Inverse problem, please help

Does anyone know the inverse of:

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

IF POSSIBLE: A step by step explanation would be better that just telling me the answer.

Multiplying:

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

y + 2y e^x = e^x

y = e^x - 2y e^x

y = (1-2y) e^x

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

x = ln [ y / (1-2y) ]

so the inverse is:

y = ln [ x / (1-2x) ]

Switch x and y, then solve for y

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