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To determine whether the function \( f(x) = \frac{2x}{x^2 + 1} \) is even, odd, or neither, we follow these steps:

### Step 1: Definition of Even and Odd Functions
- A function is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function is **odd** if \( f(-x) = -f(x) \) for all \( x \).
- If neither condition is met, the function is neither even nor odd.

### Step 2: Calculate \( f(-x) \)
We substitute \( -x \) into the function:
\[
f(-x) = \frac{2(-x)}{(-x)^2 + 1} = \frac{-2x}{x^2 + 1}
\]
(Note that \( (-x)^2 = x^2 \), so the denominator remains the same.)

### Step 3: 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 is **odd**.

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

Would you like further details or clarification? Here are 5 related questions to explore:

1. How do you test whether a function is neither even nor odd?
2. What are the properties of odd functions?
3. Can you give an example of an even function?
4. Why is the denominator \( x^2 + 1 \) important for determining the function's parity?
5. What happens when a function is neither even nor odd?

**Tip**: Graphing a function can visually confirm whether it is even or odd—odd functions have rotational symmetry about the origin!

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

Learn how to determine if the function f(x) = 2x / (x^2 + 1) is even, odd, or neither. This problem explores the concepts of even and odd functions, using algebraic symmetry to test the parity. Suitable for students in Grades 10-12.

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