---
title: "Interval Notation — AP Precalculus Definition & Guide"
description: "Interval notation writes ranges of values using brackets and parentheses. It's how you state domain and range restrictions on AP Precalc models in Topic 1.13."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/interval-notation"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 1"
---

# Interval Notation — AP Precalculus Definition & Guide

## Definition

Interval notation is a way of writing a continuous set of real numbers using two endpoints, where square brackets [ ] mean the endpoint is included and parentheses ( ) mean it is excluded; in AP Precalculus, it's the standard format for stating domain and range restrictions on function models.

## What It Is

Interval notation is the math world's shorthand for "all the numbers between here and there." You write the smaller endpoint first, the larger one second, and the symbols around them tell you whether the endpoints count. A square bracket includes the endpoint, so [2, 5] means every number from 2 to 5, including both 2 and 5. A parenthesis excludes it, so (2, 5) means everything strictly between them. You can mix them, like [0, 10), and you always use parentheses next to infinity, like [3, ∞), because infinity isn't a number you can actually reach.

In AP Precalculus, interval notation shows up everywhere a [function](/ap-pre-calc/unit-1/change-tandem/study-guide/eQFiTo22fpkDFsnj "fv-autolink") lives on a piece of the number line. The big one in [Unit 1](/ap-pre-calc/unit-1 "fv-autolink") is Topic 1.13, where you build function models and then restrict them. A model for the height of a ball might only make sense for t in [0, 4], because negative time and post-landing time are meaningless. Interval notation is how you say that cleanly and precisely.

## Why It Matters

Interval notation is the language behind learning objective [AP Pre Calc](/ap-pre-calc "fv-autolink") 1.13.B, which asks you to describe assumptions and restrictions when building a function model. Essential knowledge 1.13.B.3 says a model may require **[domain restrictions](/ap-pre-calc/key-terms/domain-restrictions "fv-autolink")** based on mathematical clues, contextual clues, or extreme values, and 1.13.B.4 says the same for **range restrictions**. When the CED says "restrict the domain," your answer is an interval. A quadratic modeling a rectangle's area only works for side lengths in (0, 20), not all real numbers. Beyond Topic 1.13, you'll use interval notation constantly across the course to state where a function is increasing or decreasing, where it's positive or negative, and where its concavity changes. Getting the brackets versus parentheses right is a small detail that separates a clean, full-credit answer from a sloppy one.

## Connections

### [Domain of a Function (Unit 1)](/ap-pre-calc/key-terms/domain-of-a-function)

The [domain](/ap-pre-calc/key-terms/domain "fv-autolink") is the set of allowed inputs, and interval notation is how you write it. A rational function with a vertical asymptote at x = 3 has domain (-∞, 3) ∪ (3, ∞), where the union symbol glues two intervals together around the hole.

### [Real Zero (Unit 1)](/ap-pre-calc/key-terms/real-zero)

[Real zeros](/ap-pre-calc/key-terms/real-zero "fv-autolink") are the boundary points that chop the number line into intervals. When you build a sign chart to find where a polynomial is positive or negative, your answer comes out in interval notation, like "f(x) > 0 on (-2, 5)."

### [Quadratic Function (Unit 1)](/ap-pre-calc/key-terms/quadratic-function)

A quadratic's [range](/ap-pre-calc/key-terms/range "fv-autolink") is read straight off its vertex. If the parabola opens up with a minimum value of -4, the range is [-4, ∞), bracket on the -4 because the vertex is actually reached, parenthesis on infinity because it never is.

### [Cubic Function (Unit 1)](/ap-pre-calc/key-terms/cubic-function)

Per 1.13.A.3, [cubic functions](/ap-pre-calc/key-terms/cubic-function "fv-autolink") model volume contexts, and those models need interval restrictions too. A box's side length might only make sense on (0, 6), so the cubic volume model gets a restricted domain even though a plain cubic has domain (-∞, ∞).

## On the AP Exam

No released FRQ asks you to define interval notation, but it's baked into how you answer. The modeling FRQ (Question 2 on the exam) regularly asks for the domain or range of a model in context, and interval notation is the expected format. Multiple-choice stems also lean on it, asking things like "on which interval is f increasing?" with answer choices written as intervals. Two skills matter: choosing the right endpoint symbols (does the boundary value actually count?) and reading context for restrictions, like time starting at 0 or a length staying positive. Writing (0, 5] when the answer is [0, 5] is exactly the kind of detail that costs points.

## interval notation vs ordered pair (coordinate point)

The interval (2, 5) and the point (2, 5) look identical on paper but mean totally different things. The interval is a stretch of the number line, every real number between 2 and 5. The point is a single location on the coordinate plane, where x = 2 and y = 5. Context tells you which is which. If the question asks about domain, range, or where a function is increasing, it's an interval. If it asks about a location on a graph, it's a point.

## Key Takeaways

- Interval notation describes a continuous range of real numbers, with square brackets meaning the endpoint is included and parentheses meaning it is excluded.
- Infinity always gets a parenthesis, never a bracket, because no function ever actually reaches infinity.
- In Topic 1.13, interval notation is how you state domain restrictions (1.13.B.3) and range restrictions (1.13.B.4) on a function model, based on context like time or length being non-negative.
- Use the union symbol ∪ to combine separate intervals, such as the domain of a rational function that skips over a vertical asymptote.
- When stating where a function is increasing, decreasing, positive, or negative, your boundaries come from real zeros and critical points, and your answer is written in interval notation.
- Always write the smaller endpoint first; (5, 2) is not a valid interval.

## FAQs

### What is interval notation in AP Precalculus?

Interval notation is a way to write a continuous set of numbers using endpoints, where [a, b] includes both endpoints and (a, b) excludes them. In AP Precalc you use it to state domains, ranges, and intervals where a function is increasing, decreasing, positive, or negative.

### Do brackets or parentheses include the endpoint?

Square brackets include the endpoint and parentheses exclude it. So [0, 5] contains both 0 and 5, while (0, 5) contains neither. A mixed interval like [0, 5) includes 0 but not 5.

### Is (2, 5) an interval or a point?

It can be either, and context decides. If a question asks about domain, range, or where a function is increasing, (2, 5) means all real numbers between 2 and 5. If it asks about a location on a graph, it's the point where x = 2 and y = 5.

### Can you use a bracket with infinity, like [3, ∞]?

No. Infinity always takes a parenthesis because it's not a number you can include, so the correct form is [3, ∞). Writing a bracket next to infinity is a classic error that can cost you on the exam.

### How is interval notation different from inequality notation?

They say the same thing in different formats. The inequality 0 ≤ x < 5 and the interval [0, 5) describe the exact same set of numbers. AP Precalc answers are usually expected in interval notation, especially for domain and range restrictions in modeling problems.

## Related Study Guides

- [1.13 Function Model Selection and Assumption Articulation](/ap-pre-calc/unit-1/function-model-selection-assumption-articulation/study-guide/tuHPqpA5XkfN1iRD)

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