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Increasing & Decreasing Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 8 April 2026

Increasing & decreasing functions

What is an increasing function?

A function is increasing over an interval of its domain
- if as the input values increase
- the output values always increase
This can be expressed formally as:
- A function $f$ is increasing on an interval of its domain
  - if for all $a$ and $b$ in the interval
  - $a < b$ implies $f (a) < f (b)$
On a graph an increasing function on an interval will go up as it moves to the right
- i.e. from bottom left to top right

What is a decreasing function?

A function is decreasing over an interval of its domain
- if as the input values increase
- the output values always decrease
This can be expressed formally as:
- A function $f$ is decreasing on an interval of its domain
  - if for all $a$ and $b$ in the interval
  - $a < b$ implies $f (a) > f (b)$
On a graph a decreasing function on an interval will go down as it moves to the right
- i.e. from top left to bottom right

Graph of y equals f(x) with a curve showing decreasing (blue), then increasing (red), and decreasing (blue) sections, marked by text along the curve. — **Graph of a function showing intervals where it is increasing and decreasing**

Examiner Tips and Tricks

In your AP® Precalculus course you will not have to worry about the endpoints on intervals where a function is increasing or decreasing.

E.g. the point in between an interval where a function is increasing and an interval where it is decreasing

Behavior at such endpoints is studied in AP® Calculus.

Worked Example

Graph of a function showing curves with left and right endpoints when x=a and x=e respectively, a local minimum when x=b, a local maximum when x=c, and another local minimum when x=d.

The figure shows the graph of $f$ on its domain $a \leq x \leq e$ . The $x$ coordinates of local maximum and minimum points are indicated. On which interval(s) in its domain is $f$ increasing?

(A) $(b, c) only$

(B) $(c, d) only$

(D) $(b, c) and (d, e)$

Answer:

Look for regions where the value of $y$ is increasing (i.e. the graph goes up) as the value of $x$ increases (i.e. the graph goes to the right)

This happens between $b$ and $c$ , AND between $d$ and $e$

(D) $(b, c) and (d, e)$

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