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 f(x)=x5+7x7f(x) = x^5 + 7x^7 is even, odd, or neither, we need to check the following:

  1. Even Function: A function f(x)f(x) is even if f(x)=f(x)f(-x) = f(x) for all xx.
  2. Odd Function: A function f(x)f(x) is odd if f(x)=f(x)f(-x) = -f(x) for all xx.
  3. Neither: If neither of the above conditions is satisfied, the function is neither even nor odd.

Let's compute f(x)f(-x) and compare it with f(x)f(x) and f(x)-f(x):

f(x)=x5+7x7f(x) = x^5 + 7x^7

Step 1: Compute f(x)f(-x)

f(x)=(x)5+7(x)7f(-x) = (-x)^5 + 7(-x)^7

f(x)=x57x7f(-x) = -x^5 - 7x^7

Step 2: Compare f(x)f(-x) with f(x)f(x) and f(x)-f(x)

f(x)=x5+7x7f(x) = x^5 + 7x^7 f(x)=(x5+7x7)=x57x7-f(x) = -(x^5 + 7x^7) = -x^5 - 7x^7

We see that:

f(x)=x57x7=f(x)f(-x) = -x^5 - 7x^7 = -f(x)

Conclusion

Since f(x)=f(x)f(-x) = -f(x), the function f(x)=x5+7x7f(x) = x^5 + 7x^7 is an odd function.

Would you like more details or have any questions? Here are some related questions:

  1. What is the definition of an even function and how does it apply to polynomials?
  2. Can you give an example of a function that is neither even nor odd?
  3. How does symmetry play a role in identifying even and odd functions?
  4. What happens when both even and odd terms are present in a function?
  5. How can you determine if a trigonometric function is even or odd?

Tip: For any function f(x)f(x), the symmetry around the y-axis indicates an even function, while symmetry around the origin indicates an odd function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Even and Odd Functions
Polynomial Functions

Formulas

-

Theorems

-

Suitable Grade Level

Advanced High School