To define the piecewise function based on the given graph, we will break it into two parts:

1. First branch: This branch is a ray that starts from the point $(0, 0)$ and rises to the left, passing through the point $(-2, 2)$. This suggests a line with a negative slope. We can determine the equation of the line passing through these points.

2. Second branch: This is a line segment that starts from an open circle at $(0, 3)$ and ends at a closed circle at $(3, 0)$, passing through the point $(2, 1)$. This is a line segment with a negative slope.

Finding the equations of each branch:

First Branch (for $x \leq 0$):

• Points on the line: $(0, 0)$ and $(-2, 2)$.
• Slope of the line: $m = \frac{2 - 0}{-2 - 0} = -1$
• Equation of the line: Using point-slope form $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is one of the points on the line: $y - 0 = -1(x - 0) \quad \Rightarrow \quad y = -x$ So, for $x \leq 0$, the equation is $f(x) = -x$.

Second Branch (for $0 < x \leq 3$):

• Points on the line: $(0, 3)$ and $(3, 0)$.
• Slope of the line: $m = \frac{0 - 3}{3 - 0} = -1$
• Equation of the line: Using the point-slope form again: $y - 3 = -1(x - 0) \quad \Rightarrow \quad y = -x + 3$ So, for $0 < x \leq 3$, the equation is $f(x) = -x + 3$.

The piecewise function is:

\begin{cases} -x & \text{if } x \leq 0 \\ -x + 3 & \text{if } 0 < x \leq 3 \end{cases}$$ This function describes the two branches of the graph. Would you like further details or have any questions? --- Here are 5 questions to further your understanding: 1. How would the function change if the slope of the first branch was steeper? 2. Can you verify the continuity of the piecewise function at $$x = 0$$? 3. How would the function behave if the second branch extended beyond $$x = 3$$? 4. What if the open circle at $$(0, 3)$$ were a closed circle? How would that affect the function? 5. Can you find the domain and range of this piecewise function? **Tip**: Always check if the piecewise function is continuous at the boundaries (e.g., at $$x = 0$$ in this case).