Inverse Logistic Function / Reverse Sigmoid Function

Question

I am currently coding up a fuzzy logic library in java. I have found the equations for all the standard functions - Grade, inverseGrade, Triangle, Trapezoid, Gaussian. However, I can't find the inverse of the sigmoid/ logistic function.

The way I have written the logistic function is java is :

//f(x) = 1/(1+e(-x)) public double logistic(double x){     return (1/(1+(Math.exp(-x)));  } 

But I can't work out or find the inverse anywhere. My algebraic/calculus abilities are fairly limited, hence why I haven't been able to work out the inverse of the function.

Any hints or pointers would be a big help.

Thanks

Answer 1

If

y = 1/(1+exp(-x)) 

then

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

Answer 2

Just to go through the steps:

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