I have tried a couple of approaches to inversing it, but I really don't know any strategies or methods. I don't even know if it's possible or how to figure out if it is possible, so I'm asking how to know if you can inverse or not as well. $f(x)=x^\frac{1}{x-1}$
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1Have a look at the Lagrange inversion formula & also the Lambert $W$ function https://en.wikipedia.org/wiki/Lagrange_inversion_theorem – Donald Splutterwit Nov 20 '20 at 00:37
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There is a Lambert W solution. You can also use its $k$th branch for complex solutions – Тyma Gaidash Jul 04 '23 at 12:56