Closed form of the inverse of a function

Question

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$?

Thanks in advance

Answer 1

As it is said in previous answers, the inverse function involves Lambert $W$ function, hence non-elementary. Here I try to add some lines on how the inverse is obtained. Note that for Lambert function we have $W(x)e^{W(x)}=x$. Follow the following manipulation for obtaining the inverse function $z=f^{-1}(x)$: $$ e^z+z=x \implies z=\log(x-z) \implies \log\left((x-z)e^{x-z}\right)=x, $$ the last equality means that $x-z=W(e^x)$.

Answer 2

The analytic inverse of this function does not exist as an elementary function. Sadly. As wolfram alpha tells us, the inverse is $$x-W(e^x)$$ where $W$ is the Lambert $W$ function which is not elementary.