how to know when to put parenthesis or brackets when doing features of functions

When describing the features of functions, parentheses and brackets are used to convey different types of intervals and points. Here’s how to use them:

### 1. **Intervals**
   - **Parentheses \(( \text{ and } )\):** Use these for open intervals, which indicate that endpoints are not included in the interval.
     - Example: \((a, b)\) represents all values between \(a\) and \(b\), but *not including* \(a\) and \(b\).
   - **Brackets \([ \text{ and } ]\):** Use these for closed intervals, which indicate that endpoints are included in the interval.
     - Example: \([a, b]\) represents all values between \(a\) and \(b\), including \(a\) and \(b\).

### 2. **Domain and Range**
   - In functions, intervals are often used to describe the **domain** (all possible \(x\)-values) and **range** (all possible \(y\)-values).
   - For instance:
     - If a function includes the endpoints, use brackets: \([-3, 5]\).
     - If it does not include the endpoints, use parentheses: \((-3, 5)\).

### 3. **Behavior at Infinity**
   - When functions approach positive or negative infinity (often in the domain or range), use parentheses since infinity is not a specific point that can be "included."
     - Example: \((-∞, 3]\) means the interval includes all values up to 3 but not beyond.

### 4. **Discontinuities and Exclusions**
   - Parentheses are used to exclude points in the domain or range.
   - For example, if there’s a discontinuity at \(x = 2\), you might describe the domain as \((-∞, 2) \cup (2, ∞)\).

### 5. **Piecewise Functions**
   - When writing piecewise functions, parentheses and brackets indicate which intervals each function piece applies to.
   - Example:
     \[
     f(x) = \begin{cases} 
      x^2, & x \in [0, 2) \\
      3x + 1, & x \in (2, 5]
      \end{cases}
     \]

Correctly using parentheses and brackets ensures clear communication of intervals, domain, and range, helping to precisely describe function behavior.

Would you like more details on any specific feature of functions or related examples?

### Related Questions
1. When should parentheses be used in limits and continuity?
2. How does interval notation differ from inequality notation?
3. How do you interpret parentheses versus brackets in graphing functions?
4. What role do parentheses and brackets play in calculus?
5. How do we express union of intervals using parentheses and brackets?

#### Tip
When describing intervals, always double-check the function's behavior at the endpoints to decide on parentheses or brackets.

Learn the proper use of parentheses and brackets in function notation. This guide covers how to apply interval notation, define domains and ranges, handle piecewise functions, and understand open vs. closed intervals for Grades 9-12.

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