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We are tasked with determining whether the given function \( f(x) = 3x^7 + 5x^3 \) is even, odd, or neither.

### Definitions:
1. **Even function**: A function is even if \( f(-x) = f(x) \) for all \( x \). Graphically, an even function is symmetric with respect to the y-axis.
2. **Odd function**: A function is odd if \( f(-x) = -f(x) \) for all \( x \). Graphically, an odd function has rotational symmetry about the origin.
3. **Neither**: If the function does not satisfy either of the above conditions, it is neither even nor odd.

### Solution:
We need to evaluate \( f(-x) \) for the given function.

Given \( f(x) = 3x^7 + 5x^3 \), let's calculate \( f(-x) \):
\[
f(-x) = 3(-x)^7 + 5(-x)^3
\]
\[
f(-x) = 3(-x^7) + 5(-x^3)
\]
\[
f(-x) = -3x^7 - 5x^3
\]

Now compare this with \( f(x) \):
\[
f(x) = 3x^7 + 5x^3
\]
\[
f(-x) = -(3x^7 + 5x^3) = -f(x)
\]

Since \( f(-x) = -f(x) \), the function is **odd**.

Thus, the correct answer is **A. odd**.

---

Would you like further details or have any questions? Here are some related questions to explore:

1. What are the steps for determining whether a more complex function is even or odd?
2. How does the degree of each term in a polynomial influence whether it is even or odd?
3. Can a function be both even and odd? If so, under what conditions?
4. How do even and odd functions relate to real-world applications?
5. How does the symmetry of a function's graph help in identifying its nature?

**Tip**: When determining whether a function is even or odd, always check how it behaves under substitution of \( -x \).

Determine whether the function is even, odd, or neither. f(x) = 3x^7 + 5x^3

Learn how to determine whether the function f(x) = 3x^7 + 5x^3 is even, odd, or neither. This algebraic problem uses the properties of even and odd functions, ideal for high school students in Grades 10-12.

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