Calculus Solver What is an interval in math?
An interval is the set of all real numbers between two endpoints. Square brackets include the endpoint, round brackets exclude it: [2, 5] holds every number from 2 to 5 including both, while (2, 5) holds the same numbers with 2 and 5 left out.
Closed, open and half-open
A closed interval [a, b] includes both endpoints. An open interval (a, b) excludes both. A half-open interval such as [a, b) includes the left endpoint and excludes the right, which is the shape most often produced by a real problem.
The choice is not cosmetic. On [0, 1] the function has a value at 1; on [0, 1) it never reaches it. That distinction is what makes a maximum exist or not.
Infinity always gets a round bracket
Infinity is not a number, so it can never be included as an endpoint. That is why it always takes a round bracket: (3, ∞), (-∞, 7], (-∞, ∞).
Writing [3, ∞] is an error, not a style choice. There is no last number to include.
Where intervals earn their keep
Domain and range are given as intervals, which is the most common use in a first calculus course. The domain of √x is [0, ∞), because zero works and negatives do not.
They also state where a function is increasing or decreasing, and where it is continuous. Two intervals joined by a union cover a domain with a hole in it: the domain of 1/x is (-∞, 0) ∪ (0, ∞).
Reading interval notation
| Notation | In words | Includes endpoints? |
|---|---|---|
| [2, 5] | From 2 to 5, both included | Both |
| (2, 5) | Strictly between 2 and 5 | Neither |
| [2, 5) | From 2 up to but not including 5 | Left only |
| (2, ∞) | Every number greater than 2 | Neither (∞ never is) |
| (-∞, ∞) | All real numbers | Neither |
Questions
What is the difference between (2, 5) and [2, 5]?
The round brackets exclude the endpoints, so 2 and 5 are not in the set. The square brackets include them. Everything strictly between belongs to both.
Why does infinity always use a round bracket?
Because infinity is not a number that can be reached, so it cannot be an included endpoint. [3, ∞] is simply incorrect notation.
How do you write two separate intervals?
Join them with the union symbol ∪. The set of everything except zero is (-∞, 0) ∪ (0, ∞).
More calculus reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the rule that justified each line named.
