AP®MathsCollege BoardPrecalculusStudy GuidesPolynomial & Rational FunctionsFunctions & Rates of ChangeIncreasing & Decreasing Functions

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

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 30 June 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

Discriminating between open and closed intervals for where a function is increasing or decreasing is outside the scope of the AP® Precalculus course and exam.

This means that 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) $(b, c) and (d, e)$

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent