Math Problem Statement

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

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:

  1. Odd Function Check:

    • Symmetry Test: An odd function has rotational symmetry about the origin, meaning f(x)=f(x)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 (,1](-\infty, -1] and [1,)[1, \infty).

    Answer: (iii) (,1][1,)(-\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=1x = 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: (,1][1,)(-\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.

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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