Q:

Inverse of homogenous function?

suppose that $\alpha>0$ and $f$ is an invertible function such that $f(\alpha x)=\alpha f(x)$. Does this automatically also imply that $f^{-1}(\alpha x)=\alpha f^{-1}(x)$? 
I would think yes because $ff^{-1}f(\alpha x)=f(\alpha x)$ but I am not sure this is sufficient to establish the claim.

A:

Yes it does automatically imply that $f^{-1}(\alpha x) = \alpha f^{-1}(x)$.  We can proceed as follows.  Taking the inverse of both side of the original equation gives:
$$f^{-1}(f(\alpha x)) = f^{-1}(\alpha f(x))$$
This evaluates to
$$\alpha x = f^{-1}(\alpha f(x))$$
Now if we define $y = f(x)$, then $x = f^{-1}(y)$ and
$$\alpha f^{-1}(y) = f^{-1}(\alpha y)$$

