Let's go through the graph and answer each of the questions step-by-step based on the visual information from the graph of $f(x)$:

a. Is the graph a function?

• Yes, the graph represents a function. This is because it passes the vertical line test, meaning that any vertical line drawn through the graph intersects the curve at most once. This indicates that for every $x$-value, there is only one corresponding $y$-value.

b. Evaluate the function for $f(0)$

• To evaluate $f(0)$, we locate $x = 0$ on the graph and find the corresponding $y$-value. From the graph, $f(0) = 0$.

c. For what intervals is the function increasing?

• The function is increasing where the graph moves upwards as $x$ increases. Based on the graph:
  • The function is increasing on the intervals $(-2, -1)$ and $(0, 2)$.

d. For what intervals is the function decreasing?

• The function is decreasing where the graph moves downwards as $x$ increases. From the graph:
  • The function is decreasing on the intervals $(-\infty, -2)$, $(-1, 0)$, and $(2, \infty)$.

e. What is the domain?

• The domain of the function is the set of all $x$-values for which the function is defined. Since the graph extends indefinitely to the left and right, the domain is:
  • $(-\infty, \infty)$.

f. What is the range?

• The range of the function is the set of all possible $y$-values that the graph reaches. From the graph, the lowest $y$-value is $-1$, and the highest is 3. Therefore, the range is:
  • $[-1, 3]$.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What does the vertical line test confirm about functions?
2. How can you determine where a function is increasing or decreasing algebraically?
3. What is the significance of the domain and range in real-world applications of functions?
4. Can a function have more than one interval where it increases or decreases?
5. How does the concept of a function’s inverse relate to its graph?

Tip: When analyzing graphs, always check for key features like intercepts, intervals of increase/decrease, and extrema (maximum or minimum points) to fully understand the function's behavior.