On my calculus text we were faced with the problem to find an inverse to $xe^x$ (that is, to give an explicit formula for the inverse and calculate its derivative and antiderivative).
When browsing stack exchange I found that the right answer is the Lambert $W$-function. However, our calc course never mentioned anything about the $W$-function. How would you go about solving this problem without a rigorous understanding of the Lambert $W$-function?
On
The function $xe^x$ is monotonically increasing for all $y=xe^x\ge0$ it is therefore one to one and is invertible there (the inverse being the Lambert W function already mentioned). For $-e^{-1}\lt xe^x<0$ the inverse is double valued. For example, the equation $xe^x=-0.1$ has 2 solutions $x=-3.5771..$ and $x=-0.1118...$. For $xe^x<-e^{-1}$ it is not invertible. These features can be easily seen from the graph of $xe^x$