Find a formula for the inverse of the function f(x)=(1+e^x)/(1-e^x)

You can find the inverse function for
.. y = (1 +e^x)/(1 -e^x)
by swapping the variables and solving for y.
.. x = (1 +e^y)/(1 -e^y)
.. x(1 -e^y) = 1 +e^y
.. x -1 = e^y*(1 +x)
.. e^y = (x -1)/(x +1)
.. y = ln((x -1)/(x +1)) . . . . . your inverse function

See this link for a graph.
https://www.desmos.com/calculator/r0lcd1veps

The orange dashed line is there to help you see the inverse function is the original function reflected across the line y=x.