The function given in the image is:

$f(x) = \frac{2x}{x^2 + 1}$

We need to determine whether this function is even, odd, or neither.

Step 1: Check for Oddness

A function $f(x)$ is odd if:

$f(-x) = -f(x)$

Substitute $-x$ into the function:

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

Compare this to $-f(x)$:

$-f(x) = -\left(\frac{2x}{x^2 + 1}\right) = \frac{-2x}{x^2 + 1}$

Since $f(-x) = -f(x)$, the function is odd.

Step 2: Check for Evenness (Not needed but for completeness)

A function $f(x)$ is even if:

$f(-x) = f(x)$

From the calculation in Step 1, we see that:

$f(-x) = \frac{-2x}{x^2 + 1} \neq \frac{2x}{x^2 + 1} = f(x)$

Thus, the function is not even.

Conclusion:

The function $f(x) = \frac{2x}{x^2 + 1}$ is odd.

Would you like further explanation or have any questions?

Here are some related questions to expand your understanding:

1. What is the geometric interpretation of an odd function?
2. How do we determine if a more complex function is neither even nor odd?
3. Can a function be both even and odd? If so, under what conditions?
4. What are some real-life applications of even and odd functions?
5. How do transformations like shifting or stretching affect the evenness or oddness of a function?

Tip: Odd functions have rotational symmetry around the origin, while even functions have mirror symmetry across the y-axis.