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 $g(x)$. 

$(g \circ f)(x)=g(f(x))=\frac{2}{\left(\sqrt[4]{x-3}\right)^4}$

$(g \circ f)(x)=g(f(x))=\frac{2}{x-3}$

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Here’s another example.

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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 $h(x)$.

$h(g(x))=3(x-4)^2-2$

$h(g(x))=3(x-4)(x-4)-2$

$h(g(x))=3(x^2-8x+16)-2$

$h(g(x))=3x^2-24x+48-2$

$h(g(x))=3x^2-24x+46$

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