Let's break down and solve each part of the problem step by step based on the provided description of the graphs.

(a) State the values of $f(2)$ and $g(-2)$.

• $f(2)$: From the description, the function $f$ passes through the point (2, 2), so $f(2) = 2$.
• $g(-2)$: The function $g$ passes through the point (-2, 1), so $g(-2) = 1$.

(b) Which is larger, $f(3)$ or $g(3)$?

• From the description:
  • $f(3)$ is between the points (2, 2) and (4, -1), likely slightly below the x-axis, which suggests it might be negative.
  • $g(3)$ ends at the point (3, 4), meaning $g(3) = 4$.

Thus, $g(3)$ is larger than $f(3)$.

(c) For what values of $x$ is $f(x) = g(x)$?

• From the description, the functions $f$ and $g$ intersect at the points (-2, 1) and (2, 2).

Therefore, $x = -2, 2$.

(d) On what interval(s) is $f(x) \leq g(x)$?

• $f(x)$ is less than or equal to $g(x)$ where the graph of $f$ is below or on the graph of $g$.
• From the description, $f(x)$ is below or equal to $g(x)$ on the intervals $(-4, -2]$ and $[2, 4]$.

So, the answer is $(-4, -2] \cup [2, 4]$.

(e) State the solution of the equation $f(x) = -1$.

• From the description, $f(x) = -1$ at $x = 4$.

So, $x = 4$.

(f) On what interval(s) is $g(x)$ decreasing?

• $g(x)$ is decreasing when the graph of $g$ is moving down as $x$ increases.
• From the description, $g(x)$ is decreasing on the interval $(-4, 0)$.

So, the interval is $(-4, 0)$.

(g) State the domain and range of $f$.

• Domain of $f$: The function $f$ is defined from $x = -4$ to $x = 4$, so the domain is $[-4, 4]$.
• Range of $f$: The lowest point on the graph of $f$ is $-2$ and the highest point is $3$, so the range is $[-2, 3]$.

(h) State the domain and range of $g$.

• Domain of $g$: The function $g$ is defined from $x = -4$ to $x = 3$, so the domain is $[-4, 3]$.
• Range of $g$: The lowest point on the graph of $g$ is approximately $0.5$ and the highest point is $4$, so the range is $[0.5, 4]$.

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. What is the maximum value of $g(x)$ and where does it occur?
2. On what interval(s) is $f(x)$ increasing?
3. Find $g(0)$ given the minimum point of $g$.
4. How many solutions does the equation $f(x) = 2$ have?
5. Compare $f(0)$ and $g(0)$.

Tip: When comparing values of functions at specific points, it helps to visualize the graph to see where one function is higher or lower than the other.