Graphical interpretation of $f(z)=\sqrt{9-z^2}$ at purely imaginary numbers.

Question

Consider the function $f(z)=\sqrt{9-z^2}$. When $z\in\mathbb{R}$, the graph of the function is:

When $z$ is purely imaginary, such as $z=5i$, how can we interpret the function graphically? I assume that it is possible to take such a domain of $f$ since $(9-z^2)\geq0$ when $z$ is purely imaginary.

Please let me know if my question or assumption is wrong.

Answer 1

Hint: if $z=it$ with $t \in \mathbb R$, then $f(it)=\sqrt{9+t^2}$.