Many students know how to find the inverse of a regular function. We simply let $f(x) = y$ then proceed to make $x$ the subject. Some might not know that if you want to find the inverse of the composite function, knowing that $(fg)^{-1} = g^{-1}f^{-1}$ will be useful. Do note the switch in position of the functions.

At times, it might be really impossible to make out the inverse of $fg(x)$ by doing the regular $fg(x)=y$ and then proceeding to make $x$ the subject. Thus, this formula will come in handy.

The following is a simple proof. Assuming all the following composite functions exists.

Let $\phi(x)= (fg)^{-1}(x)$

Then $\phi^{-1}(x) = fg(x)$
$f^{-1} [\phi^{-1}(x)] = f^{-1}[fg(x)] = g(x)$

$g^{-1} \{f^{-1} [\phi^{-1}(x)]\} = g^{-1}g(x) = x$

We have that $g^{-1} f^{-1} \phi^{-1}(x) = x$

Considering $g^{-1} f^{-1} \phi^{-1}[\phi (x)] = \phi(x)$

$\therefore, g^{-1} f^{-1} (x) = \phi(x)$