Suppose I have a curve $f$ in $\mathbb{R}^2$, the implicit function theorem guarantees the existence of a smooth local inverse of this function $f$.
Question: My question is is there a way to explicitly find the form of these local inverses?
Particular Case: Particularly I'm looking for the local inverses of the curve drawn-out b the function: \begin{equation} f(x):=a_1xe^{-b_1x}+a_2x^2e^{-b_2x}+...+a_nx^ne^{-b_nx} \end{equation} where $a_1,..,a_n \in \mathbb{R}$ and $b_1,..,b_n \in (0,\infty)$.