Increasing and Decreasing Functions

A function is increasing when its output rises as the input grows, and decreasing when its output falls. These concepts are essential for understanding the shape of graphs, finding maxima and minima, and applying calculus effectively.

Quick Reference

Concept	Meaning
Increasing on $I$	$x_{1} < x_{2}$ in $I$ implies $f (x_{1}) < f (x_{2})$
Decreasing on $I$	$x_{1} < x_{2}$ in $I$ implies $f (x_{1}) > f (x_{2})$
Monotonic on $I$	Either increasing or decreasing throughout $I$
Interval of increase	An interval on which $f$ is increasing
Interval of decrease	An interval on which $f$ is decreasing

Definitions

Let $f (x)$ be defined on an interval $I$ .

$f$ is increasing on $I$ if for every $x_{1}, x_{2} \in I$ with $x_{1} < x_{2}$ , we have $f (x_{1}) < f (x_{2})$ .
$f$ is decreasing on $I$ if for every $x_{1}, x_{2} \in I$ with $x_{1} < x_{2}$ , we have $f (x_{1}) > f (x_{2})$ .

In plain language: a function is increasing if, as you move left to right on its graph, the graph goes up. It is decreasing if the graph goes down.

Graph of an increasing function. — Graph of an increasing function: as $x$ grows, so does $f (x)$ .

Graph of a decreasing function. — Graph of a decreasing function: as $x$ grows, $f (x)$ decreases.

A function that is either increasing or decreasing on an interval is said to be monotonic on that interval.

Examples

Standard examples of increasing and decreasing behavior:

$f (x) = x^{2}$ is decreasing on $(- \infty, 0]$ and increasing on $[0, \infty)$ .
$g (x) = x^{3}$ is increasing on all of $ℝ$ .

Graph of f(x) = x² showing decreasing then increasing behavior. — (a) $f (x) = x^{2}$ : decreasing on $(- \infty, 0]$ and increasing on $[0, \infty)$ .

Graph of g(x) = x³ showing constantly increasing behavior. — (b) $g (x) = x^{3}$ : increasing on $(- \infty, \infty)$ .

Let

Determine whether $g$ is monotonic on $(- \infty, 0] \cup [1, \infty)$ .

Solution

From the graph,

g

is decreasing on

(- \infty, 0]

(since

x^{2}

falls as

x

increases toward 0) and decreasing on

[1, \infty)

(since

1 - x^{2}

falls as

x

increases). However, taking

x_{1} = 0

and

x_{2} = 1

g (0) = 0, g (1) = 0.

x_{1} < x_{2}

but

g (x_{1}) = g (x_{2})

, violating the strict inequality required for monotonic decrease. Therefore

g

is not monotonic on its entire domain

(- \infty, 0] \cup [1, \infty)

Graph of the piecewise function g showing two disconnected curves. — Graph of $g$ : decreasing on each piece separately, but not monotonic globally.

Frequently Asked Questions

Can a function be both increasing and constant?

No. A constant function (

f (x) = c

) satisfies

f (x_{1}) = f (x_{2})

for all

x_{1} \neq x_{2}

, so it fails both the increasing condition (

f (x_{1}) < f (x_{2})

) and the decreasing condition (

f (x_{1}) > f (x_{2})

). It is neither increasing nor decreasing.

How do I find intervals of increase and decrease without calculus?

For simple functions, sketch the graph and identify where it rises (increasing) and where it falls (decreasing). For more complex functions, calculus provides a systematic method: compute the derivative

. Where

, the function is increasing; where

, it is decreasing.

Is every monotonic function one-to-one?

Yes. If

f

is increasing on

I

and

x_{1} \neq x_{2}

I

, then either

x_{1} < x_{2}

(so

f (x_{1}) < f (x_{2})

) or

x_{1} > x_{2}

(so

f (x_{1}) > f (x_{2})

); either way

f (x_{1}) \neq f (x_{2})

. So

f

is one-to-one. A similar argument works for decreasing functions.

Can a function be increasing on a closed interval and decreasing on an adjacent one?

Yes. For example,

f (x) = - x^{2}

is increasing on

(- \infty, 0]

and decreasing on

[0, \infty)

, with the transition happening at the maximum point

x = 0

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Definitions

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Frequently Asked Questions

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