1. Problem: Given the piecewise function $$f(x) = \begin{cases} 3 - x & x \leq 1 \\ 2x & x > 1 \end{cases}$$ We need to identify which graph corresponds to this function. 2. Analyze each piece: - For $x \leq 1$, $f(x) = 3 - x$ is a line with slope $-1$ and y-intercept $3$. - For $x > 1$, $f(x) = 2x$ is a line with slope $2$ and y-intercept $0$. 3. Evaluate the function at $x=1$ to check continuity: - From left: $f(1) = 3 - 1 = 2$ - From right: $f(1) = 2 \times 1 = 2$ The function is continuous at $x=1$ with value $2$. 4. Plot key points: - For $x \leq 1$, points include $(0,3)$ and $(1,2)$ on the line $3 - x$. - For $x > 1$, points include $(1,2)$ and $(2,4)$ on the line $2x$. 5. Graph shape: - Left segment: line decreasing from $(0,3)$ to $(1,2)$. - Right segment: line increasing from $(1,2)$ upwards. 6. Match with options: - Option A: Line from $(0,2)$ sloping downward to $(1,2)$, then sharp upward slope starting at $x=1$. - Our function's left segment starts at $(0,3)$, not $(0,2)$, but the shape matches a downward slope to $(1,2)$ and then upward slope. - Other options do not match the slopes and continuity. Final answer: The graph corresponds to option A.