1. **Stating the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} x + 2 & \text{for } -4 \leq x < 1 \\ -2x + 8 & \text{for } 2 \leq x \leq 4 \end{cases}$$ We want to understand the behavior of this function and graph it based on the given intervals. 2. **Understanding piecewise functions:** A piecewise function is defined by different expressions depending on the input value $x$. Here, the function has two parts with a gap between $x=1$ and $x=2$. 3. **First piece:** For $-4 \leq x < 1$, the function is $f(x) = x + 2$. - At $x = -4$, $f(-4) = -4 + 2 = -2$. - At $x = 1$, the function approaches $f(1) = 1 + 2 = 3$, but since $x=1$ is not included (open circle), $f(1)$ is undefined here. 4. **Second piece:** For $2 \leq x \leq 4$, the function is $f(x) = -2x + 8$. - At $x = 2$, $f(2) = -2(2) + 8 = 4$ (closed circle, included). - At $x = 4$, $f(4) = -2(4) + 8 = 0$. 5. **Graph description:** The graph has two disconnected line segments: - A blue line segment from $(-4, -2)$ to just before $(1, 3)$ with an open circle at $(1, 3)$. - A green line segment from $(2, 4)$ to $(4, 0)$. 6. **Summary:** The function is not defined between $1$ and $2$, creating a gap. The left segment increases linearly, and the right segment decreases linearly. This matches the graph description provided.