{"id":2820,"date":"2025-08-13T18:44:57","date_gmt":"2025-08-13T18:44:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2820"},"modified":"2026-01-31T01:46:55","modified_gmt":"2026-01-31T01:46:55","slug":"introduction-to-calculus-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/introduction-to-calculus-background-youll-need-1\/","title":{"raw":"Introduction to Calculus: Background You'll Need 1","rendered":"Introduction to Calculus: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate piecewise functions given the graph<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Evaluating Piecewise Functions from Graphs<\/h2>\r\nA <strong>piecewise function<\/strong> is a function that is defined by different rules over different intervals of the input values. When a piecewise function is presented as a graph, each section of the graph represents a different rule.\r\n\r\nTo evaluate a piecewise function from a graph, the correct section of the graph must be identified for the given input value.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Piecewise function<\/h3>\r\nA <b>piecewise function<\/b> uses different rules for different ranges of input values. Each rule applies only on its specified interval.\r\n\r\n<\/div>\r\n<\/section>\r\n<h3>Interpreting the Graph<\/h3>\r\nWhen evaluating from a graph, pay attention to:\r\n<ul>\r\n \t<li>The interval where each piece applies<\/li>\r\n \t<li>Open circles, which indicate the value is not included<\/li>\r\n \t<li>Closed circles, which indicate the value is included<\/li>\r\n<\/ul>\r\n<section class=\"textbox example\">Consider the graph of a piecewise function shown below.\r\n<img class=\"alignnone wp-image-5462\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31013923\/Screenshot-2026-01-30-at-6.38.33%E2%80%AFPM.png\" alt=\"The graph is made of two straight line segments that meet at a peak, forming an inverted V shape. The graph increases to a maximum point and then decreases. The graph passes through the points: (-5, -2), (-3,0), (-1,2), (0,3), (1, 4), (3,0), (5, -5) The highest point of the graph is at (1, 4).\" width=\"338\" height=\"334\" \/>\r\nEvaluate [latex]f(-1)[\/latex].[reveal-answer q=\"385310\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385310\"]At [latex]x = -1[\/latex], the point lies on a slanted line segment. The corresponding [latex]y[\/latex]-value is [latex]2[\/latex]. Since the point lies on the graph and is not an endpoint, the function value is determined by reading the height of the graph at that [latex]x[\/latex]-value. Therefore, [latex]f(-1) = 2[\/latex].[\/hidden-answer]<\/section><section class=\"textbox example\">Use the graph to evaluate the function at [latex]x = -2[\/latex].\r\n<img class=\"alignnone wp-image-5463\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM.png\" alt=\"The graph shows a piecewise function made of two parts. One part is a horizontal line at y equals -1 that extends to the left and ends at an open circle at (-1, -1), indicating that point is not included. The second part is a line segment with positive slope that begins at a solid point at (-1, 1) and increases to the right. The graph includes the point (-1, 1) but does not include the point (-1, -1).\" width=\"244\" height=\"260\" \/>[reveal-answer q=\"553541\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"553541\"]\r\n\r\nLocate [latex]x = -1[\/latex] on the [latex]x[\/latex]-axis. At this value, the graph shows a filled (closed) circle at a height of [latex]1[\/latex] and an open circle at a[latex](-1,-1)[\/latex]. However, the open circle indicates that point is not included in the function. The function value is given by the filled point.\r\n\r\nTherefore, [latex]f(-1) = 1[\/latex].[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]319414[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate piecewise functions given the graph<\/li>\n<\/ul>\n<\/section>\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Evaluating Piecewise Functions from Graphs<\/h2>\n<p>A <strong>piecewise function<\/strong> is a function that is defined by different rules over different intervals of the input values. When a piecewise function is presented as a graph, each section of the graph represents a different rule.<\/p>\n<p>To evaluate a piecewise function from a graph, the correct section of the graph must be identified for the given input value.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Piecewise function<\/h3>\n<p>A <b>piecewise function<\/b> uses different rules for different ranges of input values. Each rule applies only on its specified interval.<\/p>\n<\/div>\n<\/section>\n<h3>Interpreting the Graph<\/h3>\n<p>When evaluating from a graph, pay attention to:<\/p>\n<ul>\n<li>The interval where each piece applies<\/li>\n<li>Open circles, which indicate the value is not included<\/li>\n<li>Closed circles, which indicate the value is included<\/li>\n<\/ul>\n<section class=\"textbox example\">Consider the graph of a piecewise function shown below.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5462\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31013923\/Screenshot-2026-01-30-at-6.38.33%E2%80%AFPM.png\" alt=\"The graph is made of two straight line segments that meet at a peak, forming an inverted V shape. The graph increases to a maximum point and then decreases. The graph passes through the points: (-5, -2), (-3,0), (-1,2), (0,3), (1, 4), (3,0), (5, -5) The highest point of the graph is at (1, 4).\" width=\"338\" height=\"334\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31013923\/Screenshot-2026-01-30-at-6.38.33%E2%80%AFPM.png 686w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31013923\/Screenshot-2026-01-30-at-6.38.33%E2%80%AFPM-300x297.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31013923\/Screenshot-2026-01-30-at-6.38.33%E2%80%AFPM-65x64.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31013923\/Screenshot-2026-01-30-at-6.38.33%E2%80%AFPM-225x222.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31013923\/Screenshot-2026-01-30-at-6.38.33%E2%80%AFPM-350x346.png 350w\" sizes=\"(max-width: 338px) 100vw, 338px\" \/><br \/>\nEvaluate [latex]f(-1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q385310\">Show Solution<\/button><\/p>\n<div id=\"q385310\" class=\"hidden-answer\" style=\"display: none\">At [latex]x = -1[\/latex], the point lies on a slanted line segment. The corresponding [latex]y[\/latex]-value is [latex]2[\/latex]. Since the point lies on the graph and is not an endpoint, the function value is determined by reading the height of the graph at that [latex]x[\/latex]-value. Therefore, [latex]f(-1) = 2[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Use the graph to evaluate the function at [latex]x = -2[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5463\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM.png\" alt=\"The graph shows a piecewise function made of two parts. One part is a horizontal line at y equals -1 that extends to the left and ends at an open circle at (-1, -1), indicating that point is not included. The second part is a line segment with positive slope that begins at a solid point at (-1, 1) and increases to the right. The graph includes the point (-1, 1) but does not include the point (-1, -1).\" width=\"244\" height=\"260\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM.png 824w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM-282x300.png 282w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM-768x818.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM-65x69.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM-225x240.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/08\/31014311\/Screenshot-2026-01-30-at-6.42.26%E2%80%AFPM-350x373.png 350w\" sizes=\"(max-width: 244px) 100vw, 244px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q553541\">Show Solution<\/button><\/p>\n<div id=\"q553541\" class=\"hidden-answer\" style=\"display: none\">\n<p>Locate [latex]x = -1[\/latex] on the [latex]x[\/latex]-axis. At this value, the graph shows a filled (closed) circle at a height of [latex]1[\/latex] and an open circle at a[latex](-1,-1)[\/latex]. However, the open circle indicates that point is not included in the function. The function value is given by the filled point.<\/p>\n<p>Therefore, [latex]f(-1) = 1[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm319414\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=319414&theme=lumen&iframe_resize_id=ohm319414&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"background_you_need","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2820"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2820\/revisions"}],"predecessor-version":[{"id":5465,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2820\/revisions\/5465"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2820\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2820"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2820"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2820"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}