Question

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

Answer 1

Using the Lambert W function, we find that `f^-1(x) = -W(-1/x)`

Explanation:

A typical way of finding the inverse of a function `f(x)` is to set `y = f(x)` and then solve for `x`. The result will be `x = f^(-1)(y)`

(This method should make sense intuitively, as plugging in `f(x)` for `y` gives you the definition of the inverse function).

Trying that here, however, will lead to some frustration, as no basic function will be enough to solve for `x`. Instead, what is needed is a function called the Lambert W function. The Lambert W function works like so:
`W(xe^x) = x`

With this special function as a tool, we can now find the inverse.

Set `y = f(x) = e^x/x`

`=> xy = e^x`

`=> x/e^x = 1/y`

`=> xe^(-x) = 1/y`

`=> (-x)e^(-x) = -1/y`

`=> W((-x)e^(-x)) = W(-1/y)`

`=> -x = W(-1/y)`

`=>x = -W(-1/y)`

Thus we have `f^-1(y) = -W(-1/y)`

So our inverse function is

`f^-1(x) = -W(-1/x)`