https://file.aiquickdraw.com/mathbot/file/17278712289916283ma3r.jpg

To determine which piecewise function rule matches the graph, let's analyze the graph and compare it to the given options.

### Graph Analysis:
- There are two distinct parts of the graph:
  1. **First piece**: The line seems to have a positive slope (rising) and covers the interval \( -3 < x < 1 \). The line passes through points such as \((-3, -4)\) and \((1, 0)\), suggesting it has the form \( f(x) = x - 1 \) (from slope-intercept form calculation).
  2. **Second piece**: The line appears to have a negative slope (falling) and covers the interval \( 2 \leq x \leq 4 \). It passes through points like \((2, 3)\) and \((4, -3)\), suggesting a slope of \(-3\), so the line should have the form \( f(x) = -3x + 9 \).

### Now, check the provided options:
- **Option A**:
  \[
  f(x) = 
  \begin{cases} 
  x - 1, & \text{if } -3 < x < 1 \\
  -3x + 9, & \text{if } 2 \leq x \leq 4
  \end{cases}
  \]
  This matches the graph exactly.

- **Option B**:
  \[
  f(x) = 
  \begin{cases} 
  x - 1, & \text{if } -3 < x < 1 \\
  -3x + 9, & \text{if } 2 < x \leq 4
  \end{cases}
  \]
  This doesn't match since the second piece has a strict inequality for \(x = 2\), while the graph includes \(x = 2\).

- **Option C**:
  \[
  f(x) = 
  \begin{cases} 
  x - 1, & \text{if } -3 \leq x < 1 \\
  x - 3, & \text{if } 2 \leq x \leq 4
  \end{cases}
  \]
  This is incorrect since the second part of the function doesn't match the graph.

### Conclusion:
The correct answer is **Option A**.

Would you like any further details or have any questions?

### Relative Questions:
1. What is the general form of a piecewise function?
2. How can we find the slope of a line from two points on a graph?
3. Why is it important to pay attention to open and closed intervals in piecewise functions?
4. What does the slope-intercept form of a line tell us?
5. How do you determine where the piecewise function changes its definition?

### Tip:
Always check the endpoints of intervals carefully in piecewise functions to see if the boundaries are included or excluded (open or closed intervals).

Which rule defines the function in the graph?

Learn how to identify the correct piecewise function rule based on a graph. This problem involves understanding linear equations, slope-intercept form, and inequalities, suitable for students in Grades 9-12.

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