What would be the inverse of the function $y=x\exp(-x)$ when $1\leq x\leq \infty$ and $0\leq y\leq 1/e$.
I tried to solve this with Lambert W function and the solution came out to be $-W(-y)$. But in this paper (just above section 2 Application) it is mentioned that the principle branch of the Lambert W function can not have a value lower than $-1$. Whereas for my problem the value of $W$ is starting from $-1$ and going down to $-\infty$.
So I am not sure whether my calculation is correct.