How do you find the inverse of f(x) = e^x - e^-x?

1 Answer
Jan 30, 2016

Let y=f(x) and rearrange into a quadratic in e^x to find:

f^(-1)(y) = ln((y+sqrt(y^2+4))/2)

Explanation:

Let y = e^x-e^-x

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

So (e^x)^2-y(e^x)-1 = 0

Using the quadratic formula we find:

e^x = (y+-sqrt(y^2+4))/2

One of these roots is negative, requiring x to be non-Real.

So only the positive root is useful for e^x as a Real valued function of Real values of x:

x = ln((y+sqrt(y^2+4))/2)

That is:

f^(-1)(y) = ln((y+sqrt(y^2+4))/2)