1. **Problem 8:** Determine how many solutions the equation $f(x) = g(x)$ has given the graphs of $f$ and $g$. 2. The solutions to $f(x) = g(x)$ are the $x$-values where the graphs of $f$ and $g$ intersect. 3. From the description, the graphs intersect at two points: $(-1, 1)$ and another point between $(1, y)$ and $(3, 9)$. 4. Therefore, the equation $f(x) = g(x)$ has exactly 2 solutions. --- 1. **Problem 9:** Find the solution set of the inequality $f(x) > g(x)$ using the same graphs. 2. The inequality $f(x) > g(x)$ holds where the graph of $f$ is above the graph of $g$. 3. From the graph description, $f$ is above $g$ between the two intersection points $x = -1$ and $x = 3$. 4. Hence, the solution set is the interval $(-1, 3)$. --- 1. **Problem 10:** Determine which statement about the rates of change (slopes) of $f$ and $g$ is true. 2. The rate of change of a function is its slope. 3. From the graph, $f$ passes through $(0, 3)$ and $(3, 9)$, so its slope is $$m_f = \frac{9 - 3}{3 - 0} = \frac{6}{3} = 2.$$ 4. The graph $g$ passes through $(0, 3)$ and $(3, 4)$, so its slope is $$m_g = \frac{4 - 3}{3 - 0} = \frac{1}{3}.$$ 5. Since $2 > \frac{1}{3}$, the rate of change of $f$ is greater than that of $g$. --- 1. **Problem 11:** Identify which equation relating $h$ and $g$ is true based on their graphs. 2. The graph $g$ is symmetric about the origin with points $(0,0)$, $(-2,-2)$, and $(2,2)$. 3. The graph $h$ is symmetric about the y-axis with points $(0,0)$, $(-2,2)$, and $(2,-2)$. 4. The transformation $h(x) = -g(-x)$ reflects $g$ about the y-axis and then about the x-axis, matching the description of $h$. 5. Therefore, the true equation is $h(x) = -g(-x)$. **Final answers:** - Problem 8: 2 solutions - Problem 9: $(-1, 3)$ - Problem 10: The rate of change of $f$ is greater than that of $g$. - Problem 11: $h(x) = -g(-x)$