https://file.aiquickdraw.com/mathbot/file/1727213387945tsf3enuw.jpg

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.

Determine whether the function f(x) = 2x / (x^2 + 1) is even, odd, or neither even nor odd.

This problem helps determine whether the function f(x) = 2x / (x^2 + 1) is even, odd, or neither. Follow the step-by-step solution to understand how symmetry determines whether a function is even or odd.

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