How to build composite functions
A composite function is a “function of a function”
In a composite function, one function is used as a variable in the other function.
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A composite of the two functions ???f(x)??? and ???g(x)??? can be written as ???(f \circ g)(x)??? or also as ???f(g(x))???. It means to treat the function ???g(x)??? as the variable in ???f(x)???.
How to find composite functions, including three-function composites
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Building the composite from f(x) and g(x)
Example
Find the composite function ???(g \circ f )(x)???.
???g(x)=\frac{2}{x^4}???
???f(x)=\sqrt[4]{x-3}???
When we take the composite ???(g \circ f )(x)??? it means that we plug ???f(x)??? into ???g(x)???, which means that we take the entire ???f(x)??? function and plug it into ???g(x)??? wherever ???x??? exists in ???g(x)???.
???(g \circ f)(x)=g(f(x))=\frac{2}{\left(\sqrt[4]{x-3}\right)^4}???
???(g \circ f)(x)=g(f(x))=\frac{2}{x-3}???
Here’s another example.
In a composite function, one function is used as a variable in the other function.
Example
Find ???h(g(x))???.
???h(x)=3x^2-2???
???g(x)=x-4???
When we take the composite ???h(g(x))??? it means that we plug ???g(x)??? into ???h(x)???, which means that we take the entire ???g(x)??? function and plug it into ???h(x)??? wherever ???x??? exists in ???h(x)???.
???h(g(x))=3(x-4)^2-2???
???h(g(x))=3(x-4)(x-4)-2???
???h(g(x))=3(x^2-8x+16)-2???
???h(g(x))=3x^2-24x+48-2???
???h(g(x))=3x^2-24x+46???
