1. **State the problem:** We need to understand and graph the piecewise function: $$f(x) = \begin{cases} -3x - 12 & \text{for } x \leq -3 \\ 1 & \text{for } x > 5 \end{cases}$$ 2. **Explain the function parts:** - For $x \leq -3$, the function is a linear function $f(x) = -3x - 12$. - For $x > 5$, the function is a constant function $f(x) = 1$. 3. **Evaluate the linear part at the boundary $x = -3$:** $$f(-3) = -3(-3) - 12 = 9 - 12 = -3$$ So the line segment ends at the point $(-3, -3)$. 4. **Graph the linear segment:** - The line $y = -3x - 12$ is valid for all $x \leq -3$. - For example, at $x = -4$, $$f(-4) = -3(-4) - 12 = 12 - 12 = 0$$ - So the segment goes from $(-\infty, \infty)$ but restricted to $x \leq -3$, passing through points like $(-4, 0)$ and ending at $(-3, -3)$. 5. **Graph the constant part:** - For $x > 5$, $f(x) = 1$ is a horizontal line at height 1. - This segment starts just to the right of $x=5$ (not including $x=5$). 6. **Summary:** - The graph has two disconnected parts: - A line segment from $x = -\infty$ up to and including $x = -3$ on the line $y = -3x - 12$. - A horizontal line segment for $x > 5$ at $y = 1$. This explains the function and how to graph it. Final answer: The piecewise function is graphed as described above with the two segments.