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 $f(x)^{f^{-1}(x)} > x$ 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 $f$ ?

Edit I suspect such trigonometric function would be works for the titled function,like $\tan x$ as shown here but this need a restriction of our Domain of definitions for which $\tan x$ to be defined