{"id":4869,"date":"2017-12-28T00:06:49","date_gmt":"2017-12-28T00:06:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&#038;p=4869"},"modified":"2018-01-03T16:39:57","modified_gmt":"2018-01-03T16:39:57","slug":"algebra-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/algebra-of-functions\/","title":{"raw":"Algebra of Functions","rendered":"Algebra of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Combining Functions:\r\n<ul>\r\n \t<li>Adding<\/li>\r\n \t<li>Subtracting<\/li>\r\n \t<li>Multiplying<\/li>\r\n \t<li>Dividing<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Algebra of Functions<\/h2>\r\n<p id=\"fs-id1165137446477\">We can carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.<\/p>\r\n<p id=\"fs-id1165135533159\">Suppose we need to add two columns of numbers that represent a husband and wife\u2019s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year\u2019s incomes and then collecting all the data in a new column. If [latex]w\\left(y\\right)[\/latex] is the wife\u2019s income and [latex]h\\left(y\\right)[\/latex] is the husband\u2019s income in year [latex]y[\/latex], and we want [latex]T[\/latex] to represent the total income, then we can define a new function.<\/p>\r\n\r\n<div id=\"fs-id1165137641862\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]T\\left(y\\right)=h\\left(y\\right)+w\\left(y\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165135347766\">If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write<\/p>\r\n\r\n<div id=\"fs-id1165137665765\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]T=h+w[\/latex]<\/div>\r\n<p id=\"fs-id1165132957142\">Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.<\/p>\r\n<p id=\"fs-id1165137424116\">For two functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] with real number outputs, we define new functions [latex]f+g,f-g,fg[\/latex], and [latex]\\frac{f}{g}[\/latex] by the relations<\/p>\r\n\r\n<div id=\"fs-id1165137543139\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{c}\\left(f+g\\right)\\left(x\\right)=f\\left(x\\right)+g\\left(x\\right)\\hfill \\\\ \\left(f-g\\right)\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)\\hfill \\\\ \\text{ }\\left(fg\\right)\\left(x\\right)=f\\left(x\\right)g\\left(x\\right)\\hfill \\\\ \\text{ }\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\hfill \\end{array}[\/latex]<\/div>\r\n<div id=\"Example_01_04_01\" class=\"example\">\r\n<div id=\"fs-id1165137585443\" class=\"exercise\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Performing Algebraic Operations on Functions<\/h3>\r\nFind and simplify the functions [latex]\\left(g-f\\right)\\left(x\\right)[\/latex] and [latex]\\left(\\frac{g}{f}\\right)\\left(x\\right)[\/latex], given [latex]f\\left(x\\right)=x - 1[\/latex] and [latex]g\\left(x\\right)={x}^{2}-1[\/latex]. Give the domain of your result. Are they the same function?\r\n[reveal-answer q=\"117163\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"117163\"]\r\n<p id=\"fs-id1165137466263\">Begin by writing the general form, and then substitute the given functions.<\/p>\r\n\r\n<div id=\"fs-id1165135701567\" class=\"equation unnumbered\">[latex]\\begin{array}{c}\\left(g-f\\right)\\left(x\\right)=g\\left(x\\right)-f\\left(x\\right) \\\\ \\left(g-f\\right)\\left(x\\right)={x}^{2}-1-\\left(x - 1\\right)\\\\ \\text{ }={x}^{2}-x \\\\ \\text{ }=x\\left(x - 1\\right) \\\\\\end{array}[\/latex] [latex]\\begin{array}{c}\\text{ }\\left(\\frac{g}{f}\\right)\\left(x\\right)=\\frac{g\\left(x\\right)}{f\\left(x\\right)} \\\\ \\text{ }\\left(\\frac{g}{f}\\right)\\left(x\\right)=\\frac{{x}^{2}-1}{x - 1}\\\\ \\text{ }=\\frac{\\left(x+1\\right)\\left(x - 1\\right)}{x - 1}\\text{ where }x\\ne 1 \\\\ \\text{ }=x+1 \\end{array}[\/latex]<\/div>\r\nThe domain of the result is [latex]x\\ne1[\/latex]\r\n<p id=\"fs-id1165137553743\">No, the functions are not the same.<\/p>\r\n<p id=\"fs-id1165135351612\">Note: For [latex]\\left(\\frac{g}{f}\\right)\\left(x\\right)[\/latex], the condition [latex]x\\ne 1[\/latex] is necessary because when [latex]x=1[\/latex], the denominator is equal to 0, which makes the function undefined.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind and simplify the functions [latex]\\left(fg\\right)\\left(x\\right)[\/latex] and [latex]\\left(f-g\\right)\\left(x\\right)[\/latex].\r\n<div id=\"fs-id1165137434994\" class=\"equation unnumbered\">[latex]f\\left(x\\right)=x - 1\\text{ and }g\\left(x\\right)={x}^{2}-1[\/latex]<\/div>\r\n<p id=\"fs-id1165137434911\">Are they the same function?\r\n[reveal-answer q=\"721147\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"721147\"][latex]\\begin{array}{c}\\left(fg\\right)\\left(x\\right)=f\\left(x\\right)g\\left(x\\right)=\\left(x - 1\\right)\\left({x}^{2}-1\\right)={x}^{3}-{x}^{2}-x+1\\\\ \\left(f-g\\right)\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)=\\left(x - 1\\right)-\\left({x}^{2}-1\\right)=x-{x}^{2}\\end{array}[\/latex]\r\nNo, the functions are not the same.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Combining Functions:\n<ul>\n<li>Adding<\/li>\n<li>Subtracting<\/li>\n<li>Multiplying<\/li>\n<li>Dividing<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Algebra of Functions<\/h2>\n<p id=\"fs-id1165137446477\">We can carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.<\/p>\n<p id=\"fs-id1165135533159\">Suppose we need to add two columns of numbers that represent a husband and wife\u2019s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year\u2019s incomes and then collecting all the data in a new column. If [latex]w\\left(y\\right)[\/latex] is the wife\u2019s income and [latex]h\\left(y\\right)[\/latex] is the husband\u2019s income in year [latex]y[\/latex], and we want [latex]T[\/latex] to represent the total income, then we can define a new function.<\/p>\n<div id=\"fs-id1165137641862\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]T\\left(y\\right)=h\\left(y\\right)+w\\left(y\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135347766\">If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write<\/p>\n<div id=\"fs-id1165137665765\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]T=h+w[\/latex]<\/div>\n<p id=\"fs-id1165132957142\">Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.<\/p>\n<p id=\"fs-id1165137424116\">For two functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] with real number outputs, we define new functions [latex]f+g,f-g,fg[\/latex], and [latex]\\frac{f}{g}[\/latex] by the relations<\/p>\n<div id=\"fs-id1165137543139\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{c}\\left(f+g\\right)\\left(x\\right)=f\\left(x\\right)+g\\left(x\\right)\\hfill \\\\ \\left(f-g\\right)\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)\\hfill \\\\ \\text{ }\\left(fg\\right)\\left(x\\right)=f\\left(x\\right)g\\left(x\\right)\\hfill \\\\ \\text{ }\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Example_01_04_01\" class=\"example\">\n<div id=\"fs-id1165137585443\" class=\"exercise\">\n<div class=\"textbox exercises\">\n<h3>Example: Performing Algebraic Operations on Functions<\/h3>\n<p>Find and simplify the functions [latex]\\left(g-f\\right)\\left(x\\right)[\/latex] and [latex]\\left(\\frac{g}{f}\\right)\\left(x\\right)[\/latex], given [latex]f\\left(x\\right)=x - 1[\/latex] and [latex]g\\left(x\\right)={x}^{2}-1[\/latex]. Give the domain of your result. Are they the same function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q117163\">Show Answer<\/span><\/p>\n<div id=\"q117163\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137466263\">Begin by writing the general form, and then substitute the given functions.<\/p>\n<div id=\"fs-id1165135701567\" class=\"equation unnumbered\">[latex]\\begin{array}{c}\\left(g-f\\right)\\left(x\\right)=g\\left(x\\right)-f\\left(x\\right) \\\\ \\left(g-f\\right)\\left(x\\right)={x}^{2}-1-\\left(x - 1\\right)\\\\ \\text{ }={x}^{2}-x \\\\ \\text{ }=x\\left(x - 1\\right) \\\\\\end{array}[\/latex] [latex]\\begin{array}{c}\\text{ }\\left(\\frac{g}{f}\\right)\\left(x\\right)=\\frac{g\\left(x\\right)}{f\\left(x\\right)} \\\\ \\text{ }\\left(\\frac{g}{f}\\right)\\left(x\\right)=\\frac{{x}^{2}-1}{x - 1}\\\\ \\text{ }=\\frac{\\left(x+1\\right)\\left(x - 1\\right)}{x - 1}\\text{ where }x\\ne 1 \\\\ \\text{ }=x+1 \\end{array}[\/latex]<\/div>\n<p>The domain of the result is [latex]x\\ne1[\/latex]<\/p>\n<p id=\"fs-id1165137553743\">No, the functions are not the same.<\/p>\n<p id=\"fs-id1165135351612\">Note: For [latex]\\left(\\frac{g}{f}\\right)\\left(x\\right)[\/latex], the condition [latex]x\\ne 1[\/latex] is necessary because when [latex]x=1[\/latex], the denominator is equal to 0, which makes the function undefined.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find and simplify the functions [latex]\\left(fg\\right)\\left(x\\right)[\/latex] and [latex]\\left(f-g\\right)\\left(x\\right)[\/latex].<\/p>\n<div id=\"fs-id1165137434994\" class=\"equation unnumbered\">[latex]f\\left(x\\right)=x - 1\\text{ and }g\\left(x\\right)={x}^{2}-1[\/latex]<\/div>\n<p id=\"fs-id1165137434911\">Are they the same function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q721147\">Show Answer<\/span><\/p>\n<div id=\"q721147\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{array}{c}\\left(fg\\right)\\left(x\\right)=f\\left(x\\right)g\\left(x\\right)=\\left(x - 1\\right)\\left({x}^{2}-1\\right)={x}^{3}-{x}^{2}-x+1\\\\ \\left(f-g\\right)\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)=\\left(x - 1\\right)-\\left({x}^{2}-1\\right)=x-{x}^{2}\\end{array}[\/latex]<br \/>\nNo, the functions are not the same.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4869\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al... <strong>Provided by<\/strong>: OpenStax.. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2.%20\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2.%20<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":60342,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al..\",\"organization\":\"OpenStax.\",\"url\":\" http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. \",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4869","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4869","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/60342"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4869\/revisions"}],"predecessor-version":[{"id":5165,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4869\/revisions\/5165"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4869\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=4869"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4869"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=4869"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=4869"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}