Inverse function

x = e^y/(1 + 2e^y)
x(1 + 2e^y) = e^y
x + 2xe^y = e^y
x = e^y(1 - 2x)
x/(1 - 2x) = e^y
so y = inverse function of f(x) = ln[x/(1 - 2x)]



2009-09-03 07:04:09 補充：
Note : (1 - 2x) must be greater than zero for y to be defined. That is x < 1/2.

2009-09-03 07:10:34 補充：
To be more precise, x should be : 0 < x < 1/2 for y to be defined.

Let y = f(x)

y = ex/(1 + 2ex)
y(1 + 2ex) = ex
y + 2yex = ex
ex - 2yex = y
ex(1 - 2y) = y
ex = y/(1 - 2y)
x = loge[y/(1 - 2y)]

Hence, f-1(x) = loge[x/(1 - 2x)]