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