Podcast
Questions and Answers
What characterizes an even function?
What characterizes an even function?
- For every input x, f(x) = -f(-x).
- It is symmetric about the x-axis.
- It can only contain odd degree terms.
- For every input x, f(x) = f(-x). (correct)
Which of the following functions is an example of an odd function?
Which of the following functions is an example of an odd function?
- f(x) = x^3 - 2x (correct)
- f(x) = x^4 + 2x^2 + 1
- f(x) = x^2 + 3
- f(x) = |x|
Which of the following statements is true about the function f(x) = 2x?
Which of the following statements is true about the function f(x) = 2x?
- It is an odd function.
- It is an even function.
- It is neither even nor odd. (correct)
- It is both even and odd.
Which function is guaranteed to be both even and odd?
Which function is guaranteed to be both even and odd?
How can one determine if a polynomial function is odd?
How can one determine if a polynomial function is odd?
If a function's graph is symmetric about the y-axis, what can be said about the function?
If a function's graph is symmetric about the y-axis, what can be said about the function?
Identify if the function f(x) = x^3 + 5 is even, odd, or neither.
Identify if the function f(x) = x^3 + 5 is even, odd, or neither.
Which of the following expressions represents a function that cannot be classified as even or odd?
Which of the following expressions represents a function that cannot be classified as even or odd?
Study Notes
Even and Odd Functions
- Even Functions:
- Graphs are symmetric about the y-axis.
- Reflecting the graph horizontally results in the original graph.
- Example: f(x) = x², f(x) = |x|
- Odd Functions:
- Graphs are symmetric about the origin.
- Reflecting the graph over both axes results in the original graph.
- Example: f(x) = x³, f(x) = 1/x
- Determining Even or Odd Functions:
- To determine if a function is even: f(x) = f(-x)
- To determine if a function is odd: f(x) = -f(-x)
- A function is neither even nor odd if it does not satisfy either rule.
- Example: f(x) = 2x is neither even nor odd.
- Polynomial Functions:
- A polynomial function with only odd degree terms is an odd function.
- A polynomial function with only even degree terms is an even function.
- Constant Function:
- The only function that is both even and odd is the constant function f(x) = 0.
Increasing and Decreasing Functions
- Increasing Function:
- Function values increase as input values increase within an interval.
- Average rate of change is positive.
- Decreasing Function:
- Function values decrease as input values increase within an interval.
- Average rate of change is negative.
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Description
Test your understanding of even and odd functions with this quiz. Explore the properties of these functions, how to determine their classification, and the examples that illustrate them. Challenge yourself on polynomial behavior and the uniqueness of constant functions.
