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.

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.