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