Let $f$ be a function such that   :$f:\mathbb{R}\to \mathbb{R}$  and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional  $f(x)^{f^{-1}(x)}=x^2$, I took $f(x)=x$ but it doesn't work it coincide only for $x=1$ and for $f(x)=\exp(x)$ I come up to $x^x=x^2$ then coincide only for $x=1$ and $x=2$ , so $f(x)^{f^{-1}} > x$ for $x >2$ which means no trivial solution exists probably  a formel power series exist arround $x=1$ or $x=2$,Then my question here is : How I can solve  $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$  is a compositional inverse of $f$ ?