Let's break down the given line segments and compare them with each of the answer choices.

First Line Segment:

• It starts at the point (-3, -4) and ends at (1, 0).
• The first point is inclusive for the x-value of -3, and the second point is non-inclusive for the x-value of 1.

Second Line Segment:

• It starts at the point (2, 3) and ends at (4, -3).
• Both endpoints are inclusive for the x-values 2 and 4.

Now, let's analyze each answer choice:

Option A:

x - 1, & \text{if } -3 < x \leq 1 \\ -3x + 9, & \text{if } 2 < x < 4 \end{cases}$$ - The first part describes the line $$y = x - 1$$, valid for $$-3 < x \leq 1$$. The first line segment in the graph has $$x$$-values from -3 to 1, but we need the point $$(-3, -4)$$ to be inclusive. This choice excludes $$x = -3$$, so this doesn't match the first line segment. - The second part describes the line $$y = -3x + 9$$, valid for $$2 < x < 4$$. The second segment requires inclusivity at both $$x = 2$$ and $$x = 4$$, which is not the case here. So, this option is incorrect. #### Option B: $$f(x) = \begin{cases} x - 1, & \text{if } -3 \leq x < 1 \\ -3x + 9, & \text{if } 2 \leq x \leq 4 \end{cases}$$ - The first part describes the line $$y = x - 1$$, valid for $$-3 \leq x < 1$$. This includes $$(-3, -4)$$ and excludes $$(1, 0)$$, matching the first segment exactly. - The second part describes the line $$y = -3x + 9$$, valid for $$2 \leq x \leq 4$$, including both $$(2, 3)$$ and $$(4, -3)$$. This matches the second segment perfectly. - **Option B is correct**. #### Option C: $$f(x) = \begin{cases} x - 1, & \text{if } -3 \leq x < 1 \\ x - 3, & \text{if } 2 \leq x \leq 4 \end{cases}$$ - The first part describes the line $$y = x - 1$$, which is correct for $$-3 \leq x < 1$$. - However, the second part describes the line $$y = x - 3$$, which doesn't match the slope of the second segment (we need $$y = -3x + 9$$). So this option is incorrect. #### Option D: $$f(x) = \begin{cases} x - 1, & \text{if } -3 < x \leq 1 \\ x - 3, & \text{if } 2 < x < 4 \end{cases}$$ - The first part excludes $$x = -3$$, which doesn't match the first segment. - The second part, $$y = x - 3$$, again doesn't match the equation of the second segment. - So, this option is also incorrect. ### Conclusion: The correct answer is **Option B**. Would you like any further explanation or details? Here are some additional questions you might consider: 1. How do you derive the equation of a line segment given two points? 2. Why is the condition of inclusivity important when defining piecewise functions? 3. What are the general steps to solving a piecewise function problem? 4. How would this problem change if the first segment included the point (1, 0)? 5. What methods can you use to graph piecewise functions effectively? **Tip:** Always carefully check the conditions of inclusivity and exclusivity when working with piecewise functions to ensure the intervals match the problem exactly.