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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.

Which of the following functions exhibits odd symmetry?

Find out how to determine which function exhibits odd symmetry. This problem explores algebraic concepts like odd and even functions, symmetry, and polynomial analysis. Perfect for students in Grades 9-12 learning about function properties.

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