Real Function of Two Variables/Examples/Root of 1 minus y^2 over Root of 1 minus x^2

Examples of Real Functions of Two Variables
Let $z$ denote the function defined as:
 * $z = \dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$

The domain of $z$ is:
 * $\Dom z = \openint {-1} 1 \times \closedint {-1} 1$

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:
 * $\forall \tuple {x, y} \in S: \map z {x, y} = \dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$

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:

Similarly:
 * $\size x \le 1$

But because $\sqrt {1 - x^2}$ is the denominator of the, it cannot be zero.

So it follows that for $z$ to be defined, it must be the case that:
 * $-1 \le y \le 1$

and:
 * $-1 < x < 1$

Hence the result.