1. **State the problem:** We need to sketch the graph of the piecewise function: $$f(x) = \begin{cases} 0 & \text{if } -4 \leq x \leq -1 \\ x - 1 & \text{if } -1 < x \leq 1 \\ 1 & \text{if } 1 < x < 4 \end{cases}$$ 2. **Understand the intervals and values:** - For $-4 \leq x \leq -1$, $f(x) = 0$ (a constant function). - For $-1 < x \leq 1$, $f(x) = x - 1$ (a linear function). - For $1 < x < 4$, $f(x) = 1$ (a constant function). 3. **Plot the first interval $-4 \leq x \leq -1$:** - Since $f(x) = 0$, the graph is a horizontal line at $y=0$ from $x=-4$ to $x=-1$. - Closed dots at $(-4,0)$ and $(-1,0)$ because the interval includes these endpoints. 4. **Plot the second interval $-1 < x \leq 1$:** - The function is $f(x) = x - 1$. - At $x=-1$, $f(-1) = -1 - 1 = -2$, but since $x=-1$ is not included, we have an open dot at $(-1,-2)$. - At $x=1$, $f(1) = 1 - 1 = 0$, included endpoint, so closed dot at $(1,0)$. - The graph is a line segment connecting these points. 5. **Plot the third interval $1 < x < 4$:** - $f(x) = 1$, a horizontal line at $y=1$. - Open dots at $(1,1)$ and $(4,1)$ because the interval excludes endpoints. 6. **Summary of dots and lines:** - Closed dots: $(-4,0)$, $(-1,0)$, $(1,0)$. - Open dots: $(-1,-2)$, $(1,1)$, $(4,1)$. - Horizontal line from $(-4,0)$ to $(-1,0)$. - Line segment from open dot $(-1,-2)$ to closed dot $(1,0)$. - Horizontal line from open dot $(1,1)$ to open dot $(4,1)$. **Note:** The user description mentions open dot at $(-1,-1)$ and line segment from $(-1,-1)$ to $(1,0)$, but according to the function $f(-1) = -2$, so the open dot should be at $(-1,-2)$. Assuming the problem's function is correct, the open dot is at $(-1,-2)$. **Final answer:** The graph consists of: - A horizontal line segment at $y=0$ from $x=-4$ to $x=-1$ with closed dots at both ends. - A line segment from open dot $(-1,-2)$ to closed dot $(1,0)$. - A horizontal line segment at $y=1$ from $x=1$ to $x=4$ with open dots at both ends.