Math Problem Statement
Determine whether the function f, of, x, equals, x, to the power 5 , plus, 7, x, to the power 7f(x)=x 5 +7x 7 is even, odd or neither.
Solution
To determine whether the function is even, odd, or neither, we need to check the following:
- Even Function: A function is even if for all .
- Odd Function: A function is odd if for all .
- Neither: If neither of the above conditions is satisfied, the function is neither even nor odd.
Let's compute and compare it with and :
Step 1: Compute
Step 2: Compare with and
We see that:
Conclusion
Since , the function is an odd function.
Would you like more details or have any questions? Here are some related questions:
- What is the definition of an even function and how does it apply to polynomials?
- Can you give an example of a function that is neither even nor odd?
- How does symmetry play a role in identifying even and odd functions?
- What happens when both even and odd terms are present in a function?
- How can you determine if a trigonometric function is even or odd?
Tip: For any function , the symmetry around the y-axis indicates an even function, while symmetry around the origin indicates an odd function.
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Math Problem Analysis
Mathematical Concepts
Even and Odd Functions
Polynomial Functions
Formulas
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Theorems
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Suitable Grade Level
Advanced High School
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