Example 3. Find the domain of the function $z=\sqrt{x y}$.

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Solution. The function is defined when the expression under the root is non-negative, i.e., $x y \geq 0$. This is possible in two cases: 1) $x \geq 0, y \geq 0 ; 2) x \leq 0, y \leq 0$. The first condition is satisfied by the coordinates of all points lying in the first quadrant and on the coordinate axes, the second by the coordinates of points lying in the third quadrant and on the coordinate axes. Therefore, the domain of definition is the set of points located in the first and third coordinate angles and on the coordinate axes.