How to find the inverse of an expotential

Question

Please tell me how to find the solution for this.

$$y= 3 + x + e^{x}$$

The range or the domain is not given. I'm just asked to find the inverse function of this.

Answer 1

Since $y'=1+e^{x}>0$, then $y$ is monotonically increasing on all of $\mathbb{R}$. That's why they don't state a domain and range-because the inverse function is defined on all of $\mathbb{R}$. As others have alluded to, there isn't a solution in terms of elementary functions. However, the following Mathematica commands get the inverse function in terms of the Lambert W function: \begin{align*} &f[x\_]:=3+x+\mathrm{Exp}[x]; \\ &\mathrm{InverseFunction}[f] \end{align*} returns $f^{-1}(x) = -3-W(\exp(x-3))+x$. (Mathematica calls the W function "$\rm{ProductLog}$") I'm a bit surprised that any homework problem would require you to pull out the Lambert W function. In any case, you can use the fact that $z=W(z)e^{W(z)}$ to verify that this is indeed the inverse function.

I have no clue how one would have gotten this result without a computer algebra system.