The use of brackets versus parentheses for increasing and decreasing intervals depends on whether the function is continuous at the endpoint. According to the reference, if the function is continuous at the endpoint, brackets are encouraged.
Brackets vs. Parentheses: Increasing/Decreasing Intervals
The crucial factor in determining whether to use a bracket or a parenthesis is the continuity of the function at the endpoint of the interval.
When to use brackets [ ]:
- Continuity: If the function is continuous at the endpoint and the endpoint is included in the interval where the function is increasing or decreasing, use a bracket. This indicates that the endpoint is part of the interval.
- Example: If a function is increasing up to and including x = 2, the interval would be written as
[a, 2].
When to use parentheses ( ):
- Discontinuity: If the function is discontinuous at the endpoint (e.g., a vertical asymptote or a hole), use a parenthesis.
- Exclusion: Parentheses are always used for infinity (
∞or-∞) because infinity is not a number and therefore cannot be included in the interval. Parentheses are also used if the endpoint is not included in the interval (even if the function is continuous). - Example: If a function is increasing but approaches x = 2 without ever quite reaching it, the interval would be written as
(a, 2).
Summary Table:
| Condition | Notation | Explanation |
|---|---|---|
| Function continuous, endpoint included | [ ] |
The endpoint is part of the interval of increasing/decreasing. |
| Function discontinuous, endpoint excluded | ( ) |
The endpoint is not part of the interval. |
Interval extends to infinity (∞ or -∞) |
( ) |
Infinity is not a number and cannot be included; therefore, use a parenthesis. |
Key Considerations
- Continuity is Key: Determine if the function is continuous at the endpoint.
- Inclusion Matters: Determine if the endpoint should be included in the increasing/decreasing interval.
- Standard Conventions: While conventions may vary slightly, the continuity principle provides a solid guideline.
