Finding the inverse of x^(e/x)

Question

While messing around on my graphing calculator, I stumbled upon this function:

$$f(x)=x^{\frac{e}{x}}$$

I was unable to find the inverse of this function using online calculators. What is the inverse of this function, and how do I go about finding it?

Answer 1

$$f^{-1}(y) = - \dfrac{e W(-\ln(y)/e)}{\ln(y)}$$ where $W$ is the Lambert W function.