https://file.aiquickdraw.com/mathbot/file/1726445055076w4kt9srx.jpg

Let's analyze the graph to answer the questions one by one:

### (a) **Domain of \( f \)**:
The domain refers to all the possible values of \( x \) for which the function is defined.

- From the graph, \( f(x) \) starts at \( x = -6 \) and ends at \( x = 6 \).
  
  **Domain:** \( [-6, 6] \)

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### (b) **Range of \( f \)**:
The range refers to all the possible values of \( f(x) \).

- The lowest point on the graph is at \( y = -6 \) and the highest point is at \( y = 4 \).
  
  **Range:** \( [-6, 4] \)

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### (c) **Zeros of \( f \)**:
Zeros are the values of \( x \) where \( f(x) = 0 \).

- The graph intersects the \( x \)-axis at \( x = -4 \), \( x = 2 \), and \( x = 6 \).
  
  **Zeros:** \( x = -4, 2, 6 \)

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### (d) **\( f(3.5) \)**:
To find \( f(3.5) \), locate \( x = 3.5 \) on the graph and observe the corresponding \( y \)-value.

- At \( x = 3.5 \), \( f(x) = 3 \).
  
  **\( f(3.5) = 3 \)**

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### (e) **Intervals on which \( f \) is increasing**:
A function is increasing where the graph moves upwards as \( x \) increases.

- From \( x = -6 \) to \( x = 2 \), the graph is increasing.
  
  **Increasing on:** \( (-6, 2) \)

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### (f) **Intervals on which \( f \) is decreasing**:
A function is decreasing where the graph moves downwards as \( x \) increases.

- From \( x = 2 \) to \( x = 6 \), the graph is decreasing.
  
  **Decreasing on:** \( (2, 6) \)

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### (g) **Values for which \( f(x) \leq 0 \)**:
These are the values of \( x \) where the graph lies below or on the \( x \)-axis.

- The graph is at or below the \( x \)-axis for \( x \in [-6, -4] \cup [2, 6] \).

  **Values where \( f(x) \leq 0 \):** \( x \in [-6, -4] \cup [2, 6] \)

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### (h) **Any relative maxima or minima**:
- **Relative Maximum:** The highest point in a region, occurs at \( x = 2 \) with \( f(2) = 4 \).
- **Relative Minimum:** The lowest point in a region, occurs at \( x = -6 \) and \( x = 6 \) with \( f(-6) = f(6) = -6 \).

  **Relative Max:** \( (2, 4) \)  
  **Relative Min:** \( (-6, -6) \) and \( (6, -6) \)

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### (i) **Value(s) of \( x \) for which \( f(x) = 3 \)**:
Find the points where the graph intersects the line \( y = 3 \).

- \( f(x) = 3 \) at \( x = 3.5 \) and \( x = 0 \).

  **\( f(x) = 3 \) at \( x = 0 \) and \( x = 3.5 \)**

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### (j) **Is \( f(2) \) positive or negative?**
- From the graph, at \( x = 2 \), \( f(2) = 4 \), which is positive.
  
  **\( f(2) \) is positive.**

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Let me know if you need further details on any of these answers. Here are some related questions to expand your understanding:

1. What is the significance of the domain and range in graph analysis?
2. How can we determine if a function is increasing or decreasing using its derivative?
3. What are the implications of having multiple zeros in a function?
4. How can the concept of relative maxima and minima be used in real-world problems?
5. How would you algebraically determine the zeros of a function from its equation?

**Tip:** Understanding the shape and behavior of the graph can help in identifying intervals of increase and decrease. Always look for key points like turning points and intersections with the axes.

Use the graph of f to determine each of the following: (a) the domain of f, (b) the range of f, (c) the zeros of f, (d) f(3.5), (e) the intervals on which f is increasing, (f) the intervals on which f is decreasing, (g) the values for which f(x) ≤ 0, (h) any relative maxima or minima, (i) the value(s) of x for which f(x) = 3, (j) is f(2) positive or negative?

Learn how to analyze a function's graph, including determining its domain, range, zeros, and identifying intervals of increase and decrease. This problem also covers relative maxima and minima, as well as values where the function equals a specific value. Suitable for Grades 9-12.

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