Question

Graph the Piecewise function
f (x) = { x + 5 if x < or = -2, 2x + 3 if x > -2
Will the graph have any open endpoints? If yes, where?

Answers
Yes, at (-2, 3)
No, all endpoints are closed.
Yes, at (-2, 3), and (-2, -1).
Yes, at (-2, -1).

Answer 1

To examine the given piecewise function \( f(x) \):

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

### Step 1: Evaluate the function at the boundary point \( x = -2 \):

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

2. For \( x > -2 \):
- When \( x \) approaches -2 from the right (e.g., \( x = -1.9 \)):
\[
f(-1.9) = 2(-1.9) + 3 = -3.8 + 3 = -0.8
\]

### Step 2: Determine the type of endpoint at \( x = -2 \):
- Since \( f(-2) = 3 \) is included in the function for \( x \leq -2\), the point (-2, 3) will be a closed endpoint.
- For values approaching -2 from the right, the function value approaches -0.8. However, at \( x = -2 \) from the right, the function value does not include -2, thus it is an open endpoint at that point.

### Conclusion:
- **At \( x = -2 \)**: The function includes the point (-2, 3) which is a solid point on the graph, and does not include the value just greater than -2 (where the output of the function approaches -0.8).
- Therefore, the graph does have open endpoints and the correct answer will reflect that.

**Final Answer**: Yes, at (-2, 3). (which corresponds to the closed endpoint) and an open endpoint at \( ( -2, -1)\).
The response options provided could be interpreted in various ways based on context and clarity. Given the above analysis, the best response to the question regarding open endpoints would be:

**Yes, at (-2, 3), and (-2, -1).**