https://file.aiquickdraw.com/mathbot/file/1724890839563gljwobah.jpg

Use the graph below to answer the following questions:
1. Is this a graph of an odd function?
(i) yes (ii) no
2. List the intervals of increase
(i) (−∞, 3] ∪ [−1,∞)
(ii) [−1, 1]
(iii) (−∞, −1] ∪ [1,∞)
(iv) [−1, 3]
(v) not listed
3. Find the relative maxima
(i) 3
(ii) -1
(iii) 1
(iv) there are no relative maxima
(v) not listed

I'll analyze the image to extract the graph and answer the questions based on it. Please give me a moment.### Analysis of the Graph and Questions:

1. **Odd Function Check:**
   - **Symmetry Test:** An odd function has rotational symmetry about the origin, meaning \( f(-x) = -f(x) \). If the graph has this symmetry, it is an odd function.
   - **Observation:** The graph appears to have rotational symmetry about the origin, indicating it is an odd function.

   **Answer:** (i) Yes

2. **Intervals of Increase:**
   - **Observation:** The graph increases as it moves from left to right in certain intervals.
   - **Correct Interval:** The graph increases on the intervals \( (-\infty, -1] \) and \( [1, \infty) \).

   **Answer:** (iii) \( (-\infty, -1] \cup [1, \infty) \)

3. **Relative Maxima:**
   - **Relative Maxima:** Points where the graph changes from increasing to decreasing.
   - **Observation:** The graph reaches a relative maximum at \( x = 1 \).

   **Answer:** (iii) 1

### Summary of Answers:
1. **Is this a graph of an odd function?**
   - **Answer:** Yes (i)
2. **List the intervals of increase.**
   - **Answer:** \( (-\infty, -1] \cup [1, \infty) \) (iii)
3. **Find the relative maxima.**
   - **Answer:** 1 (iii)

Would you like more details or have any questions? Here are some related questions for further practice:

1. What are the characteristics of even functions?
2. How can you determine the intervals of decrease from a graph?
3. How do you find relative minima on a graph?
4. What is the difference between absolute maxima and relative maxima?
5. How do you verify if a function is odd algebraically?

**Tip:** When analyzing a graph, always look for symmetry and changes in direction (increasing vs. decreasing) to identify key properties like oddness, evenness, and relative extrema.

Explore the analysis of a graph to determine if it represents an odd function. Identify intervals of increase and locate relative maxima on the graph. Suitable for high school students studying graph properties.

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