The question is asking which of the following functions exhibits odd symmetry.

For odd symmetry:

A function $f(x)$ has odd symmetry if: $f(-x) = -f(x)$ This means that when you plug in $-x$, the result is the negative of the original function. Odd symmetry is generally associated with functions made entirely of odd powers of $x$.

Let's analyze each function:

1. $f(x) = x^5 + 2x^4$:

   • $f(-x) = (-x)^5 + 2(-x)^4 = -x^5 + 2x^4$
   • This is not equal to $-f(x) = -(x^5 + 2x^4) = -x^5 - 2x^4$, so no odd symmetry.

2. $f(x) = x^7 - x^2$:

   • $f(-x) = (-x)^7 - (-x)^2 = -x^7 - x^2$
   • This equals $-f(x) = -(x^7 - x^2) = -x^7 + x^2$, so no odd symmetry.

3. $f(x) = x^5 + 3x^3$:

   • $f(-x) = (-x)^5 + 3(-x)^3 = -x^5 - 3x^3$
   • This is equal to $-f(x) = -(x^5 + 3x^3) = -x^5 - 3x^3$, so this function has odd symmetry.

4. $f(x) = x^3 + 1$:

   • $f(-x) = (-x)^3 + 1 = -x^3 + 1$
   • This is not equal to $-f(x) = -(x^3 + 1) = -x^3 - 1$, so no odd symmetry.

5. $f(x) = x^3 + x - 2$:

   • $f(-x) = (-x)^3 + (-x) - 2 = -x^3 - x - 2$
   • This is not equal to $-f(x) = -(x^3 + x - 2) = -x^3 - x + 2$, so no odd symmetry.

Conclusion:

The function with odd symmetry is: $\boxed{f(x) = x^5 + 3x^3}$

Would you like further details or explanations?

Here are 5 related questions you might want to explore:

1. What is the difference between even and odd symmetry in functions?
2. How do transformations (like shifting) affect symmetry?
3. Can a function have both even and odd symmetry?
4. How does symmetry relate to the graph of a function?
5. What are some real-world examples of odd-symmetric functions?

Tip: Odd symmetry in a function implies that its graph is symmetric about the origin.