{"id":793,"date":"2022-02-18T21:11:18","date_gmt":"2022-02-18T21:11:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=793"},"modified":"2026-01-16T23:18:07","modified_gmt":"2026-01-16T23:18:07","slug":"1-1-3-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/1-1-3-functions\/","title":{"raw":"1.1.2: Functions","rendered":"1.1.2: Functions"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Define a function<\/li>\r\n \t<li>Determine if a relation is a function<\/li>\r\n \t<li><span class=\"normaltextrun\">Determine if a function is one-to-one\u00a0<\/span><\/li>\r\n \t<li>Write a function using function notation<\/li>\r\n \t<li>Determine an inverse function from a mapping or set of ordered pairs<\/li>\r\n \t<li>Determine the domain of an inverse function given the domain and range of a one-to-one function.<\/li>\r\n \t<li>Write an inverse function using function notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Definition of a Function<\/h2>\r\nA <em><strong>function<\/strong><\/em> is defined as a <em>one-to-one<\/em> or <em>many-to-one<\/em> mapping. One-to-many and many-to-many mappings are not functions. A one-to-one mapping means every element in the domain is mapped to exactly one element in the range, and every element in the range is mapped by only one element in the domain (See Figure 2). A many-to-one mapping means in addition to one-to-one, there are elements in the domain that are mapped to the same element in the range (See Figure 3).\r\n\r\n<img class=\"aligncenter wp-image-639 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-300x216.png\" alt=\"An example of a one to one mapping, described below.\" width=\"300\" height=\"216\" \/>\r\n<p style=\"text-align: center;\">Figure 2: A one-to-one mapping is a function<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter wp-image-640 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-300x216.png\" alt=\"An example of a many to one mapping, described below.\" width=\"300\" height=\"216\" \/>\r\n<p style=\"text-align: center;\">Figure 3: A many-to-one mapping is a function<\/p>\r\nThis means that for a relation to be a function, every element in the domain has exactly one mapping arrow coming from it. And, for a function to be one-to-one, every element in the range has only one mapping arrow coming to it.\r\n\r\nOne-to-many and many-to-many mappings are not functions. A one-to-many mapping means, there is at least one element in the domain that is mapped to two or more elements in the range (See Figure 4). A many-to-many mapping means there are many-to-one and one-to-many cases in the mapping (See Figure 5).\r\n\r\n<img class=\"aligncenter wp-image-641 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-300x216.png\" alt=\"An example of a one to many mapping, described above.\" width=\"300\" height=\"216\" \/>\r\n<p style=\"text-align: center;\">Figure 4: A one-to many mapping is NOT a function<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter wp-image-642 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-300x216.png\" alt=\"An example of a many to many mapping, described above.\" width=\"300\" height=\"216\" \/>\r\n<p style=\"text-align: center;\">Figure 5: A many-to-many mapping is NOT a function<\/p>\r\nRelations are NOT functions if there are more than one mapping arrows coming from a least one element in the domain.\r\n\r\nFor example, the relation given by the set of ordered pairs {(1, 2), (4, 8), (5, -3), (4, 6)}, it\u00a0is not a function because it is a one-to-many mapping. The element 4 in the domain is mapped to 6 and 8 in the range. In order for a relation to be a function, every element in the domain must map to only one element in the range.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nDetermine if the relations are functions:\r\n<ol>\r\n \t<li>\u00a0{(0, 6), (1, 7), (2, 9), (4, 6)}<\/li>\r\n \t<li>\u00a0{(-6, 9), (7, 3), (-8, 5), (5, 4)}<\/li>\r\n \t<li>\u00a0{(-8, 4), (7, 9), (-8, 6), (0, 5)}<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm942\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm942\"]\r\n<ol>\r\n \t<li>\u00a0This relation is a function: each [latex]x[\/latex] appears only once. It is a many-to-one mapping since 0 and 4 map to 6.<\/li>\r\n \t<li>\u00a0This relation is a function: each [latex]x[\/latex] appears only once. It is a one-to-one mapping.<\/li>\r\n \t<li>\u00a0This relation is not a function: It is a one-to-many mapping since -8\u00a0 maps to 4 and 6.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Definition of a function<\/h3>\r\n<ul>\r\n \t<li>For each value in the domain, there is one and only one corresponding value in the range.<\/li>\r\n \t<li>Every element in the domain must be mapped to one and only one corresponding element in the range.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Inverse Functions<\/h2>\r\nWhen a function is one-to-one, it has an <em><strong>inverse function<\/strong><\/em>. An inverse function maps the range elements back to the original domain elements. The function must be one-to-one so that every range element has only one domain element to go back to. This ensures that the inverse relation is a function.\r\n\r\nConsider the function that maps people to their social security number. That is a one-to-one function because every person has a unique social security number. The inverse function takes any given social security number and maps it back to the person it belongs to. The original function takes the domain elements of given people and maps to their unique social security number in the range. The inverse function takes its domain elements of social security number and maps it to the person that it belongs to in the functions range. So the domain of the inverse function is the range of the original function. And the range of the inverse function is the domain of the original function.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-778 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-1024x250.png\" alt=\"Example of inverse functions showing the domain and range swap.\" width=\"1024\" height=\"250\" \/><\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nState the inverse function of [latex]\\{ (2, 1), (3, 4), (4, 7), (5, 9)\\}[\/latex].\r\n<h4>Solution<\/h4>\r\nFirst we have to check that this function is one-to-one. It is because every [latex]x[\/latex]-value maps to only one [latex]y[\/latex]-value and every\u00a0[latex]y[\/latex]-value is mapped from only one\u00a0[latex]x[\/latex]-value.\r\n\r\nThe domain of the inverse function is the range of the original function, and vice versa. This means that we just need to switch the\u00a0[latex]x[\/latex]-values and\u00a0[latex]y[\/latex]-values.\r\n\r\nInverse function = [latex]\\{ (1, 2), (4, 3), (7, 4), (9, 5)\\}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nState the inverse function, if it exists:\r\n<ol>\r\n \t<li>Original function = [latex]\\{ (1, 2), (2, 3), (3, 4), (4, 5), (5, 6)\\}[\/latex]<\/li>\r\n \t<li>Original function =\u00a0[latex]\\{ (1, 2), (2, 3), (3, 2), (4, 3)\\}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm307\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm307\"]\r\n<ol>\r\n \t<li>Inverse function =\u00a0[latex]\\{ (2, 1), (3, 2), (4, 3), (5, 4), (6, 5)\\}[\/latex]<\/li>\r\n \t<li>There is no inverse function since the original function is not one-to-one.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nIf a one-to-one function has a domain of [latex](-\\infty, +\\infty)[\/latex] and a range of [latex](-\\infty, 6)[\/latex], state the domain and range of the inverse function.\r\n\r\n[reveal-answer q=\"hjm284\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm284\"]\r\n\r\nDomain = [latex](-\\infty, 6)[\/latex]\r\n\r\nRange = [latex](-\\infty, +\\infty)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nIf [latex]f[\/latex] is a function that says. \"add 3 to any real number\",\r\n<ol>\r\n \t<li>\u00a0Create a mapping, using at least four numbers and a variable, that describes the function [latex]f[\/latex].<\/li>\r\n \t<li>\u00a0Create a mapping, using at least four numbers and a variable, that describes the inverse function of [latex]f[\/latex].<\/li>\r\n \t<li>\u00a0Describe the inverse function of [latex]f[\/latex] in words.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nChoose any four real numbers for the domain of the function\u00a0[latex]f[\/latex], then add 3 to map them to the range. For a variable domain element [latex]x[\/latex], the mapping is [latex]x+3[\/latex].<img class=\"aligncenter wp-image-784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18192047\/inverse-function-example-1-300x98.png\" alt=\"Inverse function mappings showing a possible solution.\" width=\"502\" height=\"164\" \/>For the inverse function, the domain of the original function becomes the range of the inverse function and vice versa. For a variable domain element [latex]x[\/latex], the mapping for the inverse function is [latex]x-3[\/latex].\r\n\r\nInverse function of [latex]f[\/latex] = subtract 3.\r\n\r\n<\/div>\r\nNotice that the inverse function \"undoes\" what the original function \"does\". The original function \"adds 3\" to every element in the domain. The inverse function \"undoes\" \"adding 3\" by \"subtracting 3\". If we apply a function to a domain element [latex]x[\/latex], then apply the inverse function to the result, we end up back at [latex]x[\/latex].\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nIf [latex]f[\/latex] is a function that says. \"multiply any real number <span style=\"font-size: 1rem; text-align: initial;\">by 4\",<\/span>\r\n<ol>\r\n \t<li>\u00a0create a mapping that using at least four numbers and a variable that describes the function [latex]f[\/latex].<\/li>\r\n \t<li>\u00a0create a mapping that using at least four numbers and a variable that describes the inverse function of [latex]f[\/latex].<\/li>\r\n \t<li>\u00a0describe the inverse function of [latex]f[\/latex] in words.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm008\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm008\"]\r\n\r\n<img class=\"aligncenter wp-image-1719\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22183513\/INVERSE-MAPPING-300x100.png\" alt=\"Inverse mapping of original function f and its inverse function, showing the domain and range have flipped.\" width=\"393\" height=\"131\" \/>\r\n\r\n&nbsp;\r\n\r\nInverse function of [latex]g = \\frac{x}{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Function Notation<\/h2>\r\nFunction notation describes how the elements in the range are obtained from the elements in the domain. Function notation is used because it allows us to immediately recognize the domain variable. The function is given a name that is usually a letter of the alphabet (e.g. f, g, R, C, etc). In parentheses next to the function's name we write the domain variable. The notation [latex]f(x)[\/latex], which is read, \"f of x\", means we are going to apply the function named [latex]f[\/latex] to the domain variable [latex]x[\/latex]. The function [latex]f[\/latex] is a rule that tells us how to map the elements in the domain to the elements in the range. It can be written in words, but is usually written in mathematical notation.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-720 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping-300x141.png\" alt=\"function as a mapping showing that the domain is tied to x, and the range tied to f of x.\" width=\"300\" height=\"141\" \/><\/p>\r\nFor example, the rule that says, \"take any real number and double it\" is written in function notation as [latex]f(x)=2x[\/latex]. Any element [latex]x[\/latex] in the domain is doubled and becomes part of the range.\r\n\r\n<img class=\"aligncenter wp-image-722 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping-300x149.png\" alt=\"f(x) = 2x mapping with domain x, and range 2x.\" width=\"300\" height=\"149\" \/>\r\n\r\nThe notation [latex]f(x)[\/latex] specifies that the range element depends on the domain element, [latex]x[\/latex]. As [latex]x[\/latex] changes value, so does[latex]f(x)[\/latex]. Note that the parentheses in function notation do NOT imply multiplication. They simply separate the domain variable from the function name.\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nWrite function notation that describes the function:\r\n<ol>\r\n \t<li>\u00a0Function [latex]f[\/latex] when applied to any domain element [latex]x[\/latex] adds 4 to the domain element.<\/li>\r\n \t<li>\u00a0Function [latex]g[\/latex] when applied to any domain element [latex]x[\/latex] multiplies the domain element by 7.<\/li>\r\n \t<li>\u00a0Function [latex]C[\/latex] when applied to any domain element [latex]t[\/latex] squares the domain element.<\/li>\r\n \t<li>\u00a0Function [latex]H[\/latex] when applied to any domain element [latex]z[\/latex] multiplies the domain element by -3 and adds 8 to the product.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>\u00a0[latex]f(x) = x+4[\/latex]\u00a0 \u00a0 The function is named [latex]f[\/latex], the variable is [latex]x[\/latex] and the rule is \"add 4\": [latex]x+4[\/latex]<\/li>\r\n \t<li>\u00a0[latex]g(x) = 7x[\/latex]\u00a0 \u00a0 \u00a0The function is named [latex]g[\/latex], the variable is [latex]x[\/latex] and the rule is \"multiply by 7\": [latex]7x[\/latex]<\/li>\r\n \t<li>\u00a0[latex]C(t) = t^2[\/latex]\u00a0 \u00a0 The function is named [latex]C[\/latex], the variable is [latex]t[\/latex] and the rule is \"square\": [latex]t^2[\/latex]<\/li>\r\n \t<li>\u00a0[latex]H(z) = -3z+8[\/latex]\u00a0 \u00a0 The function is named [latex]H[\/latex], the variable is [latex]z[\/latex] and the rule is \"multiply by -3 then add 8\": [latex]-3z+8[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nWrite function notation that describes the function:\r\n<ol>\r\n \t<li>\u00a0Function [latex]f[\/latex] when applied to any domain element [latex]x[\/latex] subtracts 5 from the domain element.<\/li>\r\n \t<li>\u00a0Function [latex]h[\/latex] when applied to any domain element [latex]z[\/latex] multiplies the domain element by 3.<\/li>\r\n \t<li>\u00a0Function [latex]T[\/latex] when applied to any domain element [latex]t[\/latex] cubes the domain element then add 8.<\/li>\r\n \t<li>\u00a0Function [latex]G[\/latex] when applied to any domain element [latex]x[\/latex] adds 7 to the domain element then squares the sum.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm864\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm864\"]\r\n<ol>\r\n \t<li>[latex]f(x)=x-5[\/latex]<\/li>\r\n \t<li>[latex]h(x)=3z[\/latex]<\/li>\r\n \t<li>[latex]T(t)=t^3+8[\/latex]<\/li>\r\n \t<li>[latex]G(x)=(x+7)^2[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Inverse Function Notation<\/h2>\r\nWe have already learned that for an inverse function to exist the original function must be one-to-one. This ensures that the inverse is also a\u00a0<span style=\"font-size: 1rem; text-align: initial;\">function. Rather than having to say, \"the inverse function is...\", we use the following notation: The inverse of [latex]f(x)[\/latex] is [latex]{f}^{-1}(x)[\/latex], which is read, \"the inverse of [latex]f(x)[\/latex]. This may seem like odd notation as the [latex]-1[\/latex] is NOT an exponent. This is because the notation comes from set theory and is a bit of an anomaly. All we need to know right now is that when working with any one-to-one function [latex]f(x),\\;{f}^{-1}(x)[\/latex] is the inverse of [latex]f(x)[\/latex].<\/span>\r\n\r\nConsider the function [latex]f(x)=x+3[\/latex]. We already looked at the mapping for this function and its inverse in a previous example:\r\n\r\n<img class=\"aligncenter wp-image-784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18192047\/inverse-function-example-1-300x98.png\" alt=\"Inverse function mapping of f of x equals x plus 3 and inverse of f seen before.\" width=\"514\" height=\"168\" \/>\r\n\r\nWe write the inverse of [latex]f(x)[\/latex] as [latex]{f}^{-1}(x)=x-3[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nA one-to-one function [latex]g(x)[\/latex] takes whole numbers and squares them.\r\n<ol>\r\n \t<li>\u00a0Write the function using function notation.<\/li>\r\n \t<li>\u00a0Create a mapping that shows the function [latex]g[\/latex] applied to at least 4 whole numbers and a variable term.<\/li>\r\n \t<li>\u00a0Create a mapping that shows the inverse of the function [latex]g[\/latex] applied to at least 4 whole numbers and a variable term.<\/li>\r\n \t<li>\u00a0Write the inverse function using function notation.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>\u00a0[latex]g(x)=x^2[\/latex]<\/li>\r\n \t<li>\u00a0and 3.<img class=\"aligncenter wp-image-787\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18201219\/Inverse-function-example-3-300x99.png\" alt=\"mapping of g, and g inverse.\" width=\"491\" height=\"162\" \/><\/li>\r\n \t<li value=\"4\">\u00a0[latex]{g}^{-1}(x)=\\sqrt{x}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nA one-to-one function [latex]h(x)[\/latex] takes integers and divides them by \u20132.\r\n<ol>\r\n \t<li>\u00a0Write the function using function notation.<\/li>\r\n \t<li>\u00a0Create a mapping that shows the function [latex]h[\/latex] applied to at least 4 integers and a variable term.<\/li>\r\n \t<li>\u00a0Create a mapping that shows the inverse of the function [latex]h[\/latex] applied to at least 4 integers and a variable term.<\/li>\r\n \t<li>\u00a0Write the inverse function using function notation.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm464\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm464\"]\r\n<ol>\r\n \t<li>\u00a0[latex]h(x)=\\frac{x}{-2}[\/latex]<\/li>\r\n \t<li>\u00a0and 3.<img class=\"aligncenter wp-image-788\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18202327\/inverse-example-4-300x108.png\" alt=\"inverse mappings\" width=\"464\" height=\"167\" \/><\/li>\r\n \t<li value=\"4\">\u00a0[latex]{h}^{-1}(x)=-2x[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nIf a one-to-one function [latex]f(x)[\/latex] has a domain of all real numbers and a range of [latex][0, +\\infty)[\/latex], determine the domain and range of [latex]{f}^{-1}(x)[\/latex].\r\n\r\n[reveal-answer q=\"hjm438\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm438\"]\r\n\r\nDomain of\u00a0[latex]{f}^{-1}(x)[\/latex] is\u00a0[latex][0, +\\infty)[\/latex].\r\n\r\nRange of\u00a0[latex]{f}^{-1}(x)[\/latex] is all real numbers.\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Define a function<\/li>\n<li>Determine if a relation is a function<\/li>\n<li><span class=\"normaltextrun\">Determine if a function is one-to-one\u00a0<\/span><\/li>\n<li>Write a function using function notation<\/li>\n<li>Determine an inverse function from a mapping or set of ordered pairs<\/li>\n<li>Determine the domain of an inverse function given the domain and range of a one-to-one function.<\/li>\n<li>Write an inverse function using function notation<\/li>\n<\/ul>\n<\/div>\n<h2>Definition of a Function<\/h2>\n<p>A <em><strong>function<\/strong><\/em> is defined as a <em>one-to-one<\/em> or <em>many-to-one<\/em> mapping. One-to-many and many-to-many mappings are not functions. A one-to-one mapping means every element in the domain is mapped to exactly one element in the range, and every element in the range is mapped by only one element in the domain (See Figure 2). A many-to-one mapping means in addition to one-to-one, there are elements in the domain that are mapped to the same element in the range (See Figure 3).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-639 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-300x216.png\" alt=\"An example of a one to one mapping, described below.\" width=\"300\" height=\"216\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-300x216.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-768x554.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-1024x738.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-65x47.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-225x162.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic1-350x252.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center;\">Figure 2: A one-to-one mapping is a function<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-640 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-300x216.png\" alt=\"An example of a many to one mapping, described below.\" width=\"300\" height=\"216\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-300x216.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-768x554.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-1024x738.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-65x47.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-225x162.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-Pic2-350x252.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center;\">Figure 3: A many-to-one mapping is a function<\/p>\n<p>This means that for a relation to be a function, every element in the domain has exactly one mapping arrow coming from it. And, for a function to be one-to-one, every element in the range has only one mapping arrow coming to it.<\/p>\n<p>One-to-many and many-to-many mappings are not functions. A one-to-many mapping means, there is at least one element in the domain that is mapped to two or more elements in the range (See Figure 4). A many-to-many mapping means there are many-to-one and one-to-many cases in the mapping (See Figure 5).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-641 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-300x216.png\" alt=\"An example of a one to many mapping, described above.\" width=\"300\" height=\"216\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-300x216.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-768x554.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-1024x738.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-65x47.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-225x162.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc3-350x252.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center;\">Figure 4: A one-to many mapping is NOT a function<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-642 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-300x216.png\" alt=\"An example of a many to many mapping, described above.\" width=\"300\" height=\"216\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-300x216.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-768x554.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-1024x738.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-65x47.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-225x162.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-1-1-PIc4-350x252.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center;\">Figure 5: A many-to-many mapping is NOT a function<\/p>\n<p>Relations are NOT functions if there are more than one mapping arrows coming from a least one element in the domain.<\/p>\n<p>For example, the relation given by the set of ordered pairs {(1, 2), (4, 8), (5, -3), (4, 6)}, it\u00a0is not a function because it is a one-to-many mapping. The element 4 in the domain is mapped to 6 and 8 in the range. In order for a relation to be a function, every element in the domain must map to only one element in the range.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Determine if the relations are functions:<\/p>\n<ol>\n<li>\u00a0{(0, 6), (1, 7), (2, 9), (4, 6)}<\/li>\n<li>\u00a0{(-6, 9), (7, 3), (-8, 5), (5, 4)}<\/li>\n<li>\u00a0{(-8, 4), (7, 9), (-8, 6), (0, 5)}<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm942\">Show Answer<\/span><\/p>\n<div id=\"qhjm942\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a0This relation is a function: each [latex]x[\/latex] appears only once. It is a many-to-one mapping since 0 and 4 map to 6.<\/li>\n<li>\u00a0This relation is a function: each [latex]x[\/latex] appears only once. It is a one-to-one mapping.<\/li>\n<li>\u00a0This relation is not a function: It is a one-to-many mapping since -8\u00a0 maps to 4 and 6.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Definition of a function<\/h3>\n<ul>\n<li>For each value in the domain, there is one and only one corresponding value in the range.<\/li>\n<li>Every element in the domain must be mapped to one and only one corresponding element in the range.<\/li>\n<\/ul>\n<\/div>\n<h2>Inverse Functions<\/h2>\n<p>When a function is one-to-one, it has an <em><strong>inverse function<\/strong><\/em>. An inverse function maps the range elements back to the original domain elements. The function must be one-to-one so that every range element has only one domain element to go back to. This ensures that the inverse relation is a function.<\/p>\n<p>Consider the function that maps people to their social security number. That is a one-to-one function because every person has a unique social security number. The inverse function takes any given social security number and maps it back to the person it belongs to. The original function takes the domain elements of given people and maps to their unique social security number in the range. The inverse function takes its domain elements of social security number and maps it to the person that it belongs to in the functions range. So the domain of the inverse function is the range of the original function. And the range of the inverse function is the domain of the original function.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-778 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-1024x250.png\" alt=\"Example of inverse functions showing the domain and range swap.\" width=\"1024\" height=\"250\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-1024x250.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-300x73.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-768x188.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-65x16.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-225x55.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Inverse-functions-350x85.png 350w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>State the inverse function of [latex]\\{ (2, 1), (3, 4), (4, 7), (5, 9)\\}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>First we have to check that this function is one-to-one. It is because every [latex]x[\/latex]-value maps to only one [latex]y[\/latex]-value and every\u00a0[latex]y[\/latex]-value is mapped from only one\u00a0[latex]x[\/latex]-value.<\/p>\n<p>The domain of the inverse function is the range of the original function, and vice versa. This means that we just need to switch the\u00a0[latex]x[\/latex]-values and\u00a0[latex]y[\/latex]-values.<\/p>\n<p>Inverse function = [latex]\\{ (1, 2), (4, 3), (7, 4), (9, 5)\\}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>State the inverse function, if it exists:<\/p>\n<ol>\n<li>Original function = [latex]\\{ (1, 2), (2, 3), (3, 4), (4, 5), (5, 6)\\}[\/latex]<\/li>\n<li>Original function =\u00a0[latex]\\{ (1, 2), (2, 3), (3, 2), (4, 3)\\}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm307\">Show Answer<\/span><\/p>\n<div id=\"qhjm307\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Inverse function =\u00a0[latex]\\{ (2, 1), (3, 2), (4, 3), (5, 4), (6, 5)\\}[\/latex]<\/li>\n<li>There is no inverse function since the original function is not one-to-one.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>If a one-to-one function has a domain of [latex](-\\infty, +\\infty)[\/latex] and a range of [latex](-\\infty, 6)[\/latex], state the domain and range of the inverse function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm284\">Show Answer<\/span><\/p>\n<div id=\"qhjm284\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain = [latex](-\\infty, 6)[\/latex]<\/p>\n<p>Range = [latex](-\\infty, +\\infty)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>If [latex]f[\/latex] is a function that says. &#8220;add 3 to any real number&#8221;,<\/p>\n<ol>\n<li>\u00a0Create a mapping, using at least four numbers and a variable, that describes the function [latex]f[\/latex].<\/li>\n<li>\u00a0Create a mapping, using at least four numbers and a variable, that describes the inverse function of [latex]f[\/latex].<\/li>\n<li>\u00a0Describe the inverse function of [latex]f[\/latex] in words.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>Choose any four real numbers for the domain of the function\u00a0[latex]f[\/latex], then add 3 to map them to the range. For a variable domain element [latex]x[\/latex], the mapping is [latex]x+3[\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18192047\/inverse-function-example-1-300x98.png\" alt=\"Inverse function mappings showing a possible solution.\" width=\"502\" height=\"164\" \/>For the inverse function, the domain of the original function becomes the range of the inverse function and vice versa. For a variable domain element [latex]x[\/latex], the mapping for the inverse function is [latex]x-3[\/latex].<\/p>\n<p>Inverse function of [latex]f[\/latex] = subtract 3.<\/p>\n<\/div>\n<p>Notice that the inverse function &#8220;undoes&#8221; what the original function &#8220;does&#8221;. The original function &#8220;adds 3&#8221; to every element in the domain. The inverse function &#8220;undoes&#8221; &#8220;adding 3&#8221; by &#8220;subtracting 3&#8221;. If we apply a function to a domain element [latex]x[\/latex], then apply the inverse function to the result, we end up back at [latex]x[\/latex].<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>If [latex]f[\/latex] is a function that says. &#8220;multiply any real number <span style=\"font-size: 1rem; text-align: initial;\">by 4&#8243;,<\/span><\/p>\n<ol>\n<li>\u00a0create a mapping that using at least four numbers and a variable that describes the function [latex]f[\/latex].<\/li>\n<li>\u00a0create a mapping that using at least four numbers and a variable that describes the inverse function of [latex]f[\/latex].<\/li>\n<li>\u00a0describe the inverse function of [latex]f[\/latex] in words.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm008\">Show Answer<\/span><\/p>\n<div id=\"qhjm008\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1719\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/22183513\/INVERSE-MAPPING-300x100.png\" alt=\"Inverse mapping of original function f and its inverse function, showing the domain and range have flipped.\" width=\"393\" height=\"131\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Inverse function of [latex]g = \\frac{x}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Function Notation<\/h2>\n<p>Function notation describes how the elements in the range are obtained from the elements in the domain. Function notation is used because it allows us to immediately recognize the domain variable. The function is given a name that is usually a letter of the alphabet (e.g. f, g, R, C, etc). In parentheses next to the function&#8217;s name we write the domain variable. The notation [latex]f(x)[\/latex], which is read, &#8220;f of x&#8221;, means we are going to apply the function named [latex]f[\/latex] to the domain variable [latex]x[\/latex]. The function [latex]f[\/latex] is a rule that tells us how to map the elements in the domain to the elements in the range. It can be written in words, but is usually written in mathematical notation.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-720 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping-300x141.png\" alt=\"function as a mapping showing that the domain is tied to x, and the range tied to f of x.\" width=\"300\" height=\"141\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping-300x141.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping-768x361.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping-65x31.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping-225x106.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping-350x164.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/function-as-a-mapping.png 890w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>For example, the rule that says, &#8220;take any real number and double it&#8221; is written in function notation as [latex]f(x)=2x[\/latex]. Any element [latex]x[\/latex] in the domain is doubled and becomes part of the range.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-722 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping-300x149.png\" alt=\"f(x) = 2x mapping with domain x, and range 2x.\" width=\"300\" height=\"149\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping-300x149.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping-768x382.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping-65x32.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping-225x112.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping-350x174.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/fx-mapping.png 832w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The notation [latex]f(x)[\/latex] specifies that the range element depends on the domain element, [latex]x[\/latex]. As [latex]x[\/latex] changes value, so does[latex]f(x)[\/latex]. Note that the parentheses in function notation do NOT imply multiplication. They simply separate the domain variable from the function name.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Write function notation that describes the function:<\/p>\n<ol>\n<li>\u00a0Function [latex]f[\/latex] when applied to any domain element [latex]x[\/latex] adds 4 to the domain element.<\/li>\n<li>\u00a0Function [latex]g[\/latex] when applied to any domain element [latex]x[\/latex] multiplies the domain element by 7.<\/li>\n<li>\u00a0Function [latex]C[\/latex] when applied to any domain element [latex]t[\/latex] squares the domain element.<\/li>\n<li>\u00a0Function [latex]H[\/latex] when applied to any domain element [latex]z[\/latex] multiplies the domain element by -3 and adds 8 to the product.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>\u00a0[latex]f(x) = x+4[\/latex]\u00a0 \u00a0 The function is named [latex]f[\/latex], the variable is [latex]x[\/latex] and the rule is &#8220;add 4&#8221;: [latex]x+4[\/latex]<\/li>\n<li>\u00a0[latex]g(x) = 7x[\/latex]\u00a0 \u00a0 \u00a0The function is named [latex]g[\/latex], the variable is [latex]x[\/latex] and the rule is &#8220;multiply by 7&#8221;: [latex]7x[\/latex]<\/li>\n<li>\u00a0[latex]C(t) = t^2[\/latex]\u00a0 \u00a0 The function is named [latex]C[\/latex], the variable is [latex]t[\/latex] and the rule is &#8220;square&#8221;: [latex]t^2[\/latex]<\/li>\n<li>\u00a0[latex]H(z) = -3z+8[\/latex]\u00a0 \u00a0 The function is named [latex]H[\/latex], the variable is [latex]z[\/latex] and the rule is &#8220;multiply by -3 then add 8&#8221;: [latex]-3z+8[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Write function notation that describes the function:<\/p>\n<ol>\n<li>\u00a0Function [latex]f[\/latex] when applied to any domain element [latex]x[\/latex] subtracts 5 from the domain element.<\/li>\n<li>\u00a0Function [latex]h[\/latex] when applied to any domain element [latex]z[\/latex] multiplies the domain element by 3.<\/li>\n<li>\u00a0Function [latex]T[\/latex] when applied to any domain element [latex]t[\/latex] cubes the domain element then add 8.<\/li>\n<li>\u00a0Function [latex]G[\/latex] when applied to any domain element [latex]x[\/latex] adds 7 to the domain element then squares the sum.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm864\">Show Answer<\/span><\/p>\n<div id=\"qhjm864\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=x-5[\/latex]<\/li>\n<li>[latex]h(x)=3z[\/latex]<\/li>\n<li>[latex]T(t)=t^3+8[\/latex]<\/li>\n<li>[latex]G(x)=(x+7)^2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Inverse Function Notation<\/h2>\n<p>We have already learned that for an inverse function to exist the original function must be one-to-one. This ensures that the inverse is also a\u00a0<span style=\"font-size: 1rem; text-align: initial;\">function. Rather than having to say, &#8220;the inverse function is&#8230;&#8221;, we use the following notation: The inverse of [latex]f(x)[\/latex] is [latex]{f}^{-1}(x)[\/latex], which is read, &#8220;the inverse of [latex]f(x)[\/latex]. This may seem like odd notation as the [latex]-1[\/latex] is NOT an exponent. This is because the notation comes from set theory and is a bit of an anomaly. All we need to know right now is that when working with any one-to-one function [latex]f(x),\\;{f}^{-1}(x)[\/latex] is the inverse of [latex]f(x)[\/latex].<\/span><\/p>\n<p>Consider the function [latex]f(x)=x+3[\/latex]. We already looked at the mapping for this function and its inverse in a previous example:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-784\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18192047\/inverse-function-example-1-300x98.png\" alt=\"Inverse function mapping of f of x equals x plus 3 and inverse of f seen before.\" width=\"514\" height=\"168\" \/><\/p>\n<p>We write the inverse of [latex]f(x)[\/latex] as [latex]{f}^{-1}(x)=x-3[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>A one-to-one function [latex]g(x)[\/latex] takes whole numbers and squares them.<\/p>\n<ol>\n<li>\u00a0Write the function using function notation.<\/li>\n<li>\u00a0Create a mapping that shows the function [latex]g[\/latex] applied to at least 4 whole numbers and a variable term.<\/li>\n<li>\u00a0Create a mapping that shows the inverse of the function [latex]g[\/latex] applied to at least 4 whole numbers and a variable term.<\/li>\n<li>\u00a0Write the inverse function using function notation.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>\u00a0[latex]g(x)=x^2[\/latex]<\/li>\n<li>\u00a0and 3.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-787\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18201219\/Inverse-function-example-3-300x99.png\" alt=\"mapping of g, and g inverse.\" width=\"491\" height=\"162\" \/><\/li>\n<li value=\"4\">\u00a0[latex]{g}^{-1}(x)=\\sqrt{x}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>A one-to-one function [latex]h(x)[\/latex] takes integers and divides them by \u20132.<\/p>\n<ol>\n<li>\u00a0Write the function using function notation.<\/li>\n<li>\u00a0Create a mapping that shows the function [latex]h[\/latex] applied to at least 4 integers and a variable term.<\/li>\n<li>\u00a0Create a mapping that shows the inverse of the function [latex]h[\/latex] applied to at least 4 integers and a variable term.<\/li>\n<li>\u00a0Write the inverse function using function notation.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm464\">Show Answer<\/span><\/p>\n<div id=\"qhjm464\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a0[latex]h(x)=\\frac{x}{-2}[\/latex]<\/li>\n<li>\u00a0and 3.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-788\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18202327\/inverse-example-4-300x108.png\" alt=\"inverse mappings\" width=\"464\" height=\"167\" \/><\/li>\n<li value=\"4\">\u00a0[latex]{h}^{-1}(x)=-2x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>If a one-to-one function [latex]f(x)[\/latex] has a domain of all real numbers and a range of [latex][0, +\\infty)[\/latex], determine the domain and range of [latex]{f}^{-1}(x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm438\">Show Answer<\/span><\/p>\n<div id=\"qhjm438\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain of\u00a0[latex]{f}^{-1}(x)[\/latex] is\u00a0[latex][0, +\\infty)[\/latex].<\/p>\n<p>Range of\u00a0[latex]{f}^{-1}(x)[\/latex] is all real numbers.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-793\">\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>Functions. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All Examples and Try Itd. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/desmos.com\">http:\/\/desmos.com<\/a>. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":370291,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Functions\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All Examples and Try Itd\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"desmos.com\",\"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-793","chapter","type-chapter","status-publish","hentry"],"part":613,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/793","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":16,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/793\/revisions"}],"predecessor-version":[{"id":4812,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/793\/revisions\/4812"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/613"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/793\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=793"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=793"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=793"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=793"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}