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To determine whether the given function \( h(x) = \frac{-2x}{8x^2 + 7} \) is even, odd, or neither, we need to follow these steps:

### Steps to determine if a function is even, odd, or neither:
1. **Even function**: A function is even if \( h(-x) = h(x) \) for all \( x \).
2. **Odd function**: A function is odd if \( h(-x) = -h(x) \) for all \( x \).
3. If neither of these conditions holds, the function is neither even nor odd.

### Compute \( h(-x) \):
We will substitute \( -x \) into the given function \( h(x) \):
\[
h(-x) = \frac{-2(-x)}{8(-x)^2 + 7} = \frac{2x}{8x^2 + 7}
\]

### Compare \( h(-x) \) to \( h(x) \):
- The original function is \( h(x) = \frac{-2x}{8x^2 + 7} \).
- The function \( h(-x) = \frac{2x}{8x^2 + 7} \).

Clearly, \( h(-x) = -h(x) \), which shows that the function is **odd**.

### Final Answer:
The given function \( h(x) = \frac{-2x}{8x^2 + 7} \) is **odd**.

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Do you want more details on this explanation or have any other questions?

Here are 5 related questions to expand your understanding:
1. What are some real-world examples of odd functions?
2. How can you graphically identify if a function is odd?
3. What are the key differences between even and odd functions in terms of symmetry?
4. Can a function be both even and odd? Why or why not?
5. How do transformations affect the evenness or oddness of a function?

**Tip**: Odd functions are symmetric about the origin, meaning their graphs reflect across both the x-axis and y-axis.

Determine algebraically whether the given function h(x) = (-2x) / (8x^2 + 7) is even, odd, or neither.

Learn how to determine whether the function h(x) = (-2x) / (8x^2 + 7) is even, odd, or neither. This problem involves key algebraic concepts related to function symmetry and the definition of even and odd functions. Suitable for students in Grades 9-12.

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