{"id":126,"date":"2023-06-21T13:22:35","date_gmt":"2023-06-21T13:22:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary\/"},"modified":"2023-06-21T13:22:35","modified_gmt":"2023-06-21T13:22:35","slug":"summary","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary\/","title":{"raw":"Summary: Characteristics of Functions and Their Graphs","rendered":"Summary: Characteristics of Functions and Their Graphs"},"content":{"raw":"\n\n<div id=\"Example_01_01_05\" class=\"example\">\n<h2>Key Equations<\/h2>\n<\/div>\n<table id=\"eip-id1165134393730\" summary=\"..\"><colgroup> <col> <col><\/colgroup>\n<tbody>\n<tr>\n<td>Constant function<\/td>\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\n<\/tr>\n<tr>\n<td>Identity function<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Absolute value function<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Quadratic function<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Cubic function<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reciprocal function<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reciprocal squared function<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Square root function<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Cube root function<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137851183\">\n \t<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\n \t<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right)[\/latex].<\/li>\n \t<li>In table form, a function can be represented by rows or columns that relate to input and output values.<\/li>\n \t<li>To evaluate a function we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.<\/li>\n \t<li>To solve for a specific function value, we determine the input values that yield the specific output value.<\/li>\n \t<li>An algebraic form of a function can be written from an equation.<\/li>\n \t<li>Input and output values of a function can be identified from a table.<\/li>\n \t<li>Relating input values to output values on a graph is another way to evaluate a function.<\/li>\n \t<li>A function is one-to-one if each output value corresponds to only one input value.<\/li>\n \t<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\n \t<li>A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137758543\" class=\"definition\">\n \t<dt><strong>dependent variable<\/strong><\/dt>\n \t<dd id=\"fs-id1165137758548\">an output variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137758552\" class=\"definition\">\n \t<dt><strong>domain<\/strong><\/dt>\n \t<dd id=\"fs-id1165137932576\">the set of all possible input values for a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932580\" class=\"definition\">\n \t<dt><strong>function<\/strong><\/dt>\n \t<dd id=\"fs-id1165137932585\">a relation in which each input value yields a unique output value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932588\" class=\"definition\">\n \t<dt><strong>horizontal line test<\/strong><\/dt>\n \t<dd id=\"fs-id1165134149777\">a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134149782\" class=\"definition\">\n \t<dt><strong>independent variable<\/strong><\/dt>\n \t<dd id=\"fs-id1165134149787\">an input variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511353\" class=\"definition\">\n \t<dt><strong>input<\/strong><\/dt>\n \t<dd id=\"fs-id1165135511359\">each object or value in a domain that relates to another object or value by a relationship known as a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511364\" class=\"definition\">\n \t<dt><strong>one-to-one function<\/strong><\/dt>\n \t<dd id=\"fs-id1165135511369\">a function for which each value of the output is associated with a unique input value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508564\" class=\"definition\">\n \t<dt><strong>output<\/strong><\/dt>\n \t<dd id=\"fs-id1165135508569\">each object or value in the range that is produced when an input value is entered into a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508573\" class=\"definition\">\n \t<dt><strong>range<\/strong><\/dt>\n \t<dd id=\"fs-id1165135315529\">the set of output values that result from the input values in a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315533\" class=\"definition\">\n \t<dt><strong>relation<\/strong><\/dt>\n \t<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n \t<dt><strong>vertical line test<\/strong><\/dt>\n \t<dd id=\"fs-id1165134186374\">a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once<\/dd>\n<\/dl>\n<\/div>\n<h2><\/h2>\n\n","rendered":"<div id=\"Example_01_01_05\" class=\"example\">\n<h2>Key Equations<\/h2>\n<\/div>\n<table id=\"eip-id1165134393730\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>Constant function<\/td>\n<td>[latex]f\\left(x\\right)=c[\/latex], where [latex]c[\/latex] is a constant<\/td>\n<\/tr>\n<tr>\n<td>Identity function<\/td>\n<td>[latex]f\\left(x\\right)=x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Absolute value function<\/td>\n<td>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Quadratic function<\/td>\n<td>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Cubic function<\/td>\n<td>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reciprocal function<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reciprocal squared function<\/td>\n<td>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Square root function<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Cube root function<\/td>\n<td>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137851183\">\n<li>A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output.<\/li>\n<li>Function notation is a shorthand method for relating the input to the output in the form [latex]y=f\\left(x\\right)[\/latex].<\/li>\n<li>In table form, a function can be represented by rows or columns that relate to input and output values.<\/li>\n<li>To evaluate a function we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.<\/li>\n<li>To solve for a specific function value, we determine the input values that yield the specific output value.<\/li>\n<li>An algebraic form of a function can be written from an equation.<\/li>\n<li>Input and output values of a function can be identified from a table.<\/li>\n<li>Relating input values to output values on a graph is another way to evaluate a function.<\/li>\n<li>A function is one-to-one if each output value corresponds to only one input value.<\/li>\n<li>A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point.<\/li>\n<li>A graph represents a one-to-one function if any horizontal line drawn on the graph intersects the graph at no more than one point.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137758543\" class=\"definition\">\n<dt><strong>dependent variable<\/strong><\/dt>\n<dd id=\"fs-id1165137758548\">an output variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137758552\" class=\"definition\">\n<dt><strong>domain<\/strong><\/dt>\n<dd id=\"fs-id1165137932576\">the set of all possible input values for a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932580\" class=\"definition\">\n<dt><strong>function<\/strong><\/dt>\n<dd id=\"fs-id1165137932585\">a relation in which each input value yields a unique output value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137932588\" class=\"definition\">\n<dt><strong>horizontal line test<\/strong><\/dt>\n<dd id=\"fs-id1165134149777\">a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134149782\" class=\"definition\">\n<dt><strong>independent variable<\/strong><\/dt>\n<dd id=\"fs-id1165134149787\">an input variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511353\" class=\"definition\">\n<dt><strong>input<\/strong><\/dt>\n<dd id=\"fs-id1165135511359\">each object or value in a domain that relates to another object or value by a relationship known as a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135511364\" class=\"definition\">\n<dt><strong>one-to-one function<\/strong><\/dt>\n<dd id=\"fs-id1165135511369\">a function for which each value of the output is associated with a unique input value<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508564\" class=\"definition\">\n<dt><strong>output<\/strong><\/dt>\n<dd id=\"fs-id1165135508569\">each object or value in the range that is produced when an input value is entered into a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135508573\" class=\"definition\">\n<dt><strong>range<\/strong><\/dt>\n<dd id=\"fs-id1165135315529\">the set of output values that result from the input values in a relation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315533\" class=\"definition\">\n<dt><strong>relation<\/strong><\/dt>\n<dd id=\"fs-id1165135315539\">a set of ordered pairs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n<dt><strong>vertical line test<\/strong><\/dt>\n<dd id=\"fs-id1165134186374\">a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once<\/dd>\n<\/dl>\n<\/div>\n<h2><\/h2>\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-126\">\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\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":395986,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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