Writing Domain and Range in Interval Notation

You have already calculated the correct domain and range interval notation, but your paper is bleeding points for bad formatting. The problem is not the math; it is the symbol system. Fixing notation errors takes five minutes of rule memorization and saves two to three letter grades per test.

A function like f(x) = √(x - 3) has domain [3, ∞). Write it as (3, ∞) or [3, ∞) and you lose the point. Write it as x ≥ 3 without the bracket and you may get half credit. Write it as 3 to infinity and you get zero. The scoring is binary because interval notation has exactly one correct form per answer.

The four notation conventions that cause the most lost points are brackets versus parentheses, the infinity rule, the union symbol, and set-builder alternatives. It also includes a conversion table and a three-minute practice set. Work them now and stop leaving marks on the table.

OpenStax Precalculus 2e, Section 1.2 defines the convention exactly: a square bracket [ or ] means the endpoint is included. A parenthesis ( or ) means the endpoint is excluded. The symbol is not a stylistic choice. It is the only difference between a fully correct answer and a half-point deduction.

If the domain includes x = 2, use a bracket at 2. If the range includes y = -1, use a bracket at -1. If the function never actually reaches the endpoint because the value would make a denominator zero or a radicand negative, use a parenthesis.

Where Students Get the Rule Backward

Students often assume brackets mean 'the interval is big' and parentheses mean 'the interval is small.' That is wrong. Brackets mean the endpoint value is a valid input or output. Parentheses mean it is not. For f(x) = x², the range is [0, ∞), not (0, ∞), because zero is a valid output (when x = 0). For f(x) = 1/x, the domain is (-∞, 0) ∪ (0, ∞), not (-∞, 0] ∪ [0, ∞), because zero is not a valid input.

The Table: Every Combination

A half-open interval has one bracket and one parenthesis. The interval [2, 5) includes 2 but excludes 5. The interval (-3, 1] excludes -3 but includes 1. Neither form is rarer than the other; every rational function with a vertical asymptote produces a half-open endpoint at the asymptote value.

Infinity is not a number. It is a concept meaning 'unbounded in the positive or negative direction.' Because you cannot reach infinity, you never use a bracket next to it. The interval (2, ∞) means all numbers greater than 2, with no upper bound. The interval (-∞, 5] means all numbers less than or equal to 5, with no lower bound.

Write [∞, 5] or [2, ∞] and the notation is invalid. No function has a domain or range that includes infinity as a value. Infinity always takes a parenthesis, even when the other endpoint takes a bracket.

When a function has a single excluded point or a gap, you write the domain as two separate intervals joined by the union symbol ∪. The union symbol domain (-∞, 2) ∪ (2, ∞) means 'all real numbers except 2.' Write it as (-∞, 2) (2, ∞) without the union and you lose the point. Write it as x ≠ 2 and you may get partial credit, but the test expects interval notation.

The union symbol also handles gaps. The domain of f(x) = √(x² - 4) is (-∞, -2] ∪ [2, ∞). The gap between -2 and 2 is excluded because the radicand is negative there. The two intervals are separate; they require the union symbol.

Three or More Exclusions

A rational function with three vertical asymptotes produces a domain with four intervals and three union symbols: (-∞, a) ∪ (a, b) ∪ (b, c) ∪ (c, ∞). Each interval is a separate region where the function exists. Missing any one of them or omitting a union symbol is a notation error that costs the full point.

Some problems accept set-builder notation or an inequality as the final answer. The set builder notation domain form {x | x ≥ 3} means 'the set of all x such that x is greater than or equal to 3.' The inequality form x ≥ 3 means the same thing. Either is correct if the problem specifies the format, but interval notation is the default for most precalculus and calculus textbooks.

OpenStax Precalculus 2e, Section 3.2 'Domain and Range' uses interval notation throughout, with an occasional set-builder alternative. The Common Core State Standard HSF-IF.B.5 expects students to relate the domain to its graph, which means reading interval endpoints off the picture. If you can write the answer in interval notation, you can translate it into any other form in under ten seconds.

The table below shows the same domain and range result written in all three common forms. Practice covering the interval column and reconstructing it from the inequality or set-builder form.

Interval Notation	Inequality	Set-Builder	Number Line
(-2, 5)	-2 < x < 5	{x | -2 < x < 5}	Open circle at -2, open circle at 5, shaded between
[-2, 5]	-2 ≤ x ≤ 5	{x | -2 ≤ x ≤ 5}	Closed circle at -2, closed circle at 5, shaded between
(-∞, 3)	x < 3	{x | x < 3}	Arrow from left, open circle at 3, shaded left
[3, ∞)	x ≥ 3	{x | x ≥ 3}	Closed circle at 3, arrow to right
(-∞, 2) ∪ (2, ∞)	x ≠ 2	{x | x ≠ 2}	Arrow from left, open circle at 2, arrow to right
(-∞, -1] ∪ [1, ∞)	x ≤ -1 or x ≥ 1	{x | x ≤ -1 or x ≥ 1}	Arrow from left, closed circle at -1, gap, closed circle at 1, arrow to right

Teachers and tutors report the same five notation errors across every class. Each one costs at least one point per problem. Fixing them raises your test score by two to three points per ten-question set.

Mistake 1: Using a Bracket at Infinity

Writing [3, ∞] or [-∞, 5] is invalid. Infinity always takes a parenthesis. This error appears on roughly one out of four student papers for radical and exponential functions.

Mistake 2: Omitting the Union Symbol

Writing (-∞, 2) (2, ∞) without the ∪ is a formatting error. The two intervals are separate sets; they require the union symbol to combine them into one domain. Write it as (-∞, 2) ∪ (2, ∞).

Mistake 3: Writing a Single Interval When Two Are Needed

For f(x) = 1/(x - 2), the domain is (-∞, 2) ∪ (2, ∞). Writing (-∞, ∞) is wrong because 2 is excluded. Writing x ≠ 2 is technically correct but not in the requested format.

Mistake 4: Confusing Open and Closed at the Vertex

For f(x) = x² + 1, the range is [1, ∞). Students write (1, ∞) because they think the vertex is an asymptote. It is not; the function reaches exactly 1 at x = 0. Check by evaluating the function at the endpoint value.

Mistake 5: Reversing Endpoint Order

Writing (5, -3) instead of (-3, 5) is a formatting error. The smaller number always comes first in interval notation. This happens most often when students rush and copy from a number line without checking direction.

Write each answer in interval notation. Then check your work against the solutions below. Each problem is from the function families covered in OpenStax Precalculus 2e, Section 3.2.

Problem 1: Find the domain of f(x) = √(5 - x).

Problem 2: Find the range of g(x) = -x² + 4.

Problem 3: Find the domain of h(x) = 3/(x² - 9).

Solutions

Solution 1: The radicand 5 - x must be ≥ 0. Solve 5 - x ≥ 0 to get x ≤ 5. Interval notation: (-∞, 5]. The bracket at 5 because 5 makes the radicand zero, which is valid for a square root.

Solution 2: The vertex is at x = 0, y = 4. The parabola opens downward, so the range is (-∞, 4]. The bracket at 4 because the function reaches exactly 4 at the vertex.

Solution 3: The denominator x² - 9 = 0 at x = 3 and x = -3. Exclude both. The domain is (-∞, -3) ∪ (-3, 3) ∪ (3, ∞). Two union symbols, four intervals, three parentheses at the excluded values.

Take any problem you have already solved for domain or range. Rewrite the answer using interval notation. Check every endpoint against the bracket-versus-parenthesis rule. If the answer involves a gap, confirm you used the union symbol. Repeat for ten problems. That is a ten-minute exercise that eliminates the notation errors that cost you points.