Example 2. Find the domain of the function $x^{2}+y^{2}+z^{2}=9$.
Solution. Solving this equation for $z$, we obtain two functions

$$
z= \pm \sqrt{9-x^{2}-y^{2}}
$$

These functions are defined when the expression under the square root is non-negative, i.e.,

$$
9-x^{2}-y^{2} \geq 0 \text { or } x^{2}+y^{2} \leq 9
$$

The last inequality is satisfied by the coordinates of all points lying inside and on the boundary of a circle of radius $R=3$ centered at the origin. Thus, the domain of these functions will be a circle of radius $R=3$. (The functions themselves represent a sphere of radius $R=3$ centered at the origin.)