Composite functions are formed by combining one function with another.
To form a composite, one function is substituted into the input of a second function.
In function notation, if \( g(x) \) and \( h(x) \) are two functions, then the composite of \(g\) and \(h\) is the function \(f(x)\) defined by
\(f(x) = (g \circ h)(x) = \textcolor{NavyBlue}{g(}\textcolor{BrickRed}{h(x)}\textcolor{NavyBlue}{)}\)
where \(\textcolor{NavyBlue}{g(x)}\) is the outer function and \(\textcolor{BrickRed}{h(x)}\) is the inner function. It is read either as \(g\) composed with \(h\) or as \(g\) of \(h(x)\).
For example, the function \( f(x) = (3x^2 - 7x)^5 \) may be written as the composition of two other functions.
It can be decomposed into two separate component functions: \( g(x) = x^5 \) and \( h(x) = 3x^2 - 7x \).
Note:
The order of the composition is important.
The function, \(f(x)\), is the composition of \(g\) and \(h\), which is different than the composition of \(h\) and \(g\).
The function \( h \) composed with \( g \) would be
\[\begin{align*} (h \circ g)(x) &= h(g(x)) \\& = 3(g(x))^2 - 7(g(x)) \\& = 3(x^5)^2 - 7(x^5) \\ & = 3x^{10} - 7x^5 \end{align*}\]