A function<\/strong> is a special type of relation where each input (domain<\/strong> value) corresponds to exactly one output (range<\/strong> value).<\/p>\r\n The notation [latex]f(x)[\/latex] is read as \"f of x\" and represents the output value when x is the input. For example, if [latex]f(x) = x^2 + 1[\/latex], then [latex]f(3) = 3^2 + 1 = 10[\/latex].<\/p>\r\n Determining if a Relation is a Function<\/strong><\/p>\r\n\r\n Evaluating Functions<\/strong><\/p>\r\n\r\n A function is one-to-one<\/strong> if each output corresponds to exactly one input. Use the horizontal line test<\/strong>: if any horizontal line intersects the graph more than once, the function is not one-to-one.<\/p>\r\n\r\n The\u00a0domain\u00a0<\/strong>is the set of all possible input values (x-values) for a function<\/p>\r\n The range <\/strong>is the set of all possible output values (y-values) for a function<\/p>\r\n Finding Domain from an Equation<\/strong><\/p>\r\n Exclude values that would cause:<\/p>\r\n\r\n Finding Domain and Range from a Graph<\/strong><\/p>\r\n\r\n Interval Notation Tips<\/strong><\/p>\r\n\r\n A piecewise function<\/strong> uses different formulas for different parts of the domain. To evaluate:<\/p>\r\n\r\n To graph a piecewise function, graph each piece on its specified interval, using open circles for excluded endpoints and closed circles for included endpoints.<\/p>\r\n\r\n The a<\/strong>verage rate of change<\/strong> measures how quickly a function's output changes relative to its input over an interval:<\/p>\r\n [latex]\\text{Average Rate of Change} = \\dfrac{f(x_2) - f(x_1)}{x_2 - x_1} = \\dfrac{\\Delta y}{\\Delta x}[\/latex]<\/p>\r\n The units are \"output units per input unit\" (like miles per hour or dollars per year).<\/p>\r\n\r\n Local and Absolute Extrema<\/strong><\/p>\r\n\r\n Note: When stating where extrema occur, give the x-value. When stating the extrema value, give the y-value (or [latex]f(x)[\/latex] value).<\/p>\r\n\r\n Average Rate of Change:<\/strong> [latex]\\dfrac{f(x_2) - f(x_1)}{x_2 - x_1}[\/latex]<\/p>\r\n\r\n absolute maximum<\/strong><\/p>\r\n the largest output value over the entire domain of a function<\/p>\r\n absolute minimum<\/strong><\/p>\r\n the smallest output value over the entire domain of a function<\/p>\r\n average rate of change<\/strong><\/p>\r\n the change in output divided by the change in input over a specified interval<\/p>\r\n decreasing function<\/strong><\/p>\r\n a function where output values decrease as input values increase<\/p>\r\n dependent variable<\/strong><\/p>\r\n the output variable in a function, often represented by y or [latex]f(x)[\/latex]<\/p>\r\n domain<\/strong><\/p>\r\n the set of all possible input values for a function<\/p>\r\n function<\/strong><\/p>\r\n a relation where each input corresponds to exactly one output<\/p>\r\n horizontal line test<\/strong><\/p>\r\n a test to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it's not one-to-one<\/p>\r\n increasing function<\/strong><\/p>\r\n a function where output values increase as input values increase<\/p>\r\n independent variable<\/strong><\/p>\r\n the input variable in a function, often represented by x<\/p>\r\n input<\/strong><\/p>\r\n a value from the domain that is entered into a function<\/p>\r\n interval notation<\/strong><\/p>\r\n a way of writing sets of numbers using brackets and parentheses to show included and excluded endpoints<\/p>\r\n local maximum<\/strong><\/p>\r\n a point where the function value is greater than all nearby points<\/p>\r\n local minimum<\/strong><\/p>\r\n a point where the function value is less than all nearby points<\/p>\r\n one-to-one function<\/strong><\/p>\r\n a function where each output value corresponds to exactly one input value<\/p>\r\n output<\/strong><\/p>\r\n the value produced when an input is entered into a function<\/p>\r\n piecewise function<\/strong><\/p>\r\n a function defined by different formulas over different parts of the domain<\/p>\r\n range<\/strong><\/p>\r\n the set of all possible output values for a function<\/p>\r\n rate of change<\/strong><\/p>\r\n a measure of how one quantity changes relative to another<\/p>\r\n relation<\/strong><\/p>\r\n a relationship between two sets of numbers<\/p>\r\n vertical line test<\/strong><\/p>\r\n a test to determine if a graph represents a function; if any vertical line intersects the graph more than once, it's not a function<\/p>","rendered":" A function<\/strong> is a special type of relation where each input (domain<\/strong> value) corresponds to exactly one output (range<\/strong> value).<\/p>\n The notation [latex]f(x)[\/latex] is read as “f of x” and represents the output value when x is the input. For example, if [latex]f(x) = x^2 + 1[\/latex], then [latex]f(3) = 3^2 + 1 = 10[\/latex].<\/p>\n Determining if a Relation is a Function<\/strong><\/p>\n Evaluating Functions<\/strong><\/p>\n A function is one-to-one<\/strong> if each output corresponds to exactly one input. Use the horizontal line test<\/strong>: if any horizontal line intersects the graph more than once, the function is not one-to-one.<\/p>\n The\u00a0domain\u00a0<\/strong>is the set of all possible input values (x-values) for a function<\/p>\n The range <\/strong>is the set of all possible output values (y-values) for a function<\/p>\n Finding Domain from an Equation<\/strong><\/p>\n Exclude values that would cause:<\/p>\n Finding Domain and Range from a Graph<\/strong><\/p>\n Interval Notation Tips<\/strong><\/p>\n A piecewise function<\/strong> uses different formulas for different parts of the domain. To evaluate:<\/p>\n To graph a piecewise function, graph each piece on its specified interval, using open circles for excluded endpoints and closed circles for included endpoints.<\/p>\n The a<\/strong>verage rate of change<\/strong> measures how quickly a function’s output changes relative to its input over an interval:<\/p>\n [latex]\\text{Average Rate of Change} = \\dfrac{f(x_2) - f(x_1)}{x_2 - x_1} = \\dfrac{\\Delta y}{\\Delta x}[\/latex]<\/p>\n The units are “output units per input unit” (like miles per hour or dollars per year).<\/p>\n Local and Absolute Extrema<\/strong><\/p>\n Note: When stating where extrema occur, give the x-value. When stating the extrema value, give the y-value (or [latex]f(x)[\/latex] value).<\/p>\n Average Rate of Change:<\/strong> [latex]\\dfrac{f(x_2) - f(x_1)}{x_2 - x_1}[\/latex]<\/p>\n absolute maximum<\/strong><\/p>\n the largest output value over the entire domain of a function<\/p>\n absolute minimum<\/strong><\/p>\n the smallest output value over the entire domain of a function<\/p>\n average rate of change<\/strong><\/p>\n the change in output divided by the change in input over a specified interval<\/p>\n decreasing function<\/strong><\/p>\n a function where output values decrease as input values increase<\/p>\n dependent variable<\/strong><\/p>\n the output variable in a function, often represented by y or [latex]f(x)[\/latex]<\/p>\n domain<\/strong><\/p>\n the set of all possible input values for a function<\/p>\n function<\/strong><\/p>\n a relation where each input corresponds to exactly one output<\/p>\n horizontal line test<\/strong><\/p>\n a test to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it’s not one-to-one<\/p>\n increasing function<\/strong><\/p>\n a function where output values increase as input values increase<\/p>\n independent variable<\/strong><\/p>\n the input variable in a function, often represented by x<\/p>\n input<\/strong><\/p>\n a value from the domain that is entered into a function<\/p>\n interval notation<\/strong><\/p>\n a way of writing sets of numbers using brackets and parentheses to show included and excluded endpoints<\/p>\n local maximum<\/strong><\/p>\n a point where the function value is greater than all nearby points<\/p>\n local minimum<\/strong><\/p>\n a point where the function value is less than all nearby points<\/p>\n one-to-one function<\/strong><\/p>\n a function where each output value corresponds to exactly one input value<\/p>\n output<\/strong><\/p>\n the value produced when an input is entered into a function<\/p>\n piecewise function<\/strong><\/p>\n a function defined by different formulas over different parts of the domain<\/p>\n range<\/strong><\/p>\n the set of all possible output values for a function<\/p>\n rate of change<\/strong><\/p>\n a measure of how one quantity changes relative to another<\/p>\n relation<\/strong><\/p>\n a relationship between two sets of numbers<\/p>\n vertical line test<\/strong><\/p>\n a test to determine if a graph represents a function; if any vertical line intersects the graph more than once, it’s not a function<\/p>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":36,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/673"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/673\/revisions"}],"predecessor-version":[{"id":5085,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/673\/revisions\/5085"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/673\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=673"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=673"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=673"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\r\n \t
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Domain and Range<\/h3>\r\n
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Rates of Change and Behaviors of Graphs<\/h3>\r\n
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Key Equations<\/h2>\r\n
Glossary<\/h2>\r\n
Essential Concepts<\/h2>\n
Functions and Function Notation<\/h3>\n
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Domain and Range<\/h3>\n
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Rates of Change and Behaviors of Graphs<\/h3>\n
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Key Equations<\/h2>\n
Glossary<\/h2>\n