This function has domain
[tex]D=\mathbb{R}[/tex]
but it can be inverted only in one of the two sub-domains
[tex]D_1=\{x \in \mathbb{R}\ :\ x\geq0\},\quad D_2=\{x \in \mathbb{R}\ :\ x\leq0\}[/tex]
In fact, if we invert the function we have
[tex]y=x^2-9\mapsto x=y^2-9 \iff y^2=x+9 \iff y=\pm\sqrt{x+9}[/tex]
And you choose the [tex]\pm[/tex] sign depending on which subdomain you're in.
