Example 3.20. Find the domain of the function

$$
z=\frac{1}{\sqrt{4-x^{2}-y^{2}}}
$$
Solution. The function has real values if $4-x^{2}-$ $-y^{2}>0$ or $x^{2}+y^{2}<4$. The latter inequality is satisfied by the coordinates of points lying inside a circle of radius 2 centered at the origin. The domain of existence of the function is the set of points inside this circle (Fig. 44).

![](https://cdn.mathpix.com/cropped/2024_05_22_e915d089cc29fd681be5g-169.jpg?height=444&width=530&top_left_y=308&top_left_x=447)

Fig. 44