Real Function of Two Variables/Examples/Root of x^2 + y^2 - 25

Examples of Real Functions of Two Variables
Let $z$ denote the function defined as:
 * $z = \sqrt {x^2 + y^2 - 25}$

The domain of $z$ is:
 * $\Dom z = C$

where $C$ consists of the set of points outside and on the circumference of the circle of radius $5$ whose center is at $\tuple {0, 0}$ in the Cartesian plane.

Proof
The domain of $z$ is given implicitly and conventionally.

What is meant is:
 * $z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
 * $\forall \tuple {x, y} \in S: \map z {x, y} = \sqrt {x^2 + y^2 - 25}$

From Domain of Real Square Root Function, in order for the real square root function to be defined, its argument must be non-negative.

Hence for $z$ to be defined, it is necessary for:

From Equation of Circle center Origin, $x^2 + y^2 = 25$ is the equation for the circle of radius $5$ whose center is at $\tuple {0, 0}$ in the Cartesian plane.

Points inside this circle correspond are such that $x^2 + y^2 < 25$.

Hence the domain of $z$ is the set of points consisting of the exterior of that circle and the points on its circumference.

Hence the result.