To determine which piecewise function matches the graph described, we need to analyze the given function options in relation to the characteristics of the graph, specifically the endpoints and slopes.
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Examine Endpoints:
- Since one line has an open endpoint and the other has a closed endpoint, we should focus on how the functions are defined at certain points (like x = -2 or x = 0).
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Check Each Option:
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Option 1: \( f(x) = \begin{cases} x + 5 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases} \)
- Appears to have a potential closed endpoint at \( x = -2 \) (since it's included in the first case) but the slope could lead to an open endpoint analysis depending on the values given.
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Option 2: \( f(x) = \begin{cases} 2 - x & \text{if } x \leq 0 \ x - 5 & \text{if } x > 0 \end{cases} \)
- The closed endpoint does not have values based on the graph described.
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Option 3: \( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} \)
- Here, x = 1 is a closed endpoint since it includes \( \geq 1 \), so this can potentially match.
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Option 4: \( f(x) = \begin{cases} -5 & \text{if } x < 0 \ -x + 1 & \text{if } x \geq 0 \end{cases} \)
- This piece has a clear transition at \( x = 0 \), describing a horizontal line to a negative slope after the endpoint.
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Based on analysis, since the graph has a closed endpoint and one open endpoint, we can assume the correct function should have certain behaviors that correspond to defined pieces.
Final Thought: If the graph has one line going from a point that’s included (based on \( \leq \)) and transitioning toward a slope that will lead to an eventual open endpoint, then Function Option 3 could describe such behavior effectively.
Thus, the most likely function that matches the description and behaviors you provided would be Option 3: \( f(x) = \begin{cases} -x - 1 & \text{if } x < 1 \ -x + 2 & \text{if } x \geq 1 \end{cases} \).
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