x=\frac{e^y+1}{e^y-1} \\
x \cdot (e^y-1) = e^y+1 \\
x \cdot e^y - x = e^y+1 \\
x\cdot e^y - e^y = x+1 \\
e^y \cdot (x-1) = x+1 \\
e^y = \frac{x+1}{x-1} \\
ln e^y = ln \frac{x+1}{x-1} \\
y = ln \frac{x+1}{x-1} \\
f^{-1}(x) = ln \frac{x+1}{x-1}