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I am having problems finding the inverse function of a complicated function. In this case: $$y=x|x|e^x $$

I thought I could 'split' this function but I'm not sure if that's the right way.
for $y=x$ it would be $x=y$
for $y=e^x$ it would be $x=\ln(y)$
for $y=|x|$ we need to split it in 2 cases: if $x<0, y=-x,$ if $x>0, y=x $

So all together it would be:
for $x<0$:
$$y = x \cdot (-x) \cdot \ln y $$ $$-x^2= y/\ln y$$ $$x = - \sqrt{\frac{y}{\ln y}} $$

For $x>0$:
$$y = x \cdot x \cdot ln y $$ $$x^2 = y/\ln y $$ $$x = \sqrt{\frac{y}{\ln y}} $$

Is this the right way of solving this case? I would be grateful for any help

Chill2Macht
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sophie
  • 11

1 Answers1

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The Lambert W function is defined as the inverse of $f(x) = xe^x$.

If $y=x^2e^x$, then $$\frac 12\sqrt y = \frac x2e^{\frac{x}{2}}$$ so that $x =2 W(\frac12 \sqrt y)$. Changing variables, our inverse is

$$f^{-1}(x) = 2 W(\frac12 \sqrt x)$$

Of course, the above only solves your problem for $x \geq 0$. I'm sure you can modify it for $x<0$.

MathematicsStudent1122
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  • Thank you for your answer. It's really helpful but I'm afraid the Lambert W function wasn't mentioned so I hoped there would be other way of solving it. Thank you anyway – sophie Jun 05 '16 at 17:18