What are the domain and the range of 2 variables function? f(x,y)=square root(9-x^2-y^2)

The domain of the function has to contain the values of
the variables that make the function to exist.

In this
case, because of the constraint that the radicand has to be positive or at least zero,
we'll get the domain of the function:

D = {(x,y) /9-x^2-y^2
>=0}

D = {(x,y) /x^2 + y^2 >=
9}

The domain is represented by the disc whose center is
the origin of the coordinates system and the radius is
3.

We'll determine the
range:

z = {z/z = sqrt(9-x^2-y^2), (x,y) belongs to
D}

Since z>=0 and  9-x^2-y^2 =< 9 =>
sqrt(9-x^2-y^2)=<3

The range of the function is the
closed interval [0,3].

The domain of the
function is the disc whose center is the origin of the coordinates system and the radius
is 3 andthe range is the closed interval [0,3].