Find the inverse function.

g(x)=(e^x)/(1+2e^x)

The answer is:

ln(x/(1-2x))

How?

You essentially have y = (e^x)/(1+2e^x).

To find the inverse, you switch the x and y and solve for y.

Switching x and y gives you

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

So g^-1(x) = ln(x/(1-2x))