Write a Function's Domain in Interval Notation
A worked explanation of write a function's domain in interval notation, including the key limitation and a checkable example.
Write a Function's Domain in Interval Notation
You have a function and you need to write its domain in interval notation. The rule is simple: excluded points get parentheses, included endpoints get brackets, and disconnected pieces get unions. Every interval has two endpoints. Each endpoint is either included in the set or not, and the symbol you use, parenthesis or bracket, tells the reader exactly that. Infinity is never included, so it always gets a parenthesis.
Excluded Points: The Parenthesis Rule
An excluded point is an x-value where the function is not defined. This happens at vertical asymptotes, holes, or any value that makes a denominator zero or a radicand negative. In interval notation, that excluded number is represented by a parenthesis.
Take the function f(x) = 1/(x-3). The denominator is zero at x=3, so 3 is excluded. The domain is all real numbers except 3. On the real number line, you have two separate pieces: everything left of 3 and everything right of 3. The interval notation for this is (-∞,3) ∪ (3,∞). The parenthesis at 3 tells the reader that 3 is not included. The parenthesis at -∞ and ∞ is required because infinity is not a number.
Why the Method Works
The core question is how to translate three specific features of a function's domain, excluded points, included endpoints, and disconnected pieces, into the correct symbols.
Below is each case with the rule and a concrete example.
Excluded Points
When a single x-value is excluded, you split the number line into two intervals at that value. Each interval uses a parenthesis at the excluded value. For f(x)=1/(x-3), the excluded point is 3. The two intervals are (-∞,3) and (3,∞). Joined with a union symbol, the complete domain is (-∞,3) ∪ (3,∞).
Included Endpoints
An included endpoint, a closed dot on the graph, gets a bracket. For a square root function like f(x) = √(x+2), the radicand must be ≥ 0. Solving x+2 ≥ 0 gives x ≥ -2. The endpoint -2 is included, so the domain is [-2,∞). The bracket at -2 means -2 belongs to the set. The parenthesis at ∞ is always used.
Excluded Endpoints
An excluded endpoint, an open dot on the graph, gets a parenthesis. For the inequality x < 5, the endpoint 5 is not included. The domain in interval notation is (-∞,5). The parenthesis at 5 tells the reader that 5 is not part of the set.
Disconnected Pieces
When a function has two or more separate intervals of defined x-values, you join them with the union symbol ∪. For f(x)=1/(x²-4), the denominator is zero at x=2 and x=-2. Both values are excluded. The number line is split into three intervals: (-∞,-2), (-2,2), and (2,∞). The correct domain is (-∞,-2) ∪ (-2,2) ∪ (2,∞). A common error is writing (-∞,2) ∪ (2,∞), which misses the exclusion of -2. That mistake fails the domain entirely.
A Worked Example
Work a concrete example from start to finish. Find the domain of f(x) = 1/(x²-1). The denominator is x²-1. Set it to zero: x²-1 = 0 gives x = 1 and x = -1. Both are excluded points. The number line splits into three intervals: (-∞,-1), (-1,1), and (1,∞). The domain is (-∞,-1) ∪ (-1,1) ∪ (1,∞).
Check the endpoints. At x=-1 and x=1, the function is undefined because the denominator is zero. Both get parentheses. The intervals are unbounded at both ends, so -∞ and ∞ get parentheses. The three pieces are disconnected, so two union symbols join them.
One specific failure case: a student writes (-∞,1) ∪ (1,∞). This domain includes the value -1, which makes the denominator zero. The function is undefined at x=-1, so that domain is incorrect. The correct answer requires excluding both -1 and 1, producing the three-interval union.
Calculation Steps
When you are working a problem and need to check your result, a domain and range calculator can confirm the intervals quickly. Enter the function f(x)=1/(x²-1) and it returns (-∞,-1) ∪ (-1,1) ∪ (1,∞). The calculator handles the algebra of setting denominators to zero and radicands to non-negative values, then outputs the intervals with the correct parentheses and brackets.
Always verify the calculator's output against your own work. A calculator cannot tell if you entered the function correctly. If you type 1/(x^2-1) and it returns (-∞,1) ∪ (1,∞), you know the input was wrong, the calculator is only as reliable as the expression you give it.
Translate Brackets, Parentheses, and Unions
The translation from a function's domain to interval notation follows three rules. First, excluded points, holes or asymptotes, get parentheses around the excluded value. Second, included endpoints, closed dots, get brackets. Third, disconnected pieces, separate intervals on the number line, are joined with a union symbol.
For a rational function, the domain excludes values that make the denominator zero. For a square root function, the domain requires the radicand to be ≥ 0. For a logarithmic function, the argument must be > 0. Each of these conditions produces intervals that may be bounded or unbounded, with endpoints that are included or excluded.
An unbounded interval always uses a parenthesis at ∞ or -∞. A bounded interval may use parentheses, brackets, or one of each, depending on whether each endpoint is included. A half-open interval like [a,b) includes the left endpoint a and excludes the right endpoint b.
Common Errors and How to Avoid Them
Three errors cause most of the incorrect interval notation answers. The infinity bracket error: using a bracket with ∞, like [a,∞). Infinity is not a number that can be reached, so it always gets a parenthesis. The missing union error: writing (-∞,2)(3,∞) without the ∪ symbol. Separate intervals require a union symbol. The double inclusion error: writing [a,b] when the inequality is a < x < b. Strict inequalities get parentheses, non-strict inequalities get brackets.
Another frequent error is the wrong order: writing (b,a) when a < b. Interval notation is always written left to right, smallest to largest. Check that the left endpoint is less than the right endpoint before you write the interval.
The empty set is written as ∅ or { }, not as [ ] or ( ). If no real numbers satisfy the condition, such as the domain of f(x)=√(-x²-1), the answer is ∅.
| Situation | Inequality | Interval Symbol | Example |
|---|---|---|---|
| Excluded point (hole/asymptote) | x ≠ a | Parenthesis at a | (-∞,3) ∪ (3,∞) |
| Included endpoint (closed dot) | x ≥ a | Bracket at a | [-2,∞) |
| Excluded endpoint (open dot) | x < a | Parenthesis at a | (-∞,5) |
| Both endpoints included | a ≤ x ≤ b | Brackets at a and b | [1,4] |
| Both endpoints excluded | a < x < b | Parentheses at a and b | (1,4) |
| Left included, right excluded | a ≤ x < b | Bracket at a, parenthesis at b | [1,4) |
| Left excluded, right included | a < x ≤ b | Parenthesis at a, bracket at b | (1,4] |
The Most Common Mistake to Watch For
For f(x)=1/(x²-1), the denominator is zero at x=1 and x=-1. A student writes (-∞,1) ∪ (1,∞), missing the exclusion of -1. That domain includes -1, which makes the denominator zero. The function is undefined at x=-1. The correct domain requires three intervals: (-∞,-1) ∪ (-1,1) ∪ (1,∞). Always check every value that makes the denominator zero. If there are two excluded points, you need three intervals. If there are three excluded points, you need four intervals. The pattern is n excluded points produce n+1 intervals, each joined by a union symbol.
Common Questions
How do I know when to use a parenthesis vs. a bracket for the infinity symbol?
Always use a parenthesis. Infinity is not a number that can be reached, so it cannot be included in the set. A bracket would imply that infinity is an endpoint that belongs to the interval, which is impossible.
What is the correct way to write 'all real numbers except 2' in interval notation?
Write (-∞,2) ∪ (2,∞). The parenthesis at 2 excludes that value, and the union symbol joins the two separate intervals into one set.
How do I write the domain of f(x)=1/(x²-4) in interval notation?
The denominator is zero at x=2 and x=-2. Both are excluded. The domain is (-∞,-2) ∪ (-2,2) ∪ (2,∞). A common error is writing (-∞,2) ∪ (2,∞), which misses the exclusion of -2.
What does it mean when my calculator output says '∅'?
The empty set symbol ∅ means no real numbers satisfy the condition. For example, the domain of f(x)=√(-x²-1) is ∅ because the radicand -x²-1 is always negative, and a square root of a negative number is not a real number.
How do I convert 'x < -1 or x ≥ 3' into interval notation?
This is a compound 'or' inequality. The first part x < -1 gives (-∞,-1). The second part x ≥ 3 gives [3,∞). Joined with a union, the answer is (-∞,-1) ∪ [3,∞). The parenthesis at -1 excludes that endpoint, and the bracket at 3 includes it.
What is the difference between (-∞,∞) and ℝ?
They mean the same set: all real numbers. (-∞,∞) is the interval notation for the set of all real numbers. ℝ is the set notation. Use (-∞,∞) when writing interval notation.
Can interval notation represent discrete sets like {1,2,3}?
No. Interval notation can only represent continuous sets of real numbers. A discrete set like {1,2,3} requires set-builder or roster notation. Use {1,2,3} for that set.