Why Interval Notation Is Useful
A domain or range is a set of real numbers: a domain records allowable inputs, while a range records output values. These sets can be described with words, inequalities, or set-builder notation. Interval notation is a compact way to write a set of real numbers that fills an unbroken stretch, or several such stretches joined by a union.
For example, the condition \(-2\leq x<4\) describes all real numbers from \(-2\) up to \(4\), including \(-2\) but not \(4\). The interval notation for this set is \([-2,4)\). The choice of bracket or parenthesis tells us whether the endpoint belongs to the set.
Brackets and Parentheses
A square bracket at an endpoint means that the endpoint is included. A parenthesis means that it is excluded. For a finite pair of endpoints \(a<b\), the four basic forms are:
| Interval | Equivalent condition | Endpoint meaning |
|---|---|---|
| \([a,b]\) | \(a\leq x\leq b\) | Both endpoints are included. |
| \((a,b)\) | \(a<x<b\) | Neither endpoint is included. |
| \([a,b)\) | \(a\leq x<b\) | \(a\) is included; \(b\) is excluded. |
| \((a,b]\) | \(a<x\leq b\) | \(a\) is excluded; \(b\) is included. |
The inequalities and interval notation describe exactly the same set. In particular, the symbol \(\leq\) or \(\geq\) indicates inclusion of the corresponding endpoint, while \(<\) or \(>\) indicates exclusion.
Worked Example: Translating a Double Inequality
Write the set \(\{x\in\mathbb{R}:-2\leq x<4\}\) in interval notation.
Step 1: Read the lower bound. The condition \(-2\leq x\) says that \(x\) may equal \(-2\), so the left endpoint is included and needs a square bracket.
Step 2: Read the upper bound. The condition \(x<4\) says that \(x\) cannot equal \(4\), so the right endpoint is excluded and needs a parenthesis.
Step 3: Write the interval.
Check: The interval includes \(-2\), excludes \(4\), and includes values between them. For example, \(-2\leq 0<4\), so \(0\) belongs to both descriptions. The value \(4\) satisfies neither description.
Unbounded Intervals
Sometimes a set continues indefinitely in one direction, or in both directions. The symbols \(\infty\) and \(-\infty\) describe this unbounded direction; they are not real numbers and cannot be included as endpoints. Therefore, an interval always uses a parenthesis next to infinity.
For all real numbers, the interval is \((-\infty,\infty)\), also written \(\mathbb{R}\). There is no bracket at either infinite end.
Worked Example: An Unbounded Set with Two Parts
Write the set \(\{x\in\mathbb{R}:x<-1\text{ or }x\geq 2\}\) in interval notation.
Step 1: Convert the first condition. The values \(x<-1\) extend without bound to the left and exclude \(-1\), giving \((-\infty,-1)\).
Step 2: Convert the second condition. The values \(x\geq 2\) include \(2\) and extend without bound to the right, giving \([2,\infty)\).
Step 3: Join the two parts. The word “or” means that a value may belong to either part, so use a union.
Check: The number \(-2\) is in the first interval because \(-2<-1\); \(2\) is in the second because \(2\geq2\); and \(0\) is in neither because it satisfies neither condition. The endpoint \(-1\) is excluded, while \(2\) is included, as required.
Unions and Excluded Values
A single interval describes one uninterrupted stretch of real numbers. If the set has separated parts, write an interval for each part and join them with \(\cup\). A value excluded from the middle of a stretch may split that stretch into two intervals.
For example, all real numbers except \(3\) form the set \((-\infty,3)\cup(3,\infty)\). The two intervals leave a gap at \(3\), and both use parentheses there because \(3\) is excluded. This representation does not mean that the gap is an interval of real numbers; it means that the set is the union of the values below \(3\) and the values above \(3\).
Worked Example: Writing a Set with an Excluded Value
Write \(\{x\in\mathbb{R}:x<5\text{ and }x\neq -3\}\) in interval notation.
Step 1: Start with the upper bound. The condition \(x<5\) gives all real numbers less than \(5\), namely \((-\infty,5)\).
Step 2: Remove the excluded value. The condition \(x\neq-3\) removes \(-3\) from that stretch. Since \(-3<5\), the remaining set has a left part below \(-3\) and a right part above \(-3\) but below \(5\).
Step 3: Write both parts and their union. Neither copy of \(-3\) is included, and \(5\) is also excluded.
Check: The value \(-4\) belongs to the first interval and satisfies both conditions. The value \(0\) belongs to the second and satisfies both conditions. The value \(-3\) is excluded from both, and \(5\) is excluded because the inequality is strict.
A set can also include isolated values. For example, if a condition allows all \(x\leq-2\), the single value \(1\), and all \(x>3\), its interval notation is \((-\infty,-2]\cup\{1\}\cup(3,\infty)\). Curly braces indicate the isolated one-element set. They are not endpoint brackets for an interval.
How the Notation Preserves Membership
The endpoint marks are not decoration: they encode membership. A useful way to check a translation is to test the boundary values and one value from the interior. The bracket or parenthesis must agree with whether each boundary value satisfies the original condition.
Find the number at which the condition changes, including any value explicitly excluded.
Use a bracket when the boundary value satisfies the condition; use a parenthesis when it does not.
Use infinity to show continuation without a bound, always with a parenthesis.
Use \(\cup\) when the set consists of more than one interval or includes isolated values.
These steps work because an interval lists precisely the numbers between its endpoints, while its endpoint marks state whether either boundary belongs. For an unbounded interval, the infinity symbol indicates direction rather than an attainable endpoint. Finally, a union includes every value in any of its component sets. Thus each part of the notation corresponds to a specific membership rule.
Domains and Ranges in Interval Notation
Interval notation can describe either a domain or a range; the notation itself does not decide which role the set plays. The domain is the set of inputs, and the range is the set of outputs. If a function has domain \(D=[-2,4)\) and range \(R=(-\infty,3]\), the first interval says that the allowed inputs include \(-2\) but not \(4\). The second says that output values may be arbitrarily small and may equal \(3\), but cannot exceed \(3\).
This is a way of recording sets once they are known. Writing interval notation does not, by itself, determine the domain or range of a function; it expresses the values that belong to a specified set.
A Common Endpoint Mistake
A frequent error is to use the same endpoint mark for a strict inequality and a non-strict inequality. For example, consider the set \(\{x\in\mathbb{R}:x>2\}\). A wrong translation is:
This incorrectly includes \(2\). The inequality is strict, so \(2\) does not belong. The corrected interval is:
Check the boundary directly: \(2>2\) is false, so the interval must not include \(2\). The parenthesis at infinity remains necessary because infinity is not a real endpoint that can belong to the set.
Why This Notation Matters
Interval notation lets us state domains and ranges concisely while preserving important distinctions. For example, \([0,5]\) and \((0,5)\) contain the same values between \(0\) and \(5\), but the first includes both boundary values and the second excludes both. Similarly, writing \((-\infty,-1)\cup[2,\infty)\) makes clear that the set has two separated parts and that only one of the two finite boundary values is included.
When using or reading an interval, check three things: which direction the values extend, whether the finite endpoints are included, and whether there are separate pieces. These checks prevent a compact notation from concealing a change in the set being described.
Check Your Understanding
For each question, focus on exactly which real numbers belong to the set.
- Write \(\{x\in\mathbb{R}:1<x\leq 6\}\) in interval notation.
- Write \(\{x\in\mathbb{R}:x\leq-4\}\) in interval notation.
- Write \(\{x\in\mathbb{R}:x\neq 0\}\) as a union of intervals.
- In \((-\infty,7]\), is \(7\) included? Is infinity an endpoint that is included?
- What does the union \([1,3)\cup(5,\infty)\) say about the values \(1\), \(3\), and \(5\)?
