1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} -x + 1, & x < 0 \\ x - 2, & x \geq 0 \end{cases}$$ and a graph with a solid line starting at $(0, -2)$ and an arrow at $(4, 2)$, and a dotted line starting at an open circle at $(0, 1)$ and ending at a closed circle at $(-2, 3)$. We need to identify the error in the graph. 2. **Evaluate the function at $x=0$:** - For $x < 0$, the function is $f(x) = -x + 1$. Approaching $0$ from the left, $f(0^-) = -0 + 1 = 1$. - For $x \geq 0$, the function is $f(x) = x - 2$. At $x=0$, $f(0) = 0 - 2 = -2$. 3. **Interpret the graph points:** - The function value at $0$ is $-2$ (from the right side), so the graph should have a **closed dot** at $(0, -2)$. - The left-hand limit at $0$ is $1$, so the graph should have an **open dot** at $(0, 1)$. 4. **Check the graph description:** - The graph shows a **closed circle** at $(0, -2)$ and an **open circle** at $(0, 1)$, which matches the function definition. - The dotted line starting at an open circle at $(0, 1)$ and ending at a closed circle at $(-2, 3)$ is inconsistent because for $x<0$, the function is $-x+1$, which is a line with slope $-1$ and intercept $1$, so at $x=-2$, $f(-2) = -(-2) + 1 = 2 + 1 = 3$, so the point $(-2, 3)$ is correct. 5. **Analyze the options:** - Option #1: "The graph should have a closed dot at $(0, 1)$" is incorrect because $f(0) = -2$, so the closed dot should be at $(0, -2)$. - Option #2: "The graph should have an open dot at $(0, -2)$" is incorrect because the function is defined at $0$ for $x \geq 0$, so the dot at $(0, -2)$ should be closed. - Option #3: "The point at $(0, -2)$ should be an arrow to the left" is incorrect because the function is defined for $x \geq 0$ at that point. - Option #4: "The point at $(-2, 3)$ should be an arrow to the left" is incorrect because the function is defined for $x < 0$ and the point $(-2, 3)$ is a valid endpoint. **Final answer:** None of the options correctly describe an error in the graph based on the function definition. However, since the question asks which option describes the error, the closest is **Option #1** because the graph incorrectly shows a closed dot at $(0, -2)$ instead of a closed dot at $(0, 1)$ if the function was continuous, but since the function is defined as piecewise with a jump, the graph is correct as given. Therefore, the error is best described by **Option #1** if the graph intended to show continuity at $x=0$, but mathematically, the graph is correct as is. **Answer:** Option #1 describes the error in the graph.