{"id":1583,"date":"2024-05-29T00:33:47","date_gmt":"2024-05-29T00:33:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1583"},"modified":"2025-08-13T22:57:11","modified_gmt":"2025-08-13T22:57:11","slug":"combinations-and-compositions-of-functions-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/combinations-and-compositions-of-functions-apply-it-1\/","title":{"raw":"Combinations and Compositions of Functions: Apply It 1","rendered":"Combinations and Compositions of Functions: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Use algebraic operations to combine functions and create new expressions<\/li>\r\n \t<li>Build a new function by combining two or more functions together<\/li>\r\n \t<li>Calculate the output for composite functions for given values and determine the set of inputs that work for these functions<\/li>\r\n \t<li>Break down a composite function into the original functions that were combined to make it<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Understanding Composite Functions<\/h2>\r\nNow that you have learned about composite functions and their domains, let's interpret what a composite function means.\r\n\r\n<section class=\"textbox recall\">A composite function combines two functions where the output of one function becomes the input of another.If we have two functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex], the composite function [latex](f \\circ g)(x)[\/latex] means we first apply [latex]g(x)[\/latex]and then apply [latex]f[\/latex] to the result of [latex]g(x)[\/latex].<\/section>Composite functions can model real-world scenarios where a series of processes or transformations are applied sequentially.\r\n\r\n<section class=\"textbox example\">Recall the example: Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year.Notice how we have just defined two relationships:\r\n<ul>\r\n \t<li>The cost depends on the temperature<\/li>\r\n \t<li>the temperature depends on the day<\/li>\r\n<\/ul>\r\nUsing descriptive variables, we can notate these two functions.\r\n<ul>\r\n \t<li>The function [latex]C\\left(T\\right)[\/latex] gives the cost [latex]C[\/latex] of heating a house for a given average daily temperature in [latex]T[\/latex] degrees Celsius.<\/li>\r\n \t<li>The function [latex]T\\left(d\\right)[\/latex] gives the average daily temperature on day [latex]d[\/latex] of the year.<\/li>\r\n<\/ul>\r\nFor any given day, [latex]\\text{Cost}=C\\left(T\\left(d\\right)\\right)[\/latex] means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature [latex]T\\left(d\\right)[\/latex].\r\n\r\nFor example, we could evaluate [latex]T\\left(5\\right)[\/latex] to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the <strong>cost function<\/strong> at that temperature. We would write [latex]C\\left(T\\left(5\\right)\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"376\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195613\/CNX_Precalc_Figure_01_04_0062.jpg\" alt=\"Explanation of C(T(5)), which is the cost for the temperature and T(5) is the temperature on day 5.\" width=\"376\" height=\"108\" \/> Diagram of C(T(5))[\/caption]\r\n\r\n<\/section><section class=\"textbox example\">The function [latex]c\\left(s\\right)[\/latex] gives the number of calories burned completing [latex]s[\/latex] sit-ups, and [latex]s\\left(t\\right)[\/latex] gives the number of sit-ups a person can complete in [latex]t[\/latex] minutes.Interpret [latex]c(s(3))[\/latex].\r\n\r\n<hr \/>\r\n\r\nTo interpret [latex]c(s(3))[\/latex], follow these steps:\r\n<ul>\r\n \t<li><strong>Identify [latex]s(3)[\/latex]:<\/strong> This represents the number of sit-ups a person can complete in [latex]3[\/latex] minutes. So, [latex]s(3)[\/latex]\u00a0tells us how many sit-ups are done in [latex]3[\/latex] minutes.<\/li>\r\n \t<li><strong><strong>Apply [latex]c[\/latex] <\/strong><\/strong><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">to [latex]s(3)[\/latex]:<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> Once you know the number of sit-ups completed in [latex]3[\/latex] minutes (which is [latex]s(3)[\/latex]<\/span><span class=\"math math-inline\" style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-mathml\"><semantics><mrow><mi><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">, you use the function [latex]c[\/latex] <\/span><\/mi><\/mrow><\/semantics><\/span><\/span><\/span><\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">to determine how many calories are burned from doing that number of sit-ups.<\/span><\/li>\r\n<\/ul>\r\nThus, [latex]c(s(3))[\/latex] represents the number of calories burned from the number of sit-ups that can be completed in [latex]3[\/latex] minutes.\r\n\r\nIn other words, you first calculate how many sit-ups are completed in [latex]3[\/latex] minutes, and then determine the calories burned from that amount of exercise.\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]19140[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]19141[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use algebraic operations to combine functions and create new expressions<\/li>\n<li>Build a new function by combining two or more functions together<\/li>\n<li>Calculate the output for composite functions for given values and determine the set of inputs that work for these functions<\/li>\n<li>Break down a composite function into the original functions that were combined to make it<\/li>\n<\/ul>\n<\/section>\n<h2>Understanding Composite Functions<\/h2>\n<p>Now that you have learned about composite functions and their domains, let&#8217;s interpret what a composite function means.<\/p>\n<section class=\"textbox recall\">A composite function combines two functions where the output of one function becomes the input of another.If we have two functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex], the composite function [latex](f \\circ g)(x)[\/latex] means we first apply [latex]g(x)[\/latex]and then apply [latex]f[\/latex] to the result of [latex]g(x)[\/latex].<\/section>\n<p>Composite functions can model real-world scenarios where a series of processes or transformations are applied sequentially.<\/p>\n<section class=\"textbox example\">Recall the example: Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year.Notice how we have just defined two relationships:<\/p>\n<ul>\n<li>The cost depends on the temperature<\/li>\n<li>the temperature depends on the day<\/li>\n<\/ul>\n<p>Using descriptive variables, we can notate these two functions.<\/p>\n<ul>\n<li>The function [latex]C\\left(T\\right)[\/latex] gives the cost [latex]C[\/latex] of heating a house for a given average daily temperature in [latex]T[\/latex] degrees Celsius.<\/li>\n<li>The function [latex]T\\left(d\\right)[\/latex] gives the average daily temperature on day [latex]d[\/latex] of the year.<\/li>\n<\/ul>\n<p>For any given day, [latex]\\text{Cost}=C\\left(T\\left(d\\right)\\right)[\/latex] means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature [latex]T\\left(d\\right)[\/latex].<\/p>\n<p>For example, we could evaluate [latex]T\\left(5\\right)[\/latex] to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the <strong>cost function<\/strong> at that temperature. We would write [latex]C\\left(T\\left(5\\right)\\right)[\/latex].<\/p>\n<figure style=\"width: 376px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195613\/CNX_Precalc_Figure_01_04_0062.jpg\" alt=\"Explanation of C(T(5)), which is the cost for the temperature and T(5) is the temperature on day 5.\" width=\"376\" height=\"108\" \/><figcaption class=\"wp-caption-text\">Diagram of C(T(5))<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox example\">The function [latex]c\\left(s\\right)[\/latex] gives the number of calories burned completing [latex]s[\/latex] sit-ups, and [latex]s\\left(t\\right)[\/latex] gives the number of sit-ups a person can complete in [latex]t[\/latex] minutes.Interpret [latex]c(s(3))[\/latex].<\/p>\n<hr \/>\n<p>To interpret [latex]c(s(3))[\/latex], follow these steps:<\/p>\n<ul>\n<li><strong>Identify [latex]s(3)[\/latex]:<\/strong> This represents the number of sit-ups a person can complete in [latex]3[\/latex] minutes. So, [latex]s(3)[\/latex]\u00a0tells us how many sit-ups are done in [latex]3[\/latex] minutes.<\/li>\n<li><strong><strong>Apply [latex]c[\/latex] <\/strong><\/strong><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">to [latex]s(3)[\/latex]:<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> Once you know the number of sit-ups completed in [latex]3[\/latex] minutes (which is [latex]s(3)[\/latex]<\/span><span class=\"math math-inline\" style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-mathml\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">, you use the function [latex]c[\/latex] <\/span><\/span><\/span><\/span><\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">to determine how many calories are burned from doing that number of sit-ups.<\/span><\/li>\n<\/ul>\n<p>Thus, [latex]c(s(3))[\/latex] represents the number of calories burned from the number of sit-ups that can be completed in [latex]3[\/latex] minutes.<\/p>\n<p>In other words, you first calculate how many sit-ups are completed in [latex]3[\/latex] minutes, and then determine the calories burned from that amount of exercise.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm19140\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=19140&theme=lumen&iframe_resize_id=ohm19140&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm19141\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=19141&theme=lumen&iframe_resize_id=ohm19141&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"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":142,"module-header":"apply_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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1583"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":16,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1583\/revisions"}],"predecessor-version":[{"id":8074,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1583\/revisions\/8074"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/142"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1583\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1583"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1583"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1583"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1583"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}