1. **State the problem:** Graph the piecewise function: $$f(x) = \begin{cases} -x - 2 & \text{for } -5 < x < -1 \\ -3 & \text{for } -1 < x < 2 \end{cases}$$ 2. **Understand the function:** - For $-5 < x < -1$, the function is a line with slope $-1$ and y-intercept $-2$. - For $-1 < x < 2$, the function is a constant $-3$. 3. **Plot the first segment:** - Calculate endpoints: - At $x = -5$, $f(-5) = -(-5) - 2 = 5 - 2 = 3$ - At $x = -1$, $f(-1) = -(-1) - 2 = 1 - 2 = -1$ - The segment is a line from $(-5, 3)$ to $(-1, -1)$ with open endpoints. 4. **Plot the second segment:** - Constant value $-3$ for $-1 < x < 2$. - Endpoints: - At $x = -1$, $f(-1) = -3$ - At $x = 2$, $f(2) = -3$ - The segment is a horizontal line from $(-1, -3)$ to $(2, -3)$ with open endpoints. 5. **Summary:** - Two line segments with open endpoints at $x = -1$. - First segment slopes downward from $( -5, 3 )$ to $( -1, -1 )$. - Second segment is horizontal at $y = -3$ from $( -1, -3 )$ to $( 2, -3 )$. Final answer: The graph consists of these two open line segments as described.