Does the inverse of a (monotonic) polynomial approxmiation of (monotonic) function $f(x)$ approximate the inverse of $f(x)$?

Question

Suppose $f(x)$ is monotonic and I approximate it with a polynomial. If I use some method (like in this link Inverse function of a polynomial) to get the inverse of that polynomial, should I expect that (perhaps approximated) inverse of the polynomial to approximate the inverse of the original function well?

I ask this because I have a non-closed form function f(x) and I want a closed-form inverse approximation of some kind. I'm wondering if there is some general result. Any suggestion for further reading is appreciated.

This seems to be somewhat similar: Numerical inverse of a function

Answer 1

This is an interesting problem! But for practical matters:

The data $(x_k,y_k)_{1\leq k\leq N}$ used to find the polynomial $x\mapsto p(x)$ approximating the underlying monotone function $x\mapsto y=f(x)$ can be used directly to find a polynomial $y\mapsto x=q(y)$ approximating $f^{-1}$.