What is the inverse function of $x^2$ when $x$ is negative?

Question

Since it is an injective and surjective function therefore it's inverse must exist.

NB: I know that the inverse function of $x^2$ when $x$ is positive is square root function.

Answer 1

The inverse is given by $\mathbb{R}_{\geq 0} \to \mathbb{R}_{\leq 0}$ where $x \mapsto -\sqrt{x}$.

Answer 2

The inverse of $(-\infty,0]\to [0,\infty)$, $x\mapsto x^2$ is the function $[0,\infty)\to(-\infty,0]$, $x\mapsto -\sqrt x$.