Let's analyze the graph and the corresponding questions.

(a) The domain and range of $f$:

1. Domain: The domain refers to the set of all $x$-values for which the function is defined. From the graph, we can see that the function starts somewhere in the negative $x$-direction and continues indefinitely in the positive $x$-direction without any breaks or gaps. Therefore, the domain is all real numbers: $\text{Domain: } (-\infty, \infty)$

2. Range: The range refers to the set of all $y$-values that the function attains. Observing the graph, the lowest point appears to be at $y = -2$, and the highest point is at $y = 2$. Thus, the range is: $\text{Range: } [-2, 2]$

(b) Intervals of increasing and decreasing:

1. Increasing: A function is increasing where its slope is positive (i.e., moving upwards as $x$ increases). From the graph, the function is increasing in two intervals:

   • From $x = -3$ to $x = -1$
   • From $x = 1$ to $x = 2$

   Therefore, the intervals of increasing are: $\text{Increasing: } (-3, -1) \cup (1, 2)$

2. Decreasing: A function is decreasing where its slope is negative (i.e., moving downwards as $x$ increases). From the graph, the function is decreasing in two intervals:

   • From $x = -1$ to $x = 1$
   • From $x = 2$ and beyond

   Therefore, the intervals of decreasing are: $\text{Decreasing: } (-1, 1) \cup (2, \infty)$

Summary of corrections:

• Domain: $(-\infty, \infty)$ (Your previous answer was correct, the checkmark seems misplaced.)
• Range: $[-2, 2]$ (The correct range should be revised to include $y = -2$ and $y = 2$.)
• Increasing: $(-3, -1) \cup (1, 2)$
• Decreasing: $(-1, 1) \cup (2, \infty)$

Let me know if you'd like further details or have any questions! Here are some related questions to deepen understanding:

1. How do we find the critical points of a function based on its graph?
2. What is the significance of local maxima and minima in determining increasing/decreasing intervals?
3. How do you interpret the behavior of a function as $x \to \infty$?
4. Can a function have more than one interval of increase or decrease?
5. What are the characteristics of a continuous function's graph?

Tip: When determining intervals of increase or decrease, always check the slopes of tangent lines or how the graph behaves between critical points.