Consider the fucntion $f:\mathcal{R}\rightarrow \mathcal{R}$ given by the rule
$ f(x)=(1-x)e^{-x} $
Now I want to invert this function(not just for fun but I have a data that seems to fit this form). I could see that $x$ can' be isolated. Taking the log on both sides doesn't help and I have tried other possibilities. So I tried the following. Using Taylor expansion $e^{-x}=1-x-\frac{x^{2}}{2}+...$ Now $ f(x)=(1-x)(1-x-\frac{x^{2}}{2})=-\frac{1}{2}x^{3}+\frac{3}{2}x^{2}-2x+1 $
Now I could solve the cubic equation and solve for $x$. But this looks ugly and I don't know how good of an approximation it is. I was looking for possible suggestions to glean some information or write approximately a reasonable function for $x$ in terms of $y$ . Thank you