Question

How would you find the inverse of (e^x)/(1+e^x)?

Answer 1

Solve for x:
y = e^x/(1 + e^x)
(1 + e^x) * y = e^x
ye^x + y = e^x
y = e^x - ye^x
y = e^x( 1 - y)
y/(1-y) = e^x
ln(y/(1-y)) = x
swap the variables around to make it look more consistent:
g(x) = ln(x/(1-x))
f(x) = e^x/(1+e^x)

g(x) is the inverse of f(x)