https://file.aiquickdraw.com/mathbot/file/172903406137672vze2xl.jpg

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.

The graph of a function f is given. Use the graph to estimate the following: (a) The domain and range of f, (b) The intervals on which f is increasing and decreasing.

This problem explores how to determine the domain, range, and intervals of increase and decrease of a function by analyzing its graph. Suitable for high school students, this problem reinforces key concepts related to function behavior.

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