Monotonicity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Monotonicity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.

Your age is monotonically increasing—it only goes up, never back down. A timer counting down is monotonically decreasing.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A monotonic function only ever rises (or only ever falls) across its entire domain, never reversing.

Common stuck point: The procedure for monotonicity is the easy part; the trap is judging direction from one interval. Asking "Does larger input always give a same-direction change (always up, or always down) with no turn-around?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does larger input always give a same-direction change (always up, or always down) with no turn-around?

Worked Examples

Example 1

medium

f(x) = 2x + 3

monotonically increasing? Show that if

x_1 < x_2

then

f(x_1) < f(x_2)

Answer

Yes —

f(x) = 2x+3

is strictly increasing

First step

Assume

x_1 < x_2

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Example 2

hard

Determine the intervals on which

h(x) = x^3 - 3x

is increasing and decreasing.

Example 3

medium

Determine the intervals of monotonicity for

f(x) = x^2 - 4x + 7

Example 4

medium

Is the function

g(x) = x + \sin x

monotonically increasing on

\mathbb{R}

Example 5

medium

Show that the sequence

a_n = \frac{n}{n+1}

is monotonically increasing.

Example 6

hard

Show

f(x) = \frac{2x+1}{x+3}

is monotonic on

x > -3

Example 7

hard

Find values of

a

for which

f(x) = x^3 + ax + 1

is monotonically increasing on all of

\mathbb{R}

Example 8

challenge

Prove that if a sequence

(a_n)

is monotonically increasing and bounded above, it converges.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

For

f(x) = -x + 5

, is it increasing or decreasing? Verify with two test values.

Example 2

hard

Show that

f(x) = x^2

is NOT monotone on all of

\mathbb{R}

by giving a counterexample, then state where it IS monotone.

Example 3

easy

Is the sequence

2,5,8,11,\dots

increasing or decreasing?

Example 4

easy

A timer counts down

10,8,6,4,\dots

. Is it monotonically increasing or decreasing?

Example 5

easy

f(x)=x^2

monotonic over all real numbers?

Example 6

easy

f(x)=3x+1

increasing or decreasing?

Example 7

easy

f(x)=-2x+5

increasing or decreasing?

Example 8

easy

Is your age over time monotonic?

Example 9

easy

f(x)=\frac{1}{x}

for

x>0

increasing or decreasing?

Example 10

easy

A constant function

f(x)=4

— is it increasing, decreasing, or neither?

Example 11

medium

On what interval is

f(x)=x^2

decreasing?

Example 12

medium

Is the sequence

3,3,5,4,6

monotonic?

Example 13

medium

For

f(x)=\sqrt{x}

x\ge 0

, is it increasing or decreasing?

Example 14

medium

f

is increasing and

f(2)=5

f(7)=9

, what can you say about

f(4)

Example 15

medium

f(x)=x^3

monotonic on all reals?

Example 16

medium

A function increases on

[0,3]

and decreases on

[3,6]

. Where is its maximum on

[0,6]

Example 17

medium

Is the sequence

a_n=\frac{1}{n}

for

n=1,2,3,\dots

increasing or decreasing?

Example 18

medium

f

is decreasing and

a<b

, which is larger:

f(a)

f(b)

Example 19

medium

For which values of

k

f(x)=kx+2

decreasing?

Example 20

challenge

Show

f(x)=x^3+x

is monotonic increasing, then determine how many real solutions

f(x)=10

has.

Example 21

challenge

f(x)=\frac{x}{x+1}

increasing or decreasing for

x>-1

? Justify.

Example 22

challenge

A sequence is defined by

a_1=1

a_{n+1}=\sqrt{2+a_n}

. Show it is increasing and bounded above by

2

Example 23

easy

Is the sequence

1, 4, 9, 16, 25, \dots

monotonically increasing?

Example 24

easy

f(x) = 7

(a constant) strictly increasing?

Example 25

easy

Is the sequence

5, 5, 5, 5

monotonically non-decreasing?

Example 26

easy

Is the temperature reading throughout a day (rising in morning, falling at night) monotonic?

Example 27

medium

Find the intervals where

f(x) = x^3 - 12x

is increasing.

Example 28

medium

For

f(x) = \frac{1}{x^2 + 1}

, where is

f

decreasing?

Example 29

medium

f

is strictly increasing on

\mathbb{R}

and

f(2)=5

, can

f(3)=5

Example 30

medium

Find values of

a

such that

f(x) = ax^3 + 3x

is strictly increasing on

\mathbb{R}

Example 31

medium

f

is increasing on

[0, 10]

and

f(0) = -3

f(10) = 12

, what can you conclude about the equation

f(x) = 0

Example 32

medium

For

f(x) = (x-2)^2

, on what interval is

f

increasing?

Example 33

medium

f

is increasing and

g

is increasing, must

f \circ g

be increasing?

Example 34

hard

f

is decreasing and

g

is decreasing, must

f \circ g

be increasing or decreasing?

Example 35

hard

For what value of

k

is the function

f(x) = kx - \sin x

monotonically increasing on

\mathbb{R}

Example 36

hard

A function

f

is strictly increasing on

[0, 5]

. If

f(0) = 1

and

f(5) = 11

, what is the maximum possible value of

\int_0^5 f(x)\,dx

? (Hint: no constraint other than monotonicity)

Example 37

hard

f(x) = x + \frac{1}{x}

monotonic on

(0, \infty)

? If not, where is it increasing / decreasing?

Example 38

challenge

f: \mathbb{R} \to \mathbb{R}

is monotonically increasing AND satisfies

f(x+y) = f(x) + f(y)

for all

x, y

, what form must

f

take?

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Related Concepts

Function Inverse Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition