Question

Graph the piecewise function f(x) =
x + 5 ifx ≤ -2
. Will the graph have any
2x + 3 if x > -2
open endpoints? If yes, where? (1 point)
• Yes, at (-2,3).
O Yes, at (-2, -1).
• No, all endpoints are closed
• Yes, at (-2,3) and (-2, -1).

Answer 1

To analyze the piecewise function \( f(x) \), we will consider its definition:

1. \( f(x) = x + 5 \) for \( x \leq -2 \)
2. \( f(x) = 2x + 3 \) for \( x > -2 \)

Let's evaluate the function at the critical point \( x = -2 \):

- For \( x \leq -2 \):
\[
f(-2) = -2 + 5 = 3
\]

- For \( x > -2 \):
\[
f(-2) \text{ is not defined since the condition is strictly greater than -2.}
\]

This means that at \( x = -2 \), the function value is 3 from the first piece, and does not exist from the second piece, indicating an open endpoint at that point.

### Open Endpoints:
- The endpoint at \( (-2, 3) \) is closed because it is included in the first piece of the function.
- There is no value from the second piece of the function for \( x = -2 \) (meaning there is no point at \( (-2, \text{something}) \) for the second piece).

### Conclusion:
So, the correct answer regarding whether the graph will have any open endpoints is:
- **Yes, at (-2, 3).**

Thus, the answer is:
- **Yes, at (-2, 3).**