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!