Consider a function $f(x)$ such as $x\mapsto 2e^x-\frac1{e^x}$. How do you find $f^{-1}(x)$?
I have tried, logarithms, squaring, substitution, but I wasn't able to isolate $x$. The correct answer, according to Wolfram Alpha is $f^{-1}(x) = \log{\left(\frac14\left(x+\sqrt{x^2+8}\right)\right)}$.