I'm having problems with finding the inverse equation for this function [tex]e^x/(e^x-1)[/tex]
The Attempt at a Solution
Currently, I have done about 2 methods to solve it and hitting dead-ends on each one:
Attempt 1. [tex]x=e^y/(e^y-1)[/tex] then multiply both sides by [tex](e^y-1)[/tex] so that I have [tex]xe^y-x=e^y[/tex], then subtract [tex]xe^y[/tex] from both sides to then get [tex]x=e^y-xe^y[/tex] after which I factor out [tex]e^y[/tex] and then my final solution to this attempt is: [tex]f(x)^-1= ln(x/(1-X))[/tex]
Attempt 2. I took the reciprocal of the equation so I have [tex]1/x= (e^y+1)/e^y[/tex] then split up the fraction to [tex] 1/x= 1+(1/e^y)[/tex], then multiplied both sides by [tex] e^y[/tex] to get [tex] e^y/x = 1+1[/tex] then multiplied it by [tex] x [/tex] so that my solution was [tex]f(x)^-1= ln(2x) [/tex]
Obviously by looking at the final solutions neither of them actually work. I would really appreciate any enlightenment on this problem.