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.