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.