Question
Assume that f is a function that contains the point (-2,8). Which of the given points even and odd must be included if f is: (-2,-8) (2,8-8) (-8,-2) (2,8) (8,2)
Answers
GPT-4o mini
To determine which points must be included based on whether the function \( f \) is even or odd, we need to review the definitions:
- A function \( f \) is **even** if \( f(-x) = f(x) \) for all \( x \). This means that if \( (-2, 8) \) is on the graph, then \( (2, 8) \) must also be included, since \( f(-2) = 8 \) implies \( f(2) = 8 \).
- A function \( f \) is **odd** if \( f(-x) = -f(x) \) for all \( x \). This means that if \( (-2, 8) \) is on the graph, then \( (2, -8) \) must also be included, since \( f(-2) = 8 \) implies \( f(2) = -8 \).
Given this information, let's analyze each of the provided points:
1. **(-2, -8)**: This point does not relate to the given point (-2, 8) in the context of even or odd functions.
2. **(2, 8)**: This point would be required if the function is even, because \( f(-2) = f(2) = 8 \).
3. **(2, -8)**: This point would be required if the function is odd, because \( f(-2) = -f(2) \implies f(2) = -8 \).
4. **(-8, -2)**: This point does not relate to the given point at all in the context of even or odd functions.
5. **(8, 2)**: This point does not relate to the given point at all in the context of even or odd functions.
### Summary:
- If \( f \) is **even**, then \( (2, 8) \) must be included.
- If \( f \) is **odd**, then \( (2, -8) \) must be included.
In conclusion, depending on the property of the function \( f \), at least one of the points \( (2, 8) \) or \( (2, -8) \) must be included.
- A function \( f \) is **even** if \( f(-x) = f(x) \) for all \( x \). This means that if \( (-2, 8) \) is on the graph, then \( (2, 8) \) must also be included, since \( f(-2) = 8 \) implies \( f(2) = 8 \).
- A function \( f \) is **odd** if \( f(-x) = -f(x) \) for all \( x \). This means that if \( (-2, 8) \) is on the graph, then \( (2, -8) \) must also be included, since \( f(-2) = 8 \) implies \( f(2) = -8 \).
Given this information, let's analyze each of the provided points:
1. **(-2, -8)**: This point does not relate to the given point (-2, 8) in the context of even or odd functions.
2. **(2, 8)**: This point would be required if the function is even, because \( f(-2) = f(2) = 8 \).
3. **(2, -8)**: This point would be required if the function is odd, because \( f(-2) = -f(2) \implies f(2) = -8 \).
4. **(-8, -2)**: This point does not relate to the given point at all in the context of even or odd functions.
5. **(8, 2)**: This point does not relate to the given point at all in the context of even or odd functions.
### Summary:
- If \( f \) is **even**, then \( (2, 8) \) must be included.
- If \( f \) is **odd**, then \( (2, -8) \) must be included.
In conclusion, depending on the property of the function \( f \), at least one of the points \( (2, 8) \) or \( (2, -8) \) must be included.