Question

Determine if the endpoints for the following piecewise function are opened or closed

-3 x < 2
{2 x ≤ x < 4
5 x ≥ 4

Which of the following answer are open or closed

4,2
4,5
2,2
2, -3

Answer 1

To determine whether the endpoints for the given piecewise function are open or closed, we can analyze each segment of the function.

The piecewise function can be divided as follows:

1. For \( x < 2 \): The endpoint at 2 is excluded, which means it is **open**.
2. For \( 2 \leq x < 4 \): The endpoint at 2 is included (because of the \( \leq \)), so it is **closed**; however, the endpoint at 4 is excluded (since the interval is \( x < 4 \)), so it is **open**.
3. For \( x \geq 4 \): The endpoint at 4 is included (because of the \( \geq \)), so it is **closed**.

Now let's evaluate the endpoints you've provided:

1. **4, 2**:
- 4: Closed (included in \( x \geq 4 \)).
- 2: Closed (included in \( 2 \leq x < 4 \)).
- **Result**: Closed.

2. **4, 5**:
- 4: Closed (included in \( x \geq 4 \)).
- 5: Closed (included in \( x \geq 4 \)).
- **Result**: Closed.

3. **2, 2**:
- 2: Closed (included in \( 2 \leq x < 4 \)).
- 2: Closed (included in \( 2 \leq x < 4 \)).
- **Result**: Closed.

4. **2, -3**:
- 2: Closed (included in \( 2 \leq x < 4 \)).
- -3: Open (since we have \( x < 2 \), -3 is not included).
- **Result**: Mixed (Closed, Open).

To summarize:
- **4, 2**: Closed
- **4, 5**: Closed
- **2, 2**: Closed
- **2, -3**: Mixed (Closed, Open)