Use the image to answer the question.

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:

1. **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.

2. **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.

3. **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.**