As far as I know, real polynomials of degree $\geq 5$ are known to be non-solvable. This implies that there are, in general, major algebraic difficulties in finding their roots and, provided they admit an inverse in some subspace of $\mathbb{R}$, one cannot find this inverse.
However, it is easy to use some computer software and plot a polynomial like this in the real plane. Provided we know regions in which this polynomial has an inverse, one would expect to obtain the plots of these inverses by a reflection on the line $y=x$.
My question is the following: Is there a way to obtain a closed formula for the inverse polynomial directly from its plot?