The function provided in the image is:

$f(x) = \frac{x^3 - x}{x^2 + 1}$

To determine whether the function is even, odd, or neither, we evaluate $f(-x)$ and compare it with $f(x)$.

1. Calculate $f(-x)$:

   Replace $x$ with $-x$ in the function:

   $f(-x) = \frac{(-x)^3 - (-x)}{(-x)^2 + 1}$

   Simplify each term:

   $f(-x) = \frac{-x^3 + x}{x^2 + 1}$

2. Compare $f(-x)$ with $f(x)$:

   • For the function to be even, $f(-x)$ should equal $f(x)$. In this case: $f(-x) = \frac{-x^3 + x}{x^2 + 1} \neq \frac{x^3 - x}{x^2 + 1} = f(x)$ Therefore, the function is not even.

   • For the function to be odd, $f(-x)$ should equal $-f(x)$. Let's calculate $-f(x)$: $-f(x) = -\left(\frac{x^3 - x}{x^2 + 1}\right) = \frac{-x^3 + x}{x^2 + 1}$ Since $f(-x) = -f(x)$, the function is odd.

Conclusion: The function $f(x) = \frac{x^3 - x}{x^2 + 1}$ is an odd function.

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How can you tell if a function is neither even nor odd?
2. What are some examples of functions that are both even and odd?
3. Can you determine the symmetry of a function using its graph?
4. How does the degree of the polynomial in the numerator influence whether a rational function is even or odd?
5. How do you check if a more complicated function is even or odd?

Tip: Understanding the symmetry properties of functions can simplify problem-solving, especially when integrating functions over symmetric intervals.