1. **State the problem:** We need to graph the piecewise function $$f(x) = \begin{cases} 1 - 2x & \text{if } x < 1 \\ x - 3 & \text{if } x \geq 1 \end{cases}$$ and determine which graph (A, B, C, or D) correctly represents it. 2. **Analyze each piece:** - For $x < 1$, the function is $f(x) = 1 - 2x$. This is a line with slope $-2$ and y-intercept $1$. It is decreasing because the slope is negative. - For $x \geq 1$, the function is $f(x) = x - 3$. This is a line with slope $1$ and y-intercept $-3$. It is increasing because the slope is positive. 3. **Evaluate the function at the breakpoint $x=1$:** - Left piece limit as $x \to 1^-$: $$f(1^-) = 1 - 2(1) = 1 - 2 = -1$$ - Right piece value at $x=1$: $$f(1) = 1 - 3 = -2$$ 4. **Interpret the values at $x=1$:** - Since the function is defined as $1 - 2x$ for $x < 1$, the point at $x=1$ for this piece is an open circle at $(1, -1)$. - Since the function is defined as $x - 3$ for $x \geq 1$, the point at $x=1$ for this piece is a closed circle at $(1, -2)$. 5. **Summary of graph shape:** - Left segment: decreasing line approaching $(1, -1)$ with an open circle. - Right segment: increasing line starting at $(1, -2)$ with a closed circle. 6. **Choose the correct graph:** - Graph B matches this description exactly. **Final answer:** The correct graph is **B**.