Definition:Real Function/Two Variables/Substitution for y
Definition
Let $S, T \subseteq \R$ be subsets of the set of real numbers $\R$.
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: $\map f {x, y} = x^2 + x y^2 + 5 y + 3$
Let $\map f {x, y}$ be the real function of $2$ variables defined as:
- $\forall \tuple {x, y} \in \R^2: \map f {x, y} := x^2 + x y^2 + 5 y + 3$
Substituting $2$ for $y$
Let $2$ be substituted for $y$ in $\map f {x, y}$.
Then:
\(\ds \forall x \in \R: \, \) | \(\ds \map f {x, 2}\) | \(=\) | \(\ds x^2 + x \times 2^2 + 5 \times 2 + 3\) | |||||||||||
\(\ds \) | \(=\) | \(\ds x^2 + 4 x + 13\) | simplifying |
Substituting $a$ for $y$
Let $a$ be substituted for $y$ in $\map f {x, y}$.
Then:
\(\ds \forall x \in \R: \, \) | \(\ds \map f {x, 2}\) | \(=\) | \(\ds x^2 + x \times a^2 + 5 \times a + 3\) | |||||||||||
\(\ds \) | \(=\) | \(\ds x^2 + a^2 x + 5 a + 3\) | simplifying |
Substituting $\map g x$ for $y$
Let $\map g x$ be substituted for $y$ in $\map f {x, y}$, where $\map g x$ is a real function defined on all $\R$.
Then:
\(\ds \forall x \in \R: \, \) | \(\ds \map f {x, \map g x}\) | \(=\) | \(\ds x^2 + x \times \paren {\map g x}^2 + 5 \map g x + 3\) |
Example: $\map f {x, y} = x + y$
Let $\map f {x, y}$ be the real function of $2$ variables defined on the domain $S \times T$ as:
- $\forall \tuple {x, y} \in S \times T: \map f {x, y} := x + y$
where $S$ and $T$ are the closed real intervals:
\(\ds S\) | \(:=\) | \(\ds \closedint {-1} 1\) | ||||||||||||
\(\ds T\) | \(:=\) | \(\ds \closedint 0 2\) |
Substituting $\dfrac 1 2$ for $y$
Let $\dfrac 1 2$ be substituted for $y$ in $\map f {x, y}$.
Then:
\(\ds \forall x \in \closedint {-1} 1: \, \) | \(\ds \map f {x, \dfrac 1 2}\) | \(=\) | \(\ds x + \dfrac 1 2\) |
Substituting $3$ for $y$
Let $3$ be substituted for $y$ in $\map f {x, y}$.
Then $\map f {x, 3}$ is undefined, as $3 \notin \closedint 1 2$.
Sources
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Definition $2.65$