Find the inverse

Question

Find f^-1(x) =1 / (2 - e^(-x))

Answer 1

One method for solving the original function given an inverse is:

 

1) call f^-1(x) just x, and replace the other x's with y's

2) solve new equation for y

 

Doing step 1 gives

 

x = 1 / (2 - e^(-y))

Calling x now x/1 and cross multiplying

x•(2 - e^(-y)) = 1

Dividing by x

(2 - e^-(y)) = 1/x

Subtracting 2

-e^(-y) = 1/x - 2

Dividing by -1

e^(-y) = -1/x + 2

Taking the ln of both sides

ln(e^(-y)) = ln(2 - 1/x)

-y = ln(2 - 1/x)

And dividing by -1

y = -ln(2 - 1/x)