Question

find the inverse:

y= ln (x/(x-1))

Answer 1

Follow the steps to find the inverse of a simple function:

1. interchange x and y in
y=f(x)=expression in x
2. solve for y in terms of x, if possible
3. calculate f^-1(f(x))
If the inverse is correct, the result should be x.

1.
From y=ln(x/x-1)
interchange x and y to get
x=ln(y/(y-1))
2. solve for y in terms of x
e^x=y/(y-1)
f^-1(x)=y=e^y/(e^y-1)
3. calculate
f^-1(f(x))
=x/((x-1)*(x/(x-1)-1))
=x
Therefore:
f^-1(x) = e^y/(e^y-1)