Inverse function of $x^n+e^x$

Question

I have seen posts on the inverse function of $x+e^x$ (which is $y-W(e^y)$), but (out of curiosity) how do we derive an inverse function for $x^n+e^x$ in terms of the Lambert W function? I tried to do what @MarkViola did here but I couldn't.

Answer 1

$x^n+e^x$ is not always invertible, for example when $n=2$. Therefore, your expression does not always have an inverse.

Certain values of $n$ will give elementary inverses, for example $n=0$ has the simple inverse $\log(x-1)$, and $n=1$ has $x- W(e^x)$ as its inverse, as can be seen here. It is likely, though I could not prove it, there is no elementary way to create the inverse when $n=3$ using just the $W$ function.