To determine which option represents the graph's correct piecewise function, let's analyze the features of the rays as described.
From your description:
- The first ray starts from a closed endpoint, suggesting it should be included in the function for that endpoint (i.e., use ≥ or ≤).
- The second ray starts from an open endpoint, indicating that this endpoint should not be included (i.e., use > or <).
Given this information, we can interpret the provided options:
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Option #1: \( f(x) = \begin{cases} 2x + 1 & x > 1 \ -x + 3 & x < 1 \end{cases} \)
- This option uses > for the first branch, which is incorrect because the ray starts from a closed endpoint.
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Option #2: \( f(x) = \begin{cases} 2x + 1 & x > 1 \ -x + 3 & x \leq 1 \end{cases} \)
- This option correctly uses ≥ for the second branch, indicating that the closed endpoint is included in this piece of the function.
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Option #3: \( f(x) = \begin{cases} -2x + 1 & x > 1 \ x + 3 & x \leq 1 \end{cases} \)
- This option does not match the equations provided and also switches the rays.
Based on the requirement that the function includes the closed endpoint at \( x = 1 \) and the open endpoint for \( x > 1 \), Option #2 is the correct piecewise function.
So, the answer is: Option #2.
1 answer