Question

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

Answer 1

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)`