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 $-x$.