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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?
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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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