Inverse of function with two x's

Question

So I'm trying to take the inverse of this function

$$y = x + e^x$$

I've tried switching the x's and y's and then isolate the y using a natural log. But, I can't figure out how to get the y isolated after the switch and natural log. Thank you!

Answer 1

Let $f(x) = e^x + x$. Then $f'(x) = e^x + 1 > 0$ which means that $f$ is strictly increasing and thus injective. Is it surjective? Well, $\lim_{x\to \pm\infty}f(x) = \pm\infty$, so, yes, by intermediate value theorem. So, we know that $f$ is invertible. Trouble is that the inverse can't be expressed in terms of elementary functions. We need Lambert $W$ function (do read examples on wiki, they are quite instructive).

So, \begin{align} x+e^x = y &\implies xe^{-x}+1 = ye^{-x}\\ &\implies (y-x)e^{-x} = 1\\ &\implies (y-x)e^{y-x} = e^y\quad \small\text{(apply Lambert $W$)}\\ &\implies y-x = W(e^y)\\ &\implies x = y - W(e^y) \end{align} Hence, $f^{-1}(x) = x-W(e^x)$.

Lambert $W$ is not an elementary function and one should use numerical methods to compute particular values, but then again, it's the same for $\sin$ or $\ln$.