Let's analyze the graph and the questions provided:

(a) Domain and Range of $f$:

• Domain: The domain represents all the possible values of $x$ for which the function $f(x)$ is defined. From the graph, the function exists from approximately $x = -2$ to $x = 3$. Therefore, the domain is: $\text{Domain: } [-2, 3]$

• Range: The range represents all the possible values of $y$ that the function can take. The lowest point on the graph is around $y = -1$ and the highest point is around $y = 2$. Therefore, the range is: $\text{Range: } [-1, 2]$

(b) Intervals of Increase and Decrease:

• Increasing: A function is increasing when the graph goes up as we move from left to right. The graph appears to be increasing from $x = -2$ to $x = -1$ and again from $x = 1$ to $x = 3$. So, the intervals where the function is increasing are: [ \text{Increasing: } [-2, -1] \cup [1, 3] ]

• Decreasing: A function is decreasing when the graph goes down as we move from left to right. The graph is decreasing from $x = -1$ to $x = 1$. Thus, the interval where the function is decreasing is: $\text{Decreasing: } [-1, 1]$

Let me know if you'd like further details or clarification on any part of this problem.

Relative Questions:

1. What does the term "interval notation" mean in mathematical analysis?
2. How do we determine if a function is increasing or decreasing from a graph?
3. Can a function have more than one range or domain?
4. What are the implications of a function being undefined at certain points?
5. How would the graph change if the domain was extended beyond $x = 3$?

Tip:

When using interval notation, remember that square brackets $[ ]$ indicate that the endpoints are included, while parentheses $( )$ indicate that the endpoints are not included.