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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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].

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