# Definition:Real Function/Two Variables

## Definition

Let $S, T \subseteq \R$ be subsets of the set of real numbers $\R$.

Let $f: S \times T \to \R$ be a mapping.

Then $f$ is defined as a (real) function of two (independent) variables.

The expression:

$z = \map f {x, y}$

means:

(The dependent variable) $z$ is a function of (the independent variables) $x$ and $y$.

### Substitution for $y$

Let $f: S \times T \to \R$ be a (real) function of two variables:

$z = \map f {x, y}$

Then:

$\map f {x, a}$

means the real function of $x$ obtained by replacing the independent variable $y$ with $a$.

In this context, $a$ can be:

a real constant such that $a \in T$
a real function $\map g x$ whose range is a subset of $T$.

## Examples

### Example: $y \sqrt {1 - x^2}$

Let $z$ denote the function defined as:

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

The domain of $z$ is:

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

### Example: $\dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$

Let $z$ denote the function defined as:

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

The domain of $z$ is:

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