Math Problem Statement
Use the graph below to answer the following questions:
- Is this a graph of an odd function? (i) yes (ii) no
- List the intervals of increase (i) (−∞, 3] ∪ [−1,∞) (ii) [−1, 1] (iii) (−∞, −1] ∪ [1,∞) (iv) [−1, 3] (v) not listed
- Find the relative maxima (i) 3 (ii) -1 (iii) 1 (iv) there are no relative maxima (v) not listed
Solution
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:
-
Odd Function Check:
- Symmetry Test: An odd function has rotational symmetry about the origin, meaning . 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
-
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 and .
Answer: (iii)
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Relative Maxima:
- Relative Maxima: Points where the graph changes from increasing to decreasing.
- Observation: The graph reaches a relative maximum at .
Answer: (iii) 1
Summary of Answers:
- Is this a graph of an odd function?
- Answer: Yes (i)
- List the intervals of increase.
- Answer: (iii)
- Find the relative maxima.
- Answer: 1 (iii)
Would you like more details or have any questions? Here are some related questions for further practice:
- What are the characteristics of even functions?
- How can you determine the intervals of decrease from a graph?
- How do you find relative minima on a graph?
- What is the difference between absolute maxima and relative maxima?
- 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.
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Math Problem Analysis
Mathematical Concepts
Graph Analysis
Odd Functions
Relative Extrema
Formulas
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Theorems
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Suitable Grade Level
High School
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