Inverse of $f(x) = xe^x-x$ I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal altogether. I have plotted the function and the inverse looks close to the W function. 
This is related to a problem I'm working on in which I need to find the density of the random variable $Y=Xe^X-X$, where I know the density of X. I am aware that this function is not one-to-one. For my purposes, we can restrict to $x>0$. Any help would be greatly appreciated!!
 A: This equation can be placed in the form:
$$e^x=\frac{x+f}{x}$$
We know that mixed exponential/bilinear equations can be solved by the extended Lambert function $W_r(x)$, which can be represented by the following Lagrange inverting series:
$$z(A,t,s)=t- (t-s) \sum_{n=1} \frac{L_n' (n(t-s))}{n} e^{-nt} A^n$$
which is the solution of:
$$e^z=A\frac{z-t}{z-s}$$
Related questions 
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$$e^z=A\frac{z-t}{z-s}$$
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References
On the generalization of the Lambert $W$ function with applications in theoretical physics, http://arxiv.org/abs/1408.3999
[68] C. E. Siewert and E. E. Burniston, "Solutions of the Equation $ze^z=a(z+b)$," Journal of Mathematical Analysis and Applications, 46 (1974) 329-337. http://www4.ncsu.edu/~ces/pdfversions/68.pdf
