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Monotonicity of Functions

Definition

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

Strictly increasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) < f(x_2)$
Increasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) \leq f(x_2)$
Strictly decreasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) > f(x_2)$
Decreasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) \geq f(x_2)$

Geometric Meaning

The graph of an increasing function rises from left to right
The graph of a decreasing function falls from left to right

Examples

$f(x) = x^2$ is strictly increasing on $[0, +\infty)$
$f(x) = x^2$ is strictly decreasing on $(-\infty, 0]$
$f(x) = \sin x$ is strictly increasing on $[-\frac{\pi}{2}, \frac{\pi}{2}]$

How to Test Monotonicity

Definition method: use the definition directly
Derivative method: for differentiable functions, decide monotonicity from the sign of the derivative
- If $f'(x) > 0$ , then $f(x)$ is strictly increasing
- If $f'(x) < 0$ , then $f(x)$ is strictly decreasing
- If $f'(x) \geq 0$ , then $f(x)$ is increasing (non-decreasing)
- If $f'(x) \leq 0$ , then $f(x)$ is decreasing (non-increasing)
Graphical method: read monotonicity from the graph

The derivative method may be unfamiliar—that is fine; you will meet it when you study calculus.

Exercises

Exercise 1

Determine the monotonicity of $f(x) = x^3 - 3x + 1$ on $\mathbb{R}$ .

Reference Answer

monotonicityderivativefunction graphcubic function

Approach:

Use the derivative to decide monotonicity.

Steps:

Differentiate: $f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1)$
Analyze the sign of the derivative:
- When $x < -1$ , $f'(x) > 0$ , so $f$ is strictly increasing
- When $-1 < x < 1$ , $f'(x) < 0$ , so $f$ is strictly decreasing
- When $x > 1$ , $f'(x) > 0$ , so $f$ is strictly increasing

Answer:

The function is strictly increasing on $(-\infty, -1]$ and $[1, +\infty)$ , and strictly decreasing on $[-1, 1]$ .

Exercise 2

Determine the monotonicity of $f(x) = \frac{x^2 + 1}{x}$ on $(0, +\infty)$ .

Reference Answer

monotonicityderivativefunction graphrational function

Approach:

Use the derivative to decide monotonicity.

Steps:

Differentiate: $f'(x) = \frac{2x \cdot x - (x^2 + 1) \cdot 1}{x^2} = \frac{2x^2 - x^2 - 1}{x^2} = \frac{x^2 - 1}{x^2}$
Analyze the sign of the derivative:
- When $0 < x < 1$ , $f'(x) < 0$ , so $f$ is strictly decreasing
- When $x > 1$ , $f'(x) > 0$ , so $f$ is strictly increasing

Answer:

The function is strictly decreasing on $(0, 1]$ and strictly increasing on $[1, +\infty)$ .

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$f(x)$	Mathematical symbol	f of x	A function of variable $x$
$x_1, x_2$	Mathematical symbol	x one, x two	Two points inside the interval
$f'(x)$	Mathematical symbol	f prime of x	The first derivative of the function
$I$	Mathematical symbol	I	An interval
$\mathbb{R}$	Mathematical symbol	Real numbers	The set of all real numbers

Properties of Functions

Periodicity of Functions

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