{"id":17700,"date":"2021-08-19T23:28:04","date_gmt":"2021-08-19T23:28:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=17700"},"modified":"2021-08-20T00:48:47","modified_gmt":"2021-08-20T00:48:47","slug":"section-2-2-the-graph-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/section-2-2-the-graph-of-a-function\/","title":{"raw":"Section 2.2: The Graph of a Function","rendered":"Section 2.2: The Graph of a Function"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>Find the value of a function using its graph.<\/li>\r\n \t<li>Find the domain and range from a graph.<\/li>\r\n \t<li>Find intercepts from a graph.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 style=\"text-align: left;\">Finding Function Values from a Graph<\/h2>\r\n<p id=\"fs-id1165137779152\">In this section we will evaluate functions using its graph. Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Reading Function Values from a Graph<\/h3>\r\n<p id=\"fs-id1165137469316\">Given the graph in Figure 1,<\/p>\r\n\r\n<ol id=\"fs-id1165137604039\">\r\n \t<li>Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\r\n \t<li>Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\r\n \t<li>Is [latex]f\\left(-0.5\\right)[\/latex] positive or negative?<\/li>\r\n \t<li>Find the x- and y-intercepts.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\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\/03010534\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"357788\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"357788\"]\r\n<ol id=\"fs-id1165137871522\">\r\n \t<li>To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the <em>y<\/em>-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex]. See Figure 2.\r\n<figure id=\"Figure_01_01_008\" class=\"small\">\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\/03010534\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/> <b>Figure 2<\/b>[\/caption]<\/figure>\r\n<\/li>\r\n \t<li>To solve [latex]f\\left(x\\right)=4[\/latex], we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4[\/latex], we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex]. These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]x=-1[\/latex] or [latex]x=3[\/latex]. This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4[\/latex], or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex]See Figure 3.\r\n<figure id=\"Figure_01_01_009\" class=\"small\">\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\/03010534\/CNX_Precalc_Figure_01_01_0092.jpg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/> <b>Figure 3<\/b>[\/caption]<\/figure>\r\n<\/li>\r\n \t<li>To find [latex]f\\left(-0.5\\right)[\/latex], we need to go to -0.5 on the x-axis. Then we go up until we intersect the graph. From the graph, [latex]f\\left(-0.5\\right)\\approx 2[\/latex]. Therefore, we know that [latex]f\\left(-0.5\\right)[\/latex] is positive.<\/li>\r\n \t<li>The x-intercept is where the graph crosses the x-axis. We see that graph touches the x-axis only at one place, which is the point [latex](1,0)[\/latex]. The y-intercept is where the graph crosses the y-axis. This occurs at the point [latex](0,1)[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nUsing Figure 1, solve [latex]f\\left(x\\right)=1[\/latex].\r\n\r\n[reveal-answer q=\"731542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731542\"]\r\n\r\n[latex]x=0[\/latex] or [latex]x=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Finding Domain and Range from Graphs<\/h2>\r\nNext we will determine the domain and range of functions using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em>x<\/em>-axis. The range is the set of possible output values, which are shown on the <em>y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 4.<span id=\"fs-id1165137432156\">\r\n<\/span>\r\n<div class=\"\u201ctextbox\u201d\">\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\/03010545\/CNX_Precalc_Figure_01_02_0062.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\" \/> <b>Figure 4<\/b>[\/caption]\r\n<p id=\"fs-id1165137597994\">We can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\r\n\r\n<div id=\"Example_01_02_06\" class=\"example\">\r\n<div id=\"fs-id1165137561401\" class=\"exercise\">\r\n<div id=\"fs-id1165137599824\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Finding Domain and Range from a Graph<\/h3>\r\nFind the domain and range of the function [latex]f[\/latex]\u00a0whose graph is shown in Figure 5.\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\/03010545\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"916064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"916064\"]\r\n<p id=\"fs-id1165137768165\">We can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\right][\/latex].<\/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\/03010545\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1165131968670\">The vertical extent of the graph is 0 to \u20134, so the range is [latex]\\left[-4,0\\right)[\/latex].<\/p>\r\n[\/hidden-answer]<b><\/b><span id=\"fs-id1165137937577\">\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question hide_question_numbers=1]30605[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=QAxZEelInJc\r\n<div id=\"Example_01_02_07\" class=\"example\"><\/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 difference quotient if [latex]f(x)=-\\dfrac{2}{x}[\/latex].<\/p>\r\n[reveal-answer q=\"944520\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"944520\"]\r\n<p style=\"text-align: left;\">[latex]\\dfrac{2}{x(x+h)}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Key Concepts<\/span><\/h2>\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>To evaluate a function on a graph, we determine the y-value for a corresponding x value.<\/li>\r\n \t<li>To solve for a specific x-value on the graph, we determine the x values that yield the specific y values.<\/li>\r\n \t<li>The domain of a graph includes all its x-values.<\/li>\r\n \t<li>The range of a graph includes all its y-values.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<dl id=\"fs-id1165135315542\" class=\"definition\">\r\n \t<dd><\/dd>\r\n<\/dl>\r\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 2.2 Homework Exercises<\/span><\/h2>\r\n1. Given the following graph,\r\nEvaluate\u00a0[latex]f(\u22121)[\/latex].\r\nSolve for\u00a0[latex]f(x)=3[\/latex].\r\nFind the domain.\r\nFind the range.\r\nIs [latex]f(-\\frac{3}{2})[\/latex] positive or negative?\r\nWhat is the x-intercept?\r\n<img src=\"http:\/\/www.hutchmath.com\/Images\/Sec1_3Prob_50.JPG\" alt=\"Graph of relation.\" \/>\r\n2. Given the following graph,\r\nEvaluate\u00a0[latex]f(0[\/latex]).\r\nSolve for\u00a0[latex]f(x)=\u22123[\/latex].\r\nFind the domain.\r\nFind the range.\r\nIs [latex]f(-\\frac{1}{2})[\/latex] positive or negative?\r\nIs [latex]f(1)[\/latex] positive or negative?\r\n<img src=\"http:\/\/www.hutchmath.com\/Images\/Sec1_3Prob_51.JPG\" alt=\"Graph of relation.\" \/>\r\n\r\n3. Given the following graph,\r\nEvaluate\u00a0[latex]f(4)[\/latex].\r\nSolve for [latex]f(x)=1[\/latex].\r\nFind the domain.\r\nFind the range.\r\nIs [latex]f(-2)[\/latex] positive or negative?\r\nIs [latex]f(3)[\/latex] positive or negative?\r\nWhat is the x-intercept?\r\nWhat is the y-intercept?\r\n<img src=\"http:\/\/www.hutchmath.com\/Images\/Sec1_3Prob_52.JPG\" alt=\"Graph of relation.\" \/>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Find the value of a function using its graph.<\/li>\n<li>Find the domain and range from a graph.<\/li>\n<li>Find intercepts from a graph.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2 style=\"text-align: left;\">Finding Function Values from a Graph<\/h2>\n<p id=\"fs-id1165137779152\">In this section we will evaluate functions using its graph. Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\n<div class=\"textbox shaded\">\n<h3>Example 1: Reading Function Values from a Graph<\/h3>\n<p id=\"fs-id1165137469316\">Given the graph in Figure 1,<\/p>\n<ol id=\"fs-id1165137604039\">\n<li>Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\n<li>Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\n<li>Is [latex]f\\left(-0.5\\right)[\/latex] positive or negative?<\/li>\n<li>Find the x- and y-intercepts.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/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\/03010534\/CNX_Precalc_Figure_01_01_0072.jpg\" alt=\"Graph of a positive parabola centered at (1, 0).\" width=\"487\" height=\"445\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q357788\">Show Solution<\/span><\/p>\n<div id=\"q357788\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137871522\">\n<li>To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the <em>y<\/em>-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex]. See Figure 2.<br \/>\n<figure id=\"Figure_01_01_008\" class=\"small\">\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\/03010534\/CNX_Precalc_Figure_01_01_0082.jpg\" alt=\"Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.\" width=\"487\" height=\"445\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<\/figure>\n<\/li>\n<li>To solve [latex]f\\left(x\\right)=4[\/latex], we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4[\/latex], we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex]. These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]x=-1[\/latex] or [latex]x=3[\/latex]. This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4[\/latex], or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex]See Figure 3.<br \/>\n<figure id=\"Figure_01_01_009\" class=\"small\">\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\/03010534\/CNX_Precalc_Figure_01_01_0092.jpg\" alt=\"Graph of an upward-facing\u00a0parabola with a vertex at (0,1) and\u00a0labeled points at (-1, 4) and (3,4). A\u00a0line at y = 4 intersects the parabola at the labeled points.\" width=\"487\" height=\"445\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<\/figure>\n<\/li>\n<li>To find [latex]f\\left(-0.5\\right)[\/latex], we need to go to -0.5 on the x-axis. Then we go up until we intersect the graph. From the graph, [latex]f\\left(-0.5\\right)\\approx 2[\/latex]. Therefore, we know that [latex]f\\left(-0.5\\right)[\/latex] is positive.<\/li>\n<li>The x-intercept is where the graph crosses the x-axis. We see that graph touches the x-axis only at one place, which is the point [latex](1,0)[\/latex]. The y-intercept is where the graph crosses the y-axis. This occurs at the point [latex](0,1)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Using Figure 1, solve [latex]f\\left(x\\right)=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731542\">Show Solution<\/span><\/p>\n<div id=\"q731542\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=0[\/latex] or [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Finding Domain and Range from Graphs<\/h2>\n<p>Next we will determine the domain and range of functions using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em>x<\/em>-axis. The range is the set of possible output values, which are shown on the <em>y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 4.<span id=\"fs-id1165137432156\"><br \/>\n<\/span><\/p>\n<div class=\"\u201ctextbox\u201d\">\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\/03010545\/CNX_Precalc_Figure_01_02_0062.jpg\" alt=\"Graph of a polynomial that shows the x-axis is the domain and the y-axis is the range\" width=\"487\" height=\"666\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137597994\">We can observe that the graph extends horizontally from [latex]-5[\/latex] to the right without bound, so the domain is [latex]\\left[-5,\\infty \\right)[\/latex]. The vertical extent of the graph is all range values [latex]5[\/latex] and below, so the range is [latex]\\left(\\mathrm{-\\infty },5\\right][\/latex]. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\n<div id=\"Example_01_02_06\" class=\"example\">\n<div id=\"fs-id1165137561401\" class=\"exercise\">\n<div id=\"fs-id1165137599824\" class=\"problem textbox shaded\">\n<h3>Example 2: Finding Domain and Range from a Graph<\/h3>\n<p>Find the domain and range of the function [latex]f[\/latex]\u00a0whose graph is shown in Figure 5.<\/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\/03010545\/CNX_Precalc_Figure_01_02_0072.jpg\" alt=\"Graph of a function from (-3, 1].\" width=\"487\" height=\"364\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q916064\">Show Solution<\/span><\/p>\n<div id=\"q916064\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137768165\">We can observe that the horizontal extent of the graph is \u20133 to 1, so the domain of [latex]f[\/latex]\u00a0is [latex]\\left(-3,1\\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\/03010545\/CNX_Precalc_Figure_01_02_0082.jpg\" alt=\"Graph of the previous function shows the domain and range.\" width=\"487\" height=\"365\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165131968670\">The vertical extent of the graph is 0 to \u20134, so the range is [latex]\\left[-4,0\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><span id=\"fs-id1165137937577\"><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm30605\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=30605&theme=oea&iframe_resize_id=ohm30605\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Determine the Domain and Range of the Graph of a Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QAxZEelInJc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"Example_01_02_07\" class=\"example\"><\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165134149846\">Find the difference quotient if [latex]f(x)=-\\dfrac{2}{x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q944520\">Show Solution<\/span><\/p>\n<div id=\"q944520\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\dfrac{2}{x(x+h)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Key Concepts<\/span><\/h2>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>To evaluate a function on a graph, we determine the y-value for a corresponding x value.<\/li>\n<li>To solve for a specific x-value on the graph, we determine the x values that yield the specific y values.<\/li>\n<li>The domain of a graph includes all its x-values.<\/li>\n<li>The range of a graph includes all its y-values.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n<dd><\/dd>\n<\/dl>\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 2.2 Homework Exercises<\/span><\/h2>\n<p>1. Given the following graph,<br \/>\nEvaluate\u00a0[latex]f(\u22121)[\/latex].<br \/>\nSolve for\u00a0[latex]f(x)=3[\/latex].<br \/>\nFind the domain.<br \/>\nFind the range.<br \/>\nIs [latex]f(-\\frac{3}{2})[\/latex] positive or negative?<br \/>\nWhat is the x-intercept?<br \/>\n<img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/Sec1_3Prob_50.JPG\" alt=\"Graph of relation.\" \/><br \/>\n2. Given the following graph,<br \/>\nEvaluate\u00a0[latex]f(0[\/latex]).<br \/>\nSolve for\u00a0[latex]f(x)=\u22123[\/latex].<br \/>\nFind the domain.<br \/>\nFind the range.<br \/>\nIs [latex]f(-\\frac{1}{2})[\/latex] positive or negative?<br \/>\nIs [latex]f(1)[\/latex] positive or negative?<br \/>\n<img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/Sec1_3Prob_51.JPG\" alt=\"Graph of relation.\" \/><\/p>\n<p>3. Given the following graph,<br \/>\nEvaluate\u00a0[latex]f(4)[\/latex].<br \/>\nSolve for [latex]f(x)=1[\/latex].<br \/>\nFind the domain.<br \/>\nFind the range.<br \/>\nIs [latex]f(-2)[\/latex] positive or negative?<br \/>\nIs [latex]f(3)[\/latex] positive or negative?<br \/>\nWhat is the x-intercept?<br \/>\nWhat is the y-intercept?<br \/>\n<img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/Sec1_3Prob_52.JPG\" alt=\"Graph of relation.\" \/><\/p>\n","protected":false},"author":264444,"menu_order":2,"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-17700","chapter","type-chapter","status-publish","hentry"],"part":17684,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17700","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":16,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17700\/revisions"}],"predecessor-version":[{"id":17721,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17700\/revisions\/17721"}],"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\/17700\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=17700"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=17700"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=17700"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=17700"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}