{"id":481,"date":"2025-07-10T17:07:13","date_gmt":"2025-07-10T17:07:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=481"},"modified":"2025-12-17T16:42:04","modified_gmt":"2025-12-17T16:42:04","slug":"functions-and-function-notation-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/functions-and-function-notation-learn-it-3\/","title":{"raw":"Functions and Function Notation: Learn It 4","rendered":"Functions and Function Notation: Learn It 4"},"content":{"raw":"<section id=\"fs-id1165137648450\">\r\n<h2>Evaluating a Function Given in Tabular Form<\/h2>\r\n<p id=\"fs-id1165135186424\">We can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.<\/p>\r\n<p id=\"fs-id1165135186427\">The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.<\/p>\r\n\r\n<table id=\"Table_01_01_10\" summary=\"Six rows and two columns. The first column is labeled, \">\r\n<thead>\r\n<tr>\r\n<th>Pet<\/th>\r\n<th>Memory span in hours<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Puppy<\/td>\r\n<td>0.008<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Adult dog<\/td>\r\n<td>0.083<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cat<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Goldfish<\/td>\r\n<td>2160<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Beta fish<\/td>\r\n<td>3600<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137584852\">At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[\/latex].<\/p>\r\nThe [pb_glossary id=\"3331\"]domain[\/pb_glossary] of the function is the type of pet and the [pb_glossary id=\"3332\"]range[\/pb_glossary] is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of \"goldfish.\" We would write [latex]P\\left(\\text{goldfish}\\right)=2160[\/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value by looking at the corresponding row in the table.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1165133356033\">Using the table below,<\/p>\r\n\r\n<ol id=\"fs-id1165137653327\">\r\n \t<li>Evaluate [latex]g\\left(3\\right)[\/latex].<\/li>\r\n \t<li>Solve [latex]g\\left(n\\right)=6[\/latex].<\/li>\r\n<\/ol>\r\n<table id=\"Table_01_01_11\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>n<\/strong><\/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<\/tr>\r\n<tr>\r\n<td><strong>g(n)<\/strong><\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"71183\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"71183\"]\r\n<ul>\r\n \t<li>Evaluating [latex]g\\left(3\\right)[\/latex] means determining the output value of the function [latex]g[\/latex] for the input value of [latex]n=3[\/latex]. The table output value corresponding to [latex]n=3[\/latex] is 7, so [latex]g\\left(3\\right)=7[\/latex].<\/li>\r\n \t<li>Solving [latex]g\\left(n\\right)=6[\/latex] means identifying the input values, [latex]n[\/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[\/latex] and [latex]n=4[\/latex].<\/li>\r\n<\/ul>\r\n<table id=\"Table_01_01_12\" summary=\"Two rows and six columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>n<\/strong><\/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<\/tr>\r\n<tr>\r\n<td><strong>g(n)<\/strong><\/td>\r\n<td>8<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137448125\">When we input 2 into the function [latex]g[\/latex], our output is 6. When we input 4 into the function [latex]g[\/latex], our output is also 6.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317461[\/ohm_question]<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137648450\"><section id=\"fs-id1165135696152\">\r\n<h2>Finding Function Values from a Graph<\/h2>\r\n<p id=\"fs-id1165137779152\">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<section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1165137469316\">Given the graph<\/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<\/ol>\r\n<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\" \/>\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].\r\n<figure id=\"Figure_01_01_008\" class=\"small\"><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\" \/><\/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].\r\n<figure id=\"Figure_01_01_009\" class=\"small\"><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\" \/><\/figure>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317462[\/ohm_question]<\/section><\/div>\r\n<\/section>","rendered":"<section id=\"fs-id1165137648450\">\n<h2>Evaluating a Function Given in Tabular Form<\/h2>\n<p id=\"fs-id1165135186424\">We can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.<\/p>\n<p id=\"fs-id1165135186427\">The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.<\/p>\n<table id=\"Table_01_01_10\" summary=\"Six rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Pet<\/th>\n<th>Memory span in hours<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Puppy<\/td>\n<td>0.008<\/td>\n<\/tr>\n<tr>\n<td>Adult dog<\/td>\n<td>0.083<\/td>\n<\/tr>\n<tr>\n<td>Cat<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td>Goldfish<\/td>\n<td>2160<\/td>\n<\/tr>\n<tr>\n<td>Beta fish<\/td>\n<td>3600<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137584852\">At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[\/latex].<\/p>\n<p>The <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_481_3331\">domain<\/a> of the function is the type of pet and the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_481_3332\">range<\/a> is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of &#8220;goldfish.&#8221; We would write [latex]P\\left(\\text{goldfish}\\right)=2160[\/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value by looking at the corresponding row in the table.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1165133356033\">Using the table below,<\/p>\n<ol id=\"fs-id1165137653327\">\n<li>Evaluate [latex]g\\left(3\\right)[\/latex].<\/li>\n<li>Solve [latex]g\\left(n\\right)=6[\/latex].<\/li>\n<\/ol>\n<table id=\"Table_01_01_11\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>n<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td><strong>g(n)<\/strong><\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q71183\">Show Solution<\/button><\/p>\n<div id=\"q71183\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Evaluating [latex]g\\left(3\\right)[\/latex] means determining the output value of the function [latex]g[\/latex] for the input value of [latex]n=3[\/latex]. The table output value corresponding to [latex]n=3[\/latex] is 7, so [latex]g\\left(3\\right)=7[\/latex].<\/li>\n<li>Solving [latex]g\\left(n\\right)=6[\/latex] means identifying the input values, [latex]n[\/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[\/latex] and [latex]n=4[\/latex].<\/li>\n<\/ul>\n<table id=\"Table_01_01_12\" summary=\"Two rows and six columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>n<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td><strong>g(n)<\/strong><\/td>\n<td>8<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137448125\">When we input 2 into the function [latex]g[\/latex], our output is 6. When we input 4 into the function [latex]g[\/latex], our output is also 6.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317461\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317461&theme=lumen&iframe_resize_id=ohm317461&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/section>\n<section>\n<section id=\"fs-id1165135696152\">\n<h2>Finding Function Values from a Graph<\/h2>\n<p id=\"fs-id1165137779152\">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<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1165137469316\">Given the graph<\/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<\/ol>\n<p><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<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q357788\">Show Solution<\/button><\/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].<br \/>\n<figure id=\"Figure_01_01_008\" class=\"small\"><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\" \/><\/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].<br \/>\n<figure id=\"Figure_01_01_009\" class=\"small\"><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\" \/><\/figure>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317462\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317462&theme=lumen&iframe_resize_id=ohm317462&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_481_3331\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_481_3331\"><div tabindex=\"-1\"><p>set of all input values of a function<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_481_3332\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_481_3332\"><div tabindex=\"-1\"><p>set of all output values for a function<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":13,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":36,"module-header":"learn_it","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\/481"}],"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\/13"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/481\/revisions"}],"predecessor-version":[{"id":5120,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/481\/revisions\/5120"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/481\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=481"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=481"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=481"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=481"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}