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 $x=1$, seems there is a formel power series arround $x=1$ which I can't get it  ,Then my question here is : How I can solve  $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$  is a compositional inverse of $f$ ?