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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.

tzamboiv
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Emma
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    You might see things more clearly if you pretend that $f$ is the exponential function, $f(t)=\exp(t)=e^t$, reals to positive reals. Then in effect you want to show that $\ln(xy)=\ln x+\ln y$, except that you will use $f$ for $\exp$ and $f^{-1}$ for $\ln$. – André Nicolas Nov 18 '15 at 20:47
  • That is a very helpful way of thinking about it. Thank you! – Emma Nov 18 '15 at 23:01

2 Answers2

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

Tsemo Aristide
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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 $b$.

Ben Grossmann
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