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.

To determine whether the function \(f(x) = x^5 + 7x^7\) is even, odd, or neither, we need to check the following:

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

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

\[
f(x) = x^5 + 7x^7
\]

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

\[
f(-x) = (-x)^5 + 7(-x)^7
\]

\[
f(-x) = -x^5 - 7x^7
\]

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

\[
f(x) = x^5 + 7x^7
\]
\[
-f(x) = -(x^5 + 7x^7) = -x^5 - 7x^7
\]

We see that:

\[
f(-x) = -x^5 - 7x^7 = -f(x)
\]

### Conclusion

Since \(f(-x) = -f(x)\), the function \(f(x) = x^5 + 7x^7\) 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 \(f(x)\), the symmetry around the y-axis indicates an even function, while symmetry around the origin indicates an odd function.

Explore whether the function f(x) = x^5 + 7x^7 is even, odd, or neither through detailed mathematical analysis. Understand the definitions of even and odd functions, and apply these concepts to polynomial functions. Suitable for advanced high school students and anyone interested in polynomial function properties.

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