What is Increasing, decreasing? — Mathematics

Definition 3.8 High School Mathematics · Chapter 3 — Functions

Let $f$ be defined on an interval $I$ .

$f$ is increasing on $I$ when for all $u, v \in I$ , if $u < v$ then $f(u) \leq f(v)$ : the outputs grow with the inputs, and the graph climbs from left to right.
$f$ is decreasing on $I$ when $u < v$ implies $f(u) \geq f(v)$ : the graph falls from left to right.

With strict inequalities $f(u) < f(v)$ (resp. $>$ ) we say strictly increasing (resp. decreasing).

Examples

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

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

Example 3.11 (Proving a variation)

Show that $f(x) = 3x + 1$ is strictly increasing on $\R$ . Take any $u < v$ and compare the images:

f(v) - f(u) = (3v + 1) - (3u + 1) = 3(v - u) > 0,

because $v - u > 0$ . So $f(u) < f(v)$ : the function is strictly increasing. The same computation with a negative slope, e.g. $g(x) = -2x + 5$ , gives $g(v) - g(u) = -2(v-u) < 0$ : strictly decreasing.

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Definition 11.4 High School Mathematics · Chapter 11 — Functions and Variations

Let $f$ be defined on an interval $I$ . The function $f$ is increasing on $I$ if it preserves order:

\text{for all } u, v \in I: \quad u < v \implies f(u) \leq f(v),

and decreasing if it reverses order ( $u < v \implies f(u) \geq f(v)$ ). With strict inequalities on the right, $f$ is strictly increasing (resp. decreasing). A function that is increasing or decreasing on $I$ is monotonic on $I$ .

Examples

Example 11.10

Study $f(x) = \sqrt{x^2 + 1}$ on $\R$ . The inner function $u(x) = x^2 + 1$ is positive, decreasing on $\intoc{-\infty}{0}$ and increasing on $\intco{0}{+\infty}$ (a shifted square). By point 3, $f$ has the same variations: decreasing then increasing, with minimum $f(0) = 1$ .

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Examples

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