When to use brackets or parentheses in domain and range
The Correct Answer and Explanation is:
In mathematics, brackets and parentheses are used to denote the domain and range of a function, specifically when describing intervals. The difference between using brackets and parentheses hinges on whether the endpoints of the interval are included or excluded.
- Parentheses ( ): Parentheses are used when an endpoint is not included in the interval. This is known as an open interval. For example, in the interval (3, 7), both 3 and 7 are excluded, meaning the function will take values strictly between 3 and 7, but not including those exact numbers. Parentheses are commonly used when the domain or range involves asymptotes, limits, or open-ended boundaries that do not allow the inclusion of the endpoint.
- Brackets [ ]: Brackets are used when an endpoint is included in the interval. This is called a closed interval. For instance, in the interval [3, 7], both 3 and 7 are included, meaning the function can take the values 3, 7, and everything in between. This is often used when the function is continuous and includes the endpoint values, such as when the function reaches a limit or is defined exactly at those values.
- Combination of Parentheses and Brackets: It is also possible to have a combination of parentheses and brackets in an interval. For example, the interval [3, 7) means that 3 is included but 7 is excluded. This can happen when the function reaches a value at the left endpoint but does not include the right endpoint.
In summary, parentheses indicate exclusion of an endpoint, while brackets indicate inclusion. These conventions help precisely define the range of values that a function can take, ensuring clarity in mathematical statements and problems.
