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:

$z = y \sqrt {1 - x^2}$

The domain of $z$ is:

$\Dom z = \closedint {-1} 1 \times \R$

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:

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

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

$\ds 1 - x^2$

$\ge$

$\ds 0$

$\ds \leadsto \ \ $

$\ds x^2$

$\le$

$\ds 1$

$\ds \leadsto \ \ $

$\ds \size x$

$\le$

$\ds 1$

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.

$\blacksquare$

1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables