Braces, denoted as { }, are used in domain and range notation specifically when representing a collection of discrete values, not continuous intervals.
Understanding Domain and Range Notation
Before diving into the specifics of braces, it's essential to understand the general notation used for domains and ranges:
- Parentheses ( ): Represent open intervals, indicating that the endpoints are not included in the set. For example, (2, 5) means all numbers between 2 and 5, excluding 2 and 5.
- Square Brackets [ ]: Represent closed intervals, indicating that the endpoints are included in the set. For example, [2, 5] means all numbers between 2 and 5, including 2 and 5.
When to Use Braces { }
As our reference states, braces { } are used when the domain or range consists of specific, distinct values, rather than a continuous stretch of numbers. In other words, it's used for discrete numbers. Here's a breakdown:
- Discrete Values: These are individual, separate numbers with no values in between. Examples include specific integers or the elements of a set.
- Not Intervals: Braces are not used for describing intervals, whether open or closed.
Examples
| Scenario | Notation | Explanation |
|---|---|---|
| Domain of f(x) is only 1, 3, and 7 | {1, 3, 7} | The function f(x) is only defined for x equals to 1, 3, and 7. |
| Range of g(x) is -2, 0, and 5 | {-2, 0, 5} | The function g(x) only outputs -2, 0, and 5 as possible values. |
| Domain includes all numbers from 1 to 5 | [1, 5] | All values between and including 1 and 5 are included. |
| Range includes all numbers between 0 and 10 | (0, 10) | All values between 0 and 10 but excluding 0 and 10 are included. |
Practical Insights
- Functions with Specific Inputs/Outputs: Functions that only accept certain inputs or produce specific outputs will have a domain or range described using braces.
- Data Sets: If you are describing a dataset, rather than a function, braces can be used for the domain or range when the data is finite or discrete.
- Set Notation: Braces are the standard symbols for defining a set of values in mathematics.
Conclusion
Braces { } are essential for specifying discrete values in the domain or range of a function, rather than continuous intervals. Use them carefully to correctly convey the scope of values being considered.
