Example 4. Find the domain of the function

$$
z=\frac{1}{\sqrt{9-x^{2}-y^{2}}}
$$
Solution. The function is defined when the expression under the square root is positive (unlike in the previous example, equality to zero is excluded here), i.e.

$$
9-x^{2}-y^{2}>0 \text { or } x^{2}+y^{2}<9 \text {. }
$$

The latter inequality is satisfied by points lying inside a circle of radius $R=3$ (boundary points are excluded). The domain of the function is the set of points lying inside a circle of radius $R=3$ centered at the origin.