The volume of a cube depends on the length of the sides [latex]s[\/latex].\n<ol id=\"fs-id1170572478086\" style=\"list-style-type: lower-alpha;\">\n<li>Write a function [latex]V(s)[\/latex] for the area of a square.<\/li>\n<li>Find and interpret [latex]V(11.8)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>A vehicle has a [latex]20[\/latex]-gal tank and gets [latex]15[\/latex] mpg. The number of miles [latex]N[\/latex] that can be driven depends on the amount of gas [latex]x[\/latex] in the tank.\n<ol id=\"fs-id1170572431541\" style=\"list-style-type: lower-alpha;\">\n<li>Write a formula that models this situation.<\/li>\n<li>Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) [latex]3\/4[\/latex] of a tank of gas.<\/li>\n<li>Determine the domain and range of the function.<\/li>\n<li>Determine how many times the driver had to stop for gas if she has driven a total of [latex]578[\/latex] mi.<\/li>\n<\/ol>\n<\/li>\n<li>A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by [latex]r(t)=6-\\left[\\dfrac{5}{(t^2+1)}\\right][\/latex], where [latex]t[\/latex] is time measured in hours since a circle of a [latex]1[\/latex] cm radius of the bacterium was put into the culture.\n<ol id=\"fs-id1170572171809\" style=\"list-style-type: lower-alpha;\">\n<li>Express the area of the bacteria as a function of time.<\/li>\n<li>Find the exact and approximate area of the bacterial culture in [latex]3[\/latex] hours.<\/li>\n<li>Express the circumference of the bacteria as a function of time.<\/li>\n<li>Find the exact and approximate circumference of the bacteria in [latex]3[\/latex] hours.<\/li>\n<\/ol>\n<\/li>\n<li>The manager at a skateboard shop pays his workers a monthly salary [latex]S[\/latex] of [latex]$750[\/latex] plus a commission of [latex]$8.50[\/latex] for each skateboard they sell.\n<ol id=\"fs-id1170572425360\" style=\"list-style-type: lower-alpha;\">\n<li>Write a function [latex]y=S(x)[\/latex] that models a worker\u2019s monthly salary based on the number of skateboards [latex]x[\/latex] he or she sells.<\/li>\n<li>Find the approximate monthly salary when a worker sells [latex]25[\/latex], [latex]40[\/latex], or [latex]55[\/latex] skateboards.<\/li>\n<li>Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of [latex]$1400[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h2>Basic Classes of Functions<\/h2>\n<p id=\"fs-id1170573587694\"><strong>For the following exercises (1-4), for each pair of points,<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>find the slope of the line passing through the points<\/strong><\/li>\n<li><strong>indicate whether the line is increasing, decreasing, horizontal, or vertical<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex](-2,4)[\/latex] and [latex](1,1)[\/latex]<\/li>\n<li>[latex](3,5)[\/latex] and [latex](-1,2)[\/latex]<\/li>\n<li>[latex](2,3)[\/latex] and [latex](5,7)[\/latex]<\/li>\n<li>[latex](2,4)[\/latex] and [latex](1,4)[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573588100\"><strong>For the following exercises (5-8), write the equation of the line satisfying the given conditions in slope-intercept form.<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"5\">\n<li>Slope [latex]=-6[\/latex], passes through [latex](1,3)[\/latex]<\/li>\n<li>Slope [latex]=\\dfrac{1}{3}[\/latex], passes through [latex](0,4)[\/latex]<\/li>\n<li>Passing through [latex](2,1)[\/latex] and [latex](-2,-1)[\/latex]<\/li>\n<li>[latex]x[\/latex]-intercept [latex]=5[\/latex] and [latex]y[\/latex]-intercept [latex]=-3[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573588583\"><strong>For the following exercises (9-12), for each linear equation,<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>give the slope [latex](m)[\/latex], and [latex]y[\/latex]-intercept [latex](b)[\/latex], if any<\/strong><\/li>\n<li><strong>graph the line<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\" start=\"9\">\n<li>[latex]y=2x-3[\/latex]<\/li>\n<li>[latex]f(x)=-6x[\/latex]<\/li>\n<li>[latex]4y+24=0[\/latex]<\/li>\n<li>[latex]2x+3y=6[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573589155\"><strong>For the following exercises (13-15), for each polynomial,<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>find the degree<\/strong><\/li>\n<li><strong>find the zeros, if any<\/strong><\/li>\n<li><strong>find the [latex]y[\/latex]-intercept(s), if any<\/strong><\/li>\n<li><strong>use the leading coefficient to determine the graph\u2019s end behavior<\/strong><\/li>\n<li><strong>determine algebraically whether the polynomial is even, odd, or neither.<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\" start=\"13\">\n<li>[latex]f(x)=2x^2-3x-5[\/latex]<\/li>\n<li>[latex]f(x)=\\frac{1}{2}x^2-1[\/latex]<\/li>\n<li>[latex]f(x)=3x-x^3[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573589452\"><strong>For the following exercise (16), use the graph of [latex]f(x)=x^2[\/latex] to graph the transformed function [latex]g[\/latex].<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"16\">\n<li>[latex]g(x)=(x+3)^2+1[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573589595\"><strong>For the following exercise (17), use the graph of [latex]f(x)=\\sqrt{x}[\/latex] to graph the transformed function [latex]g[\/latex].<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"17\">\n<li>[latex]g(x)=\u2212\\sqrt{x}-1[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573589723\"><strong>For the following exercise (18), use the graph of [latex]y=f(x)[\/latex] to graph the transformed function [latex]g[\/latex].<\/strong><\/p>\n<p><span id=\"fs-id1170573589753\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202403\/CNX_Calc_Figure_01_02_213.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a function that starts at point (-3, 0), where it begins to increase until the point (-1, 2). After the point (-1, 2), the function becomes a horizontal line and stays that way until the point (1, 2). After the point (1, 2), the function begins to decrease until the point (3, 0), where the function ends.\" \/><\/span><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"18\">\n<li>[latex]g(x)=f(x-1)+2[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573589885\"><strong>For the following exercises (19-20), for each of the piecewise-defined functions, <\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>Evaluate at the given values of the independent variable <\/strong><\/li>\n<li><strong>Sketch the graph<\/strong><\/li>\n<\/ol>\n<ol style=\"list-style-type: decimal;\" start=\"19\">\n<li>[latex]f(x)=\\begin{cases}x^2-3, & x < 0 \\\\ 4x-3, & x \\ge 0 \\end{cases}[\/latex];\u00a0 \u00a0[latex]f(-4); \\, f(0); \\, f(2)[\/latex]<\/li>\n<li>[latex]g(x)=\\begin{cases} \\left(\\dfrac{3}{x-2}\\right), & x \\ne 2 \\\\ 4, & x = 2 \\end{cases}[\/latex];\u00a0 \u00a0[latex]g(0); \\, g(-4); \\, g(2)[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573595284\"><strong>For the following exercises (21-22), determine whether the statement is <em>true or false<\/em>. Explain why.<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"21\">\n<li>[latex]g(x)=\\sqrt[3]{x}[\/latex] is an odd root function<\/li>\n<li>A function of the form [latex]f(x)=x^b[\/latex], where [latex]b[\/latex] is a real valued constant, is an exponential function.<\/li>\n<\/ol>\n<p><strong>For the following exercises (23-27), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.<\/strong><\/p>\n<ol style=\"list-style-type: decimal;\" start=\"23\">\n<li>A company purchases some computer equipment for [latex]$20,500[\/latex]. At the end of a [latex]3[\/latex]-year period, the value of the equipment has decreased linearly to [latex]$12,300[\/latex].\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Find a function [latex]y=V(t)[\/latex] that determines the value [latex]V[\/latex] of the equipment at the end of [latex]t[\/latex] years.<\/li>\n<li>Find and interpret the meaning of the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts for this situation.<\/li>\n<li>What is the value of the equipment at the end of [latex]5[\/latex] years?<\/li>\n<li>When will the value of the equipment be [latex]$3000[\/latex]?<\/li>\n<\/ol>\n<\/li>\n<li>A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of [latex]$125[\/latex] to set up a cupcake stand. The owner estimates that it costs [latex]$0.75[\/latex] to make each cupcake. The owner is interested in determining the total cost [latex]C[\/latex] as a function of number of cupcakes made.\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Find a linear function that relates cost [latex]C[\/latex] to [latex]x[\/latex], the number of cupcakes made.<\/li>\n<li>Find the cost to bake [latex]160[\/latex] cupcakes.<\/li>\n<li>If the owner sells the cupcakes for [latex]$1.50[\/latex] apiece, how many cupcakes does she need to sell to start making profit? (<em>Hint<\/em>: <em>Use the INTERSECTION function on a calculator to find this number.)<\/em><\/li>\n<\/ol>\n<\/li>\n<li>A car was purchased for [latex]$26,000[\/latex]. The value of the car depreciates by [latex]$1500[\/latex] per year.\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Find a linear function that models the value [latex]V[\/latex] of the car after [latex]t[\/latex] years.<\/li>\n<li>Find and interpret [latex]V(4)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>The total cost [latex]C[\/latex] (in thousands of dollars) to produce a certain item is modeled by the function [latex]C(x)=10.50x+28,500[\/latex], where [latex]x[\/latex] is the number of items produced. Determine the cost to produce [latex]175[\/latex] items.<\/li>\n<li>The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function [latex]P(t)=1.8576t+68.052[\/latex], where [latex]t[\/latex] is time in years and [latex]t=0[\/latex] corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":143,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1990"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1990\/revisions"}],"predecessor-version":[{"id":2336,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1990\/revisions\/2336"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1990\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1990"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1990"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1990"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}