Question

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

2x x < 2
f(x) = {3x x ≤ x < 4
4x x > 55

Answer 1

To determine whether the endpoints of the piecewise function are opened or closed, we need to examine the definitions of each piece of the function and their respective endpoints:

1. **For the first piece**: \( f(x) = 2x \) for \( x < 2 \)
- The endpoint is \( x = 2 \). Since the inequality is strict (\(<\)), this endpoint is **not included** in the function. Therefore, it is an **open endpoint**.

2. **For the second piece**: \( f(x) = 3x \) for \( 2 \leq x < 4 \)
- The endpoints are \( x = 2 \) and \( x = 4 \). Here, \( x = 2 \) is included in the function (due to the \(\leq\)), making it a **closed endpoint**. For \( x = 4 \), since the inequality is strict (\(<\)), this endpoint is **not included**, making it an **open endpoint**.

3. **For the third piece**: \( f(x) = 4x \) for \( x > 5 \)
- The endpoint is \( x = 5 \). The inequality is strict (\(>\)), meaning this endpoint is **not included** in the function, making it an **open endpoint**.

**Summary of the endpoints**:
- The endpoint at \( x = 2 \) is **closed**.
- The endpoint at \( x = 4 \) is **open**.
- The endpoint at \( x = 5 \) is **open**.