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We are tasked with determining whether the function \( f(x) = \frac{2x}{x^2 + 1} \) is even, odd, or neither. Let's go through the steps:

### Definitions:
- A function is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function is **odd** if \( f(-x) = -f(x) \) for all \( x \).

### Step 1: Calculate \( f(-x) \)
We will substitute \( -x \) into the function and simplify.

\[
f(-x) = \frac{2(-x)}{(-x)^2 + 1} = \frac{-2x}{x^2 + 1}
\]

### Step 2: Compare \( f(-x) \) with \( f(x) \)
- \( f(x) = \frac{2x}{x^2 + 1} \)
- \( f(-x) = \frac{-2x}{x^2 + 1} \)

Clearly, \( f(-x) = -f(x) \), which means the function satisfies the condition for being **odd**.

### Conclusion:
The function \( f(x) = \frac{2x}{x^2 + 1} \) is **odd**.

Do you want further details or have any questions? Here are some related questions:

1. What are the key differences between even and odd functions?
2. How do transformations (like translations) affect whether a function is even or odd?
3. Can a function be both even and odd? Under what circumstances?
4. How does the denominator \(x^2 + 1\) affect the symmetry of this function?
5. Can you provide another example of an odd function with a different denominator?

**Tip:** To check if a function is even or odd, substituting \( -x \) into the function is always a useful first step.

Determine whether the function f(x) = (2x) / (x^2 + 1) is even, odd, or neither. Show your work.

Explore how to determine whether the function f(x) = 2x / (x^2 + 1) is even, odd, or neither. This step-by-step solution examines symmetry and key properties of even and odd functions. Suitable for high school students in Grades 10-12.

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