Finding the inverse for a composite function

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)

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