Simplify x from x |x|

Question

I have that

$$ x |x| = -y, $$ Clearly the sign of x is determined by the sign of y. But then how to write what x is?What happens if y is discontinuous at 0?

Is it $$ x = sgn(x) \sqrt{ |y|} ?$$ or doesn't it make any sense?there is something I probably don't understand.

silly example $$ 5 |5| = - (-25)$$ x is positive when y negative and $$ - 5 |-5| = - (25)$$ x is negative when y positive.

Answer 1

I'm assuming you're asking this question in the case where $x$ and $y$ are real numbers rather than complex numbers.

It looks to me as if $$ x = -\operatorname{sgn}(y) \sqrt{|y|} $$ where you might have to do something when $y = 0$, depending on the definition of "sgn".

In other words, your conjectured answer was almost correct.

Answer 2

There are four solutions:

$x = \pm \sqrt{y}$ and $\pm i \sqrt{y}$.

Answer 3

$y = \begin{cases}x^2 & x\le 0\\-x^2 & x>0 \end{cases}$

and to invert this.

$x = \begin{cases} \sqrt {-y} & y\le 0\\ -\sqrt{y} & y> 0\end {cases}$

or $x = -\text{sgn}(y)\sqrt{|y|}$