1. **State the problem:** We need to graph the piecewise function: $$f(x) = \begin{cases} 3 - 2x & \text{if } x < 1 \\ x - 2 & \text{if } x \geq 1 \end{cases}$$ 2. **Understand the function:** This function has two parts: - For values of $x$ less than 1, the function is $3 - 2x$. - For values of $x$ greater than or equal to 1, the function is $x - 2$. 3. **Evaluate the function at the boundary $x=1$:** - From the left side (using $3 - 2x$): $$f(1^-) = 3 - 2(1) = 3 - 2 = 1$$ - From the right side (using $x - 2$): $$f(1) = 1 - 2 = -1$$ 4. **Plot key points:** - For $x < 1$, the line $3 - 2x$ is a line with slope $-2$ and y-intercept $3$. - For $x \geq 1$, the line $x - 2$ is a line with slope $1$ and y-intercept $-2$. 5. **Check continuity at $x=1$:** The function has a jump discontinuity because the left limit is 1 and the right limit is -1. 6. **Sketch the graph:** - Draw the line $3 - 2x$ for $x < 1$ ending with an open circle at $(1,1)$. - Draw the line $x - 2$ for $x \geq 1$ starting with a closed circle at $(1,-1)$. 7. **Choose the correct graph:** The correct graph shows two line segments with a jump at $x=1$ where the left segment ends at $(1,1)$ open circle and the right segment starts at $(1,-1)$ closed circle. **Final answer:** The graph matching these conditions is option B.