1. The problem is to graph the piecewise function: $$f(x) = \begin{cases} 1 - 2x & \text{if } x < 2 \\ x - 3 & \text{if } x \geq 2 \end{cases}$$ 2. For piecewise functions, we graph each piece separately and pay attention to the domain restrictions and whether endpoints are open or closed circles. 3. For the first piece, $f(x) = 1 - 2x$ when $x < 2$: - Calculate $f(2)$ to find the endpoint value: $1 - 2(2) = 1 - 4 = -3$. - Since $x=2$ is not included in this piece, the point $(2, -3)$ is an open circle. - The slope is $-2$, so the line decreases as $x$ increases. 4. For the second piece, $f(x) = x - 3$ when $x \geq 2$: - Calculate $f(2)$: $2 - 3 = -1$. - Since $x=2$ is included here, the point $(2, -1)$ is a closed circle. - The slope is $1$, so the line increases as $x$ increases. 5. The graph consists of two line segments: - A line with slope $-2$ for $x < 2$ ending at an open circle at $(2, -3)$. - A line with slope $1$ for $x \geq 2$ starting at a closed circle at $(2, -1)$. 6. Comparing the options, the correct graph is option D, which matches these conditions exactly.