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

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

Then the domain of $z$ is:
 * $\Dom z = \closedint {-1} 1 \times \R$

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

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:
 * $1 - x^2 \ge 0$

So for $z$ to be defined, it must be that $-1 \le x \le 1$.

There are no such restrictions on the value of $y$.

Hence the result.