inverse of x*e^x

We are asked to find the inverse of xe^x :
One way to find an inverse is to start with the equation y=f(x); then exchange x and y and solve the result for y.
y=xe^x
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x=ye^y
To get rid of the exponent, take the natural logarithm of each side:
lnx=ln(ye^y)
Use properties of logarithms to simplify the right hand side:
lnx=lny+y
Note that we cannot write this as a function of x using elementary functions.
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The inverse  is called the Lambert W-function. If we assume that x is real and x>-1 we get a single valued function. Without this restriction, y=xe^x does not have an inverse function as it fails the horizontal line test.
The graph of y=xe^x :

We can draw the inverse relation by reflecting the graph over the line y=x.