Explain why the range of function f(x,y)=square root[100-(x^2+y^2)] is [0;10].

Let's determine the range of 2 variables function. First,
we'll determine the domain of the function.

The domain of
the function has to contain the values of the variables thatsatisfy the function
.

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) /100-x^2-y^2
>=0}

D = {(x,y) /x^2 + y^2 >=
100}

The domain is represented by the disc whose center is
the origin of the coordinates system and the radius is
10.

We'll determine the
range:

z = {z/z = sqrt(100-x^2-y^2), (x,y) belongs to
D}

Since z>=0 and  100-x^2-y^2 =< 100
=> sqrt(100-x^2-y^2)=<10 (the values of the function could not overstep
the radius of the disc r = 10). That is why the range of the function is the closed
interval [0,10].

The the range of 2 varaible
function is indeed the closed interval [0,10].