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To determine which function exhibits odd symmetry, we need to recall the condition for odd symmetry in a function \( f(x) \). A function is considered odd if:

\[
f(-x) = -f(x)
\]

This means that for every term in the function, the powers of \( x \) must be odd. Let's analyze each option:

1. \( f(x) = x^5 + 2x^4 \)
   - \( f(-x) = (-x)^5 + 2(-x)^4 = -x^5 + 2x^4 \)
   - Since \( f(-x) \neq -f(x) \), this is not odd.

2. \( f(x) = x^7 - x^2 \)
   - \( f(-x) = (-x)^7 - (-x)^2 = -x^7 - x^2 \)
   - \( f(-x) \neq -f(x) \), so this is also not odd.

3. \( f(x) = x^5 + 3x^3 \)
   - \( f(-x) = (-x)^5 + 3(-x)^3 = -x^5 - 3x^3 \)
   - Since \( f(-x) = -f(x) \), this function is odd.

4. \( f(x) = x^3 + 1 \)
   - \( f(-x) = (-x)^3 + 1 = -x^3 + 1 \)
   - \( f(-x) \neq -f(x) \), so it is not odd.

5. \( f(x) = x^3 + x - 2 \)
   - \( f(-x) = (-x)^3 + (-x) - 2 = -x^3 - x - 2 \)
   - \( f(-x) \neq -f(x) \), so it is not odd.

Thus, the correct answer is:

**\( f(x) = x^5 + 3x^3 \)**, which exhibits odd symmetry.

Would you like further details or have any questions? 

Here are some follow-up questions:
1. What are the conditions for a function to be even?
2. How can we verify symmetry in more complex functions?
3. What is the difference between even and odd symmetry in practical applications?
4. How do power functions relate to the symmetry of a function?
5. Can a function exhibit both even and odd symmetry?

**Tip:** Functions with only odd-powered terms tend to exhibit odd symmetry.

Which of the following functions exhibits odd symmetry?

Explore which functions exhibit odd symmetry by applying the key condition f(-x) = -f(x). This problem demonstrates the process of determining symmetry through algebraic analysis, ideal for high school students learning about function properties.

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