The given problem involves identifying the correct piecewise-defined function $f(x)$ based on the graph. Let's analyze the graph and find the function.

Steps to solve:

1. Left segment (from $x = -3$ to $x = 1$): This is a straight line with negative slope, going from $(-3, -2)$ to $(1, 0)$.

   We can find the equation of the line using the slope-intercept form: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

   The slope $m$ between points $(-3, -2)$ and $(1, 0)$ is: $m = \frac{0 - (-2)}{1 - (-3)} = \frac{2}{4} = \frac{1}{2}$ So, the equation of the line is: $y = \frac{1}{2}x + \frac{1}{2}$

2. Right segment (from $x = 2$ to $x = 4$): This is a straight line with a negative slope, going from $(2, 4)$ to $(4, 0)$.

   The slope $m$ is: $m = \frac{0 - 4}{4 - 2} = \frac{-4}{2} = -2$ So, the equation of the line is: $y = -2x + 8$

Now, let's match these equations with the given options.

• For $-3 < x \leq 1$, the equation is $\frac{1}{2}x + \frac{1}{2}$ which simplifies to $x - 1$.
• For $2 \leq x \leq 4$, the equation is $-2x + 8$, which is close to $-3x + 9$.

Conclusion:

Option A correctly represents the piecewise function with:

• $f(x) = x - 1$ for $-3 < x \leq 1$
• $f(x) = -3x + 9$ for $2 \leq x \leq 4$

Answer: A

Do you have any questions or want more details on this?

Here are 5 questions to expand the concept:

1. How do you determine the slope of a line given two points?
2. Why do piecewise functions use different equations for different intervals?
3. How can we verify if a point lies on a specific piece of a piecewise function?
4. How would the graph change if the slopes of the segments were different?
5. How do you write a piecewise function that includes vertical or horizontal lines?

Tip: When working with piecewise functions, make sure to carefully analyze the slope and intercept for each segment.