# Composite Function

Let a function $f$ be $f:x\rightarrow x-3$ and a function $g$ such that $g \circ f : x \rightarrow 4x^2 - 20x + 25$. What is the function $g(x)$ ?

Solution :

To tackle problems involving composite functions, we have to clear other functions by using the property

$f\circ f^{-1} : x \rightarrow x$

\begin{aligned} g \circ f &: x \rightarrow 4x^2 - 20x + 25 \\ g \circ f &: x \rightarrow (2x-5)^2 \\ g \circ f \circ f^{-1} &: x \rightarrow \big [2f^{-1}(x) - 5\big ]^2 \end{aligned}

It’s clear that $f^{-1}(x) = x + 3$. Substitute into the expression we have

$g:x\rightarrow (2x+1)^2$