Could someone help me figure out how to find the inverse of $$f(x)=\frac{e^{x}-1}{|x|}$$ I know you can just switch $x$ and $y$, but how can you solve it for y then? And what would be the domain? I think maybe it is something simple that I'm overlooking. If you can help me it would be very much appreciated!
Thanks in advance
The inverse of $f(x)=\frac{e^x-1}{|x|}$ is $$f^{-1}(x)= \begin{cases} \frac{1}{x}-W\left(\frac{1}{x}e^{1/x}\right)&-1<x<0\\ -\frac{1}{x}-W_{-1}\left(-\frac{1}{x}e^{-1/x}\right)&x>1\\ \end{cases} $$
where $W$ is the Lambert W function.