What is Function? — Mathematics

Definition 3.1 High School Mathematics · Chapter 3 — Functions

Let $D$ be a set of real numbers. A function $f$ defined on $D$ associates to each number $x \in D$ exactly one real number, written $f(x)$ and called the image of $x$ . The set $D$ is the domain of $f$ . We write

f : x \longmapsto f(x).

If $f(x) = y$ , then $x$ is a preimage of $y$ .

Examples

Example 3.3 (Finding a domain)

When a function is given by a formula, its domain is the set of $x$ for which the formula makes sense.

$f(x) = 3x^2 - 5x + 1$ makes sense for every $x$ : the domain is $\R$ .
$g(x) = \dfrac{1}{x - 2}$ requires $x - 2 \neq 0$ : the domain is all reals except $2$ .
$h(x) = \sqrt{x - 3}$ requires $x - 3 \geq 0$ : the domain is $\intco{3}{+\infty}$ .

Example 3.4 (Computing images and preimages)

Let $f(x) = x^2 - 4x + 3$ , defined on $\R$ .

Image of $5$ : substitute $x = 5$ : $f(5) = 25 - 20 + 3 = 8$ .

Preimages of $3$ : solve $f(x) = 3$ :

x^2 - 4x + 3 = 3 \quad\Longleftrightarrow\quad x^2 - 4x = 0 \quad\Longleftrightarrow\quad x(x - 4) = 0,

so the preimages of $3$ are $0$ and $4$ .

Example 3.10

Consider the function graphed below on $\intcc{-2}{4}$ .

A function defined on [-2, 4], increasing then decreasing. — A function defined on $\intcc{-2}{4}$ , increasing then decreasing.

Reading the graph: $f$ increases from $f(-2) = -1.5$ up to its maximum $f(1) = 3$ , then decreases down to $f(4) = -1.5$ . Its variation table is

$x$	$-2$		$1$		$4$
$f$	$-1.5$	$\nearrow$	$3$	$\searrow$	$-1.5$

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Examples

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