Inverse function of y=x|x|e^x

Question

Hello, so I need to find the inverse function to y=x|x|e^x. 

 

I've never seen such a difficult function to inverse. I thought I could 'split' it and then put together. 

so: x=y, x=lny and x=y or x=-y. All together it would be:

 

y=x*x*ln y (for x>0) or y=x*x*ln y (for x<0). 

So x = sqrt(y/lny) or x= - sqrt (y/lny).

 

Is this the right way of solving this problem?

 

Thank you all in advance!

Answer 1

Josh, There is no algebraic way I know of to invert the given expression to get y = f^-1(x). You can however find an inverse, at least over perhaps a limited range of x.  First, in the x-y plane draw or graph the function y = f(x) = x|x|e^x.  Next, draw the line y=x. Now draw or graph the mirror image of f(x) with respect to the line y=x. The inverse will exist in the region where the inverse function you have drawn is single-valued and it should have the shape x = y|y|e^{y ,}  Note that sometimes double-valued regions of the inverse function may be separated into separate branches that apply to different regions of x, but perhaps not in this case. Check it out!

 

Charles