Given the composition of functions $f(g(x))=h(x)$, solve for $f(x)$.

Question

I know that $f(x)○g(x)=f(g(x))$.

In the case that $$f(g(x))=h(x)$$

then $$f(x)○g(x)○g(x)^{-1}=h(x)○g(x)^{-1}$$ $$f(x)=h(x)○g(x)^{-1}$$ My question then is how do you then convert out of the composition notation. Would it then be $f(x)=h(g(x)^{-1})$ or maybe $f(x)=g(h(x))^{-1}$. Thanks.

Answer 1

The answer would be $f(x)=h\big(g^{-1}(x)\big)$. Of course, this is assuming that $g^{-1}(x)$ exists.

It's basically the same as $f(x)\circ g(x)=f(g(x))$. You use the rightmost function as the argument of the leftmost function.