If f(x+y)=f(x)*f(y) and f is a bijection, show that its inverse satisfies this function equation

Question

I'm having trouble with this problem. I'm not even sure how to go about finding the inverse of an equation with both x and y.

Here is the problem:

If $f(x+y)=f(x)*f(y)$ and $f$ is a bijection, show that its inverse satisfies the functional equation:

$f^{-1}(xy)=f^{-1}x+f^{-1}(y)$

I appreciate any help.

Answer 1

Hint: Take $f^{-1}$ on both sides of the functional equation to find that $$ f^{-1}(f(x + y)) = f^{-1}(f(x)\cdot f(y)) $$ Note that if $f$ is a bijection, $x$ and $y$ can be written as $f^{-1}(a)$ and $f^{-1}(b)$ for some $a$ and $b$.

Answer 2

$f(f^{-1}(x)+f^{-1}(y))=f(f^{-1}(x))f(f^{-1}(y))=xy=f(f^{-1}(xy))$ since $f$ is bijective, $f^{-1}(x)+f^{-1}(y)=f^{-1}(xy)$