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Algebra 2 Relations and functions (advanced)

Interval and set notation for domain and range

20 practice questions 0 video lessons Theory + worked examples

Interval and Set Notation

California Algebra 2 • Standard F-IF.5 • Relations & Functions

Interval and Set Notation is a topic in Relations & Functions in the California Common Core State Standards. It is aligned to Standard F-IF.5, which requires students to relate the domain of a function to its graph and describe it with appropriate notation.

Interval and set notation express domains and ranges: a bracket includes an endpoint, a parenthesis excludes it, and infinity is always open.

California Algebra 2 › Relations & Functions › Interval and Set Notation  —  Standard F-IF.5

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Theory

A set of numbers can be written three equivalent ways:

  • Inequality: \(-2\le x<3\).
  • Interval: \([-2,3)\) — bracket includes the endpoint, parenthesis excludes it.
  • Set-builder: \(\{x\mid -2\le x<3\}\).
Infinity is never reached, so \(\infty\) and \(-\infty\) always take a parenthesis.
Interval on a number line The interval negative 2 to 3 includes negative 2 (closed dot) and excludes 3 (open dot). -5 -4 -3 -2 -1 0 1 2 3 4 5 [-2, 3)
\([-2,3)\): closed at \(-2\), open at \(3\).
Notation for -2 ≤ x < 3 Notation for -2 ≤ x < 3 Notation for -2 ≤ x < 3 inequality: -2 ≤ x < 3 interval: [-2, 3) set: { x | -2 ≤ x < 3 }
Three notations for the same set.

Endpoint rules:

\[[\,a,b\,]\ \text{includes ends},\quad (a,b)\ \text{excludes ends},\quad (-\infty,b\,]\ \text{one-sided}\]
brackets include endpoints, parentheses exclude them, infinity is always open
Combine disjoint pieces with a union \(\cup\).

How to convert notations

  1. Read whether each endpoint is included (\(\le,\ge\)) or excluded (\(<,>\)).
  2. Use a bracket for included, a parenthesis for excluded or infinity.
  3. Write the lower value first, then the upper.
  4. Join separate pieces with \(\cup\).
Example 1 — Inequality to interval
Write \(-1\le x<4\) in interval notation.
Solution

A closed end uses a bracket, an open end a parenthesis.

\(-1\le x<4\)\(\Rightarrow\)\([-1,\ 4)\)
the interval is negative 1 closed to 4 open
Example 2 — Interval to set notation
Write \((2,\infty)\) in set-builder notation.
Solution

Infinity always takes a parenthesis.

\((2,\infty)\)\(\Rightarrow\)\(\{x\mid x>2\}\)
the set of x such that x is greater than 2
Example 3 — Union of intervals
Write \(x<0\) or \(x\ge 5\) in interval notation.
Solution

Join the two pieces with a union.

\(x<0\ \text{or}\ x\ge5\)\(\Rightarrow\)\((-\infty,0)\cup[5,\infty)\)
negative infinity to 0 union 5 to infinity
Example 4 — Domain in interval notation
Give the domain of \(f(x)=\sqrt{x-3}\) in interval notation.
Solution

The radicand must be \(\ge 0\): \(x-3\ge0\).

\(x\)\(\ge\)\(3\)
\(\text{domain}\)\(=\)\([3,\infty)\)
the domain is 3 to infinity, including 3

Common pitfalls

Infinity always uses a parenthesis, never a bracket.
Bracket = included, parenthesis = excluded — match the inequality.
Write the smaller number first in an interval.

Frequently asked questions

What is interval notation?

A compact way to write a set of numbers using brackets and parentheses, like \([-2,3)\).

When do you use a bracket versus a parenthesis?

A bracket includes the endpoint (\(\le,\ge\)); a parenthesis excludes it (\(<,>\)).

How is infinity written?

Always with a parenthesis, since infinity is never reached.

What is set-builder notation?

A description like \(\{x\mid x>2\}\), read “all \(x\) such that \(x>2\).”

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