In the uploaded image, we have a graph representing a piecewise-defined function, and four possible answers (A, B, C, and D) are listed, each showing different piecewise functions. Let's break down the graph and check which function corresponds to the graph shown.

Analysis of the graph:

1. First segment (line from $x = -3$ to $x = 1$):

   • The line appears to pass through points $(-3, -4)$ and $(1, 0)$.
   • We can find the equation of this line using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-4)}{1 - (-3)} = \frac{4}{4} = 1$ Using point-slope form $y - y_1 = m(x - x_1)$, we can plug in $(1, 0)$: $y - 0 = 1(x - 1) \quad \Rightarrow \quad y = x - 1$ So, for $x \in [-3, 1]$, the equation is $y = x - 1$.

2. Second segment (line from $x = 2$ to $x = 4$):

   • The line passes through $(2, 3)$ and $(4, -3)$.
   • Calculating the slope: $m = \frac{-3 - 3}{4 - 2} = \frac{-6}{2} = -3$ Using point-slope form at $(2, 3)$: $y - 3 = -3(x - 2) \quad \Rightarrow \quad y = -3x + 6 + 3 = -3x + 9$ So, for $x \in (2, 4]$, the equation is $y = -3x + 9$.

Verifying against the options:

• Option A: \begin{cases} x - 1, & \text{if } -3 < x \leq 1 \\ -3x + 9, & \text{if } 2 < x < 4 \end{cases}$$ This is almost correct but has open intervals for $$x = 2$$, where the graph appears to have a closed circle.
• Option B: \begin{cases} x - 1, & \text{if } -3 \leq x < 1 \\ -3x + 9, & \text{if } 2 \leq x \leq 4 \end{cases}$$ This matches the graph correctly.
• Option C and D: Both options incorrectly represent the second part of the function with $x - 3$, which does not match the equation from the graph.

Conclusion:

The correct option is B, as it matches both the equations and the intervals on the graph.

Would you like more details on piecewise functions or have any questions?

Five related questions:

1. How do you determine the equation of a line between two points?
2. What is the difference between open and closed intervals in piecewise functions?
3. How do you graph a piecewise function on a coordinate plane?
4. Can a piecewise function have discontinuities? How do you identify them?
5. What are the steps to solve a piecewise function for a specific value of $x$?

Tip:

When solving piecewise functions, always pay attention to the boundary points to determine whether they are included (closed circle) or excluded (open circle).