To solve this problem, we need to analyze the graph and identify which piecewise function rule best matches the graph.

Step 1: Understanding the Graph

• The graph shows two distinct lines: one from $x = -3$ to $x = 1$, and another from $x = 2$ to $x = 4$.
• The first line appears to have a positive slope (rising from left to right) from $x = -3$ to $x = 1$.
• The second line appears to have a negative slope (falling from left to right) from $x = 2$ to $x = 4$.

Step 2: Matching Line Segments to Function Rules

We now match the lines to the correct equations in the options:

1. For $-3 \leq x \leq 1$:

   • The first segment is linear with a positive slope. It looks like the line can be represented by the equation $y = x - 1$. This matches options A, B, C, and D for this interval.

2. For $2 \leq x \leq 4$:

   • The second segment has a negative slope, and by looking at the intercept, it appears the line can be represented by $y = -3x + 9$. This matches options A and B.

Step 3: Verifying Endpoints and Intervals

• The first piecewise rule applies from $-3 \leq x \leq 1$, inclusive of both boundaries.
• The second rule applies from $2 \leq x \leq 4$, inclusive of both boundaries.

Step 4: Conclusion

The correct piecewise function rule that defines the graph is Option A: