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

What would be the last point graphed on the second piece of the function, f (x) = 3x

Answer 1

To determine whether the endpoints of the piecewise function

\[
f(x) =
\begin{cases}
2x & \text{if } x < 2 \\
3x & \text{if } 2 \leq x < 4 \\
4x & \text{if } x > 5
\end{cases}
\]

are open or closed, we need to analyze the conditions at the transition points of the piecewise function.

1. **For the first piece**: \( f(x) = 2x \) for \( x < 2 \)
- The endpoint \( x = 2 \) is not included in this piece, which means it is open.

2. **For the second piece**: \( f(x) = 3x \) for \( 2 \leq x < 4 \)
- Here, \( x = 2 \) is included (closed), and \( x = 4 \) is not included (open).

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

Now, to answer the question about the last point graphed on the second piece of the function \( f(x) = 3x \):
- The last point of this piece would occur at \( x = 4 \). Since the condition for this piece is \( 2 \leq x < 4 \), the function does not include \( x = 4 \). Therefore, the last point graphed on this segment is at \( x = 4 \) (open).

In conclusion:
- At \( x = 2 \): Closed
- At \( x = 4 \): Open
- At \( x = 5 \): Open

The answer to your specific question is that the last point graphed on the second piece \( f(x) = 3x \) is at \( x = 4 \) (open).