1. **Stating the problem:** We are given a piecewise function $f(x)$ defined by two segments: - A horizontal line segment from $(-3, -4)$ to $(0, -4)$ with filled endpoints. - A line segment from an open circle at $(3, -8)$ to a filled circle at $(5, -2)$. 2. **Interpreting the segments:** - The first segment is constant at $y = -4$ for $x$ between $-3$ and $0$, inclusive. - The second segment is a line connecting $(3, -8)$ (open circle, so not included) to $(5, -2)$ (filled circle, included). 3. **Writing the piecewise function:** $$ f(x) = \begin{cases} -4 & \text{for } -3 \leq x \leq 0 \\ \text{slope} \cdot (x - 3) + (-8) & \text{for } 3 < x \leq 5 \end{cases} $$ 4. **Finding the slope of the second segment:** $$ \text{slope} = \frac{-2 - (-8)}{5 - 3} = \frac{6}{2} = 3 $$ 5. **Equation of the second segment:** Using point-slope form with point $(3, -8)$: $$ f(x) = 3(x - 3) - 8 = 3x - 9 - 8 = 3x - 17 $$ 6. **Final piecewise function:** $$ f(x) = \begin{cases} -4 & -3 \leq x \leq 0 \\ 3x - 17 & 3 < x \leq 5 \end{cases} $$ This matches the graph description with the correct domain intervals and endpoint inclusions.