Function composition<\/strong> is combining two functions so that the output of one function becomes the input of another. Think of it like a two-step process: the first function processes the input, then the second function processes that result.<\/p>\r\n The composition of functions [latex]f[\/latex] and [latex]g[\/latex] is written as [latex](f \\circ g)(x) = f(g(x))[\/latex], read as \"f of g of x\" or \"f composed with g at x.\"<\/p>\r\n In general, [latex](f \\circ g)(x) \\neq (g \\circ f)(x)[\/latex]. The order in which you compose functions matters because you're applying different operations in different sequences.<\/p>\r\n Evaluating Composite Functions<\/strong><\/p>\r\n\r\n Domain of Composite Functions<\/strong> The domain of [latex](f \\circ g)(x)[\/latex] consists of all [latex]x[\/latex] values that satisfy both:<\/p>\r\n\r\n Decomposing Functions<\/strong> Sometimes you need to break down a complex function into simpler parts. Look for a \"function inside a function\" pattern. There may be multiple ways to decompose the same function.<\/p>\r\n\r\n Types of Transformations<\/strong><\/p>\r\n Vertical Shifts:<\/strong> [latex]g(x) = f(x) + k[\/latex]<\/p>\r\n\r\n Horizontal Shifts:<\/strong> [latex]g(x) = f(x - h)[\/latex]<\/p>\r\n\r\n Vertical Reflections:<\/strong> [latex]g(x) = -f(x)[\/latex]<\/p>\r\n\r\n Horizontal Reflections:<\/strong> [latex]g(x) = f(-x)[\/latex]<\/p>\r\n\r\n Vertical Stretches and Compressions:<\/strong> [latex]g(x) = a \\cdot f(x)[\/latex]<\/p>\r\n\r\n Horizontal Stretches and Compressions:<\/strong> [latex]g(x) = f(bx)[\/latex]<\/p>\r\n\r\n Even and Odd Functions<\/strong><\/p>\r\n\r\n Order of Transformations<\/strong> When applying multiple transformations to [latex]y = a \\cdot f(b(x - h)) + k[\/latex], apply in this order:<\/p>\r\n\r\n An inverse function<\/strong> reverses the operation of the original function. If [latex]f[\/latex] takes input [latex]a[\/latex] to output [latex]b[\/latex], then [latex]f^{-1}[\/latex] takes input [latex]b[\/latex] back to output [latex]a[\/latex].<\/p>\r\n The inverse of [latex]f(x)[\/latex] is written [latex]f^{-1}(x)[\/latex], read as \"f inverse of x.\" Note: This is NOT the same as [latex]\\frac{1}{f(x)}[\/latex].<\/p>\r\n A function must be one-to-one<\/strong> (pass the horizontal line test) to have an inverse. If a function is not one-to-one over its entire domain, you may be able to restrict the domain to make it one-to-one.<\/p>\r\n Two functions [latex]f[\/latex] and [latex]g[\/latex] are inverses if both of these are true:<\/p>\r\n\r\n Domain and Range of Inverses<\/strong><\/p>\r\n\r\n Finding Inverse Functions<\/strong><\/p>\r\n\r\n Graphing Inverse Functions<\/strong> The graph of [latex]f^{-1}(x)[\/latex] is the reflection of the graph of [latex]f(x)[\/latex] across the line [latex]y = x[\/latex]. If point [latex](a, b)[\/latex] is on the graph of [latex]f[\/latex], then point [latex](b, a)[\/latex] is on the graph of [latex]f^{-1}[\/latex].<\/p>\r\n\r\n composite function<\/strong><\/p>\r\n the new function formed by function composition, when the output of one function is used as the input of another<\/p>\r\n even function<\/strong><\/p>\r\n a function whose graph is unchanged by horizontal reflection, satisfying [latex]f(x) = f(-x)[\/latex], and is symmetric about the y-axis<\/p>\r\n horizontal compression<\/strong><\/p>\r\n a transformation that compresses a function's graph horizontally by multiplying the input by a constant [latex]b > 1[\/latex]<\/p>\r\n horizontal reflection<\/strong><\/p>\r\n a transformation that reflects a function's graph across the y-axis by multiplying the input by [latex]-1[\/latex]<\/p>\r\n horizontal shift<\/strong><\/p>\r\n a transformation that shifts a function's graph left or right by adding a constant to the input<\/p>\r\n horizontal stretch<\/strong><\/p>\r\n a transformation that stretches a function's graph horizontally by multiplying the input by a constant [latex]0 < b < 1[\/latex]<\/p>\r\n inverse function<\/strong><\/p>\r\n for any one-to-one function [latex]f(x)[\/latex], the inverse is a function [latex]f^{-1}(x)[\/latex] such that [latex]f^{-1}(f(x)) = x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]; this also implies that [latex]f(f^{-1}(x)) = x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f^{-1}[\/latex]<\/p>\r\n odd function<\/strong><\/p>\r\n a function whose graph is unchanged by combined horizontal and vertical reflection, satisfying [latex]f(x) = -f(-x)[\/latex], and is symmetric about the origin<\/p>\r\n one-to-one function<\/strong><\/p>\r\n a function where each output value corresponds to exactly one input value; passes the horizontal line test<\/p>\r\n vertical compression<\/strong><\/p>\r\n a transformation that compresses a function's graph vertically by multiplying the output by a constant [latex]0 < a < 1[\/latex]<\/p>\r\n vertical reflection<\/strong><\/p>\r\n a transformation that reflects a function's graph across the x-axis by multiplying the output by [latex]-1[\/latex]<\/p>\r\n vertical shift<\/strong><\/p>\r\n a transformation that shifts a function's graph up or down by adding a constant to the output<\/p>\r\n vertical stretch<\/strong><\/p>\r\n a transformation that stretches a function's graph vertically by multiplying the output by a constant [latex]a > 1[\/latex]<\/p>","rendered":" Function composition<\/strong> is combining two functions so that the output of one function becomes the input of another. Think of it like a two-step process: the first function processes the input, then the second function processes that result.<\/p>\n The composition of functions [latex]f[\/latex] and [latex]g[\/latex] is written as [latex](f \\circ g)(x) = f(g(x))[\/latex], read as “f of g of x” or “f composed with g at x.”<\/p>\n In general, [latex](f \\circ g)(x) \\neq (g \\circ f)(x)[\/latex]. The order in which you compose functions matters because you’re applying different operations in different sequences.<\/p>\n Evaluating Composite Functions<\/strong><\/p>\n Domain of Composite Functions<\/strong> The domain of [latex](f \\circ g)(x)[\/latex] consists of all [latex]x[\/latex] values that satisfy both:<\/p>\n Decomposing Functions<\/strong> Sometimes you need to break down a complex function into simpler parts. Look for a “function inside a function” pattern. There may be multiple ways to decompose the same function.<\/p>\n Types of Transformations<\/strong><\/p>\n Vertical Shifts:<\/strong> [latex]g(x) = f(x) + k[\/latex]<\/p>\n Horizontal Shifts:<\/strong> [latex]g(x) = f(x - h)[\/latex]<\/p>\n Vertical Reflections:<\/strong> [latex]g(x) = -f(x)[\/latex]<\/p>\n Horizontal Reflections:<\/strong> [latex]g(x) = f(-x)[\/latex]<\/p>\n Vertical Stretches and Compressions:<\/strong> [latex]g(x) = a \\cdot f(x)[\/latex]<\/p>\n Horizontal Stretches and Compressions:<\/strong> [latex]g(x) = f(bx)[\/latex]<\/p>\n Even and Odd Functions<\/strong><\/p>\n Order of Transformations<\/strong> When applying multiple transformations to [latex]y = a \\cdot f(b(x - h)) + k[\/latex], apply in this order:<\/p>\n An inverse function<\/strong> reverses the operation of the original function. If [latex]f[\/latex] takes input [latex]a[\/latex] to output [latex]b[\/latex], then [latex]f^{-1}[\/latex] takes input [latex]b[\/latex] back to output [latex]a[\/latex].<\/p>\n The inverse of [latex]f(x)[\/latex] is written [latex]f^{-1}(x)[\/latex], read as “f inverse of x.” Note: This is NOT the same as [latex]\\frac{1}{f(x)}[\/latex].<\/p>\n A function must be one-to-one<\/strong> (pass the horizontal line test) to have an inverse. If a function is not one-to-one over its entire domain, you may be able to restrict the domain to make it one-to-one.<\/p>\n Two functions [latex]f[\/latex] and [latex]g[\/latex] are inverses if both of these are true:<\/p>\n Domain and Range of Inverses<\/strong><\/p>\n Finding Inverse Functions<\/strong><\/p>\n Graphing Inverse Functions<\/strong> The graph of [latex]f^{-1}(x)[\/latex] is the reflection of the graph of [latex]f(x)[\/latex] across the line [latex]y = x[\/latex]. If point [latex](a, b)[\/latex] is on the graph of [latex]f[\/latex], then point [latex](b, a)[\/latex] is on the graph of [latex]f^{-1}[\/latex].<\/p>\n composite function<\/strong><\/p>\n the new function formed by function composition, when the output of one function is used as the input of another<\/p>\n even function<\/strong><\/p>\n a function whose graph is unchanged by horizontal reflection, satisfying [latex]f(x) = f(-x)[\/latex], and is symmetric about the y-axis<\/p>\n horizontal compression<\/strong><\/p>\n a transformation that compresses a function’s graph horizontally by multiplying the input by a constant [latex]b > 1[\/latex]<\/p>\n horizontal reflection<\/strong><\/p>\n a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]-1[\/latex]<\/p>\n horizontal shift<\/strong><\/p>\n a transformation that shifts a function’s graph left or right by adding a constant to the input<\/p>\n horizontal stretch<\/strong><\/p>\n a transformation that stretches a function’s graph horizontally by multiplying the input by a constant [latex]0 < b < 1[\/latex]<\/p>\n inverse function<\/strong><\/p>\n for any one-to-one function [latex]f(x)[\/latex], the inverse is a function [latex]f^{-1}(x)[\/latex] such that [latex]f^{-1}(f(x)) = x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]; this also implies that [latex]f(f^{-1}(x)) = x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f^{-1}[\/latex]<\/p>\n odd function<\/strong><\/p>\n a function whose graph is unchanged by combined horizontal and vertical reflection, satisfying [latex]f(x) = -f(-x)[\/latex], and is symmetric about the origin<\/p>\n one-to-one function<\/strong><\/p>\n a function where each output value corresponds to exactly one input value; passes the horizontal line test<\/p>\n vertical compression<\/strong><\/p>\n a transformation that compresses a function’s graph vertically by multiplying the output by a constant [latex]0 < a < 1[\/latex]<\/p>\n vertical reflection<\/strong><\/p>\n a transformation that reflects a function’s graph across the x-axis by multiplying the output by [latex]-1[\/latex]<\/p>\n vertical shift<\/strong><\/p>\n a transformation that shifts a function’s graph up or down by adding a constant to the output<\/p>\n vertical stretch<\/strong><\/p>\n a transformation that stretches a function’s graph vertically by multiplying the output by a constant [latex]a > 1[\/latex]<\/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":498,"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\/677"}],"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\/677\/revisions"}],"predecessor-version":[{"id":5089,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/677\/revisions\/5089"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/498"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/677\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=677"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=677"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=677"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=677"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\r\n \t
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Transformation of Functions<\/h3>\r\n
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Inverse Functions<\/h3>\r\n
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Glossary<\/h2>\r\n
Essential Concepts<\/h2>\n
Composition of Functions<\/h3>\n
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Transformation of Functions<\/h3>\n
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Inverse Functions<\/h3>\n
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Glossary<\/h2>\n