1. **State the problem:** We are given a piecewise linear function $f$ and a table with some missing values. We need to complete the table and explain why the equation $f(x) = 4$ has infinitely many solutions. 2. **Complete the table:** - The ordered pair $(2, )$ means $f(2)$ is needed. From the graph, at $x=2$, $f(2) = 10$. - At $x=0$, the graph shows $f(0) = 0$. - At $x=-6$, the graph is near $-3$ (rising gently from about $-10,-3$ to $0,0$), so $f(-6) \\approx -3$. - At $x=8$, the graph flattens near $y=1$, so $f(8) \\approx 1$. - For $f(x) = -6$, find $x$ where $f(x) = -6$. The graph does not reach $-6$; it goes from about $-3$ to $10$ and then flattens near $1$. So no $x$ satisfies $f(x) = -6$, leave blank. - At $x=10$, the graph is near $1$, so $f(10) \\approx 1$. 3. **Fill in the ordered pairs:** - $(2,10)$ - $(0,0)$ - $(-6,-3)$ - $(8,1)$ - $(10,1)$ 4. **Explain why $f(x) = 4$ has infinite solutions:** - The graph shows a horizontal segment at $y=4$ between approximately $x=3$ and $x=6$. - For every $x$ in this interval, $f(x) = 4$. - Since there are infinitely many $x$ values in the interval $[3,6]$, there are infinitely many solutions to $f(x) = 4$. **Final answers:** | x | f(x) | Ordered Pair | |----|------|--------------| | 2 | 10 | (2,10) | | 0 | 0 | (0,0) | | -6 | -3 | (-6,-3) | | 8 | 1 | (8,1) | | | -6 | | | 10 | 1 | (10,1) |