https://file.aiquickdraw.com/mathbot/file/1727717159437axyhy9mf.jpg

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**:
\[
f(x) = 
\begin{cases} 
x - 1, & \text{if } -3 \leq x \leq 1 \\
-3x + 9, & \text{if } 2 \leq x \leq 4 
\end{cases}
\]

Would you like more details on this solution?

Here are 5 related questions to explore further:
1. How do you find the equation of a line given two points?
2. What are piecewise functions, and how are they used in real life?
3. How do you interpret the slope of a line from a graph?
4. What is the difference between inclusive and exclusive inequalities in functions?
5. How do vertical and horizontal shifts affect the graph of a line?

**Tip:** When analyzing graphs of piecewise functions, always check for open or closed circles to identify the correct boundaries of the function intervals!

Which rule defines the function in the graph? (based on the provided graph and options in the image)

Learn how to determine which rule defines a piecewise function from a given graph. This problem focuses on understanding the linear segments of the graph and matching them to the correct piecewise-defined function. Suitable for students in Grades 9-11.

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