I'm looking for the inverse function for $f(x)=x/(1-e^{-x})$, over the domain $x>0$.
Wolfram says that the answer is $f^{-1}(x)=x+W(-xe^{-x})$, for $x>1$, where $W(x)$ is the Lambert W function. Graphically this looks correct, but I'm struggling to verify it for myself on paper. I would greatly appreciate it if someone could provide the working for this. Thanks in advance!
Best Answer
Your formulation uses $x$ for both for the variable in $f(x)$ and for the different variable in $f^{-1}(x)$, which may cause confusion. Let's instead describe this as solving $y =\frac{x}{1-e^{-x}}$ for $x$.
The Lambert W function provides the solution(s) to $we^w =z$ with $w=W_n(z)$ which suggests here to get your desired solution you should be considering $z=-ye^{-y}$ and $w = x-y$. We now need to manipulate the original equation into a suitable form:
which we can now solve using the Lambert W function as
and this is the desired solution as given by Wolfram Alpha.