To create a piecewise function based on the description provided, let's analyze the two options given:
Option #1: \[ f(x) = \begin{cases} -2x + 1 & \text{if } x < 0 \ 3x - 3 & \text{if } x \geq 0 \end{cases} \]
Option #2: \[ f(x) = \begin{cases} 5 & \text{if } x \leq 2 \ 2x - 4 & \text{if } x > 2 \end{cases} \]
Evaluating the Options
Given the description of the rays from the image (assuming we want to identify which piecewise function corresponds to them based on their characteristics):
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First Ray:
- Starts from a closed endpoint, indicating that the function includes that point. This usually corresponds to the condition involving \( x \geq \) or \( x \leq \).
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Second Ray:
- Starts from an open endpoint, meaning that the function does not include that endpoint. This generally corresponds to the condition involving strict inequalities, such as \( x < \) or \( x > \).
Analyzing the Options in Context
- In Option #1, the first piece applies when \( x < 0 \) (likely an open ray as it’s less than 0) and the second piece applies when \( x \geq 0\) (a closed ray starting at 0).
- In Option #2, the first piece is constant at \( 5 \) for \( x \leq 2\) (closed endpoint at 2) and the second piece \( 2x - 4\) for \( x > 2\) (open ray starting at 2).
Conclusion
Without the image, it is difficult to definitively declare which piecewise function is correct, but the structure of these options aligns with the endpoint descriptions as follows:
- If the closed endpoint is at \(0\) and the rays continue from there, Option #1 would correspond to that.
- If the closed endpoint is at \(2\) and the rays continue from there, then Option #2 fits.
Therefore, I would say:
Choose Option #1 if the closed endpoint is at \(0\), or choose Option #2 if the closed endpoint is at \(2\).
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