1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} -3x - 9 & \text{for } -4 < x \leq -1 \\ -x - 3 & \text{for } -1 < x \leq 5 \end{cases}$$ We want to understand and graph this function, noting the endpoints and continuity. 2. **Explain the function segments:** - For the first segment, the function is $y = -3x - 9$ valid on the interval $-4 < x \leq -1$. - For the second segment, the function is $y = -x - 3$ valid on the interval $-1 < x \leq 5$. 3. **Calculate endpoints for the first segment:** - At $x = -4$ (open endpoint), $$y = -3(-4) - 9 = 12 - 9 = 3$$ - At $x = -1$ (closed endpoint), $$y = -3(-1) - 9 = 3 - 9 = -6$$ 4. **Calculate endpoints for the second segment:** - At $x = -1$ (open endpoint), $$y = -(-1) - 3 = 1 - 3 = -2$$ - At $x = 5$ (closed endpoint), $$y = -(5) - 3 = -5 - 3 = -8$$ 5. **Note about continuity:** - The function has a jump discontinuity at $x = -1$ because the first segment ends at $y = -6$ (closed) and the second segment starts at $y = -2$ (open). 6. **Summary:** - The graph consists of two line segments: - From just greater than $-4$ to $-1$, line segment from $( -4, 3 )$ (open) to $( -1, -6 )$ (closed). - From just greater than $-1$ to $5$, line segment from $( -1, -2 )$ (open) to $( 5, -8 )$ (closed). **Final answer:** The piecewise function is graphed as described with the given endpoints and discontinuity at $x = -1$.