The image shows a piecewise-defined function question involving a graph. To determine the correct piecewise function, we must analyze the behavior of the graph in different intervals of $x$.

Analysis of the Graph:

1. First Interval (-3 ≤ x < 1):

   • The line starts at $x = -3$, $y = -4$ and ends at $x = 1$, $y = 0$.
   • The slope of this line is $\frac{0 - (-4)}{1 - (-3)} = \frac{4}{4} = 1$, indicating the equation of the line is $f(x) = x - 1$.

2. Second Interval (2 < x ≤ 4):

   • This segment starts at $x = 2$, $y = 3$ and ends at $x = 4$, $y = -3$.
   • The slope of this line is $\frac{-3 - 3}{4 - 2} = \frac{-6}{2} = -3$, and using the point-slope form, the equation is $f(x) = -3x + 9$.

Piecewise Function:

• For $-3 \leq x < 1$, the equation is $f(x) = x - 1$.
• For $2 < x \leq 4$, the equation is $f(x) = -3x + 9$.

This matches option B.

Conclusion:

The correct answer is B.

Would you like further explanation or have any other questions?

Here are 5 related questions:

1. How is the slope calculated between two points?
2. What is the point-slope form of a linear equation?
3. How do you determine the equation of a line from a graph?
4. What does it mean for a function to be piecewise-defined?
5. How can you check continuity for a piecewise function?

Tip: Always check for open and closed intervals when dealing with piecewise functions to avoid domain conflicts.