Algebra 2 question

$\text{Find the invers of }f(x)=x*|x|$

If x>=0  then  f(x)=x^2  which is a very simple  half concave up parabola in the first quadrant

if X < 0  then  f(x)=-x^2  which is a very simple  half concave down parabola in the third quadrant

To be honest, the first thing I did was sketch this. The inverse is the refection in the line y=x so I can already see where this is going.

But lets look at the algebra

let   y= f(x)  )it is just easier to work with)

If x >=0     then y is also positive

y=x^2

x=+sqrt(y)

so

f^-1(x)=+sqrt(x)

If  x<0   then y is also negative

x=-sqrt(x)

so

f^-1(x)=-sqrt(|x|)

The tricky bit was how to put these together into one function   

|x| / x = 1    if x>0        and    |x| / x = -1    if x<0     so

$f^{-1}(x)=\frac{|x|*\sqrt{|x|}}{x}$  

check: