Let's work through the given questions based on the graph of the functions $f(x)$ (red curve) and $g(x)$ (blue curve).

(a) Which is larger, $f(0)$ or $g(0)$?

• From the graph, at $x = 0$, the red curve $f(x)$ is at $y = 2$, and the blue curve $g(x)$ is at $y = 0$.
• Answer: $f(0)$ is larger because $f(0) = 2$ and $g(0) = 0$.

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

• From the graph, at $x = -3$, $f(x)$ appears to be at around $y = 1$, while $g(x)$ is at a negative value, around $y = -2$.
• Answer: $f(-3)$ is larger because $f(-3) \approx 1$ and $g(-3) \approx -2$.

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

• From the graph, the two curves intersect at two points: around $x = -2$ and $x = 2$.
• Answer: $f(x) = g(x)$ at $x = -2$ and $x = 2$.

(d) Find the values of $x$ for which $f(x) \leq g(x)$.

• $f(x)$ is less than or equal to $g(x)$ where the red curve is either below or touching the blue curve.
• From the graph, this happens in the intervals $(-\infty, -2] \cup [2, \infty)$.
• Answer: $f(x) \leq g(x)$ for $x \in (-\infty, -2] \cup [2, \infty)$.

(e) Find the values of $x$ for which $f(x) > g(x)$.

• $f(x)$ is greater than $g(x)$ where the red curve is above the blue curve.
• From the graph, this happens between $x = -2$ and $x = 2$, not including those points.
• Answer: $f(x) > g(x)$ for $x \in (-2, 2)$.

Would you like more details on any of these answers? Here are some related questions for further practice:

1. What is the behavior of $f(x)$ and $g(x)$ as $x \to \infty$?
2. At what points does the slope of $f(x)$ equal zero?
3. What is the maximum value of $g(x)$ and at what $x$-value does it occur?
4. Where does the function $g(x)$ have a minimum value?
5. Can you determine whether the functions are increasing or decreasing at $x = 1$?

Tip: When solving graphically, always focus on the relative positions of the curves to compare their values at specific points.