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Determine whether the following functions are even or odd:

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.

Explore how to determine if a function is odd or even with an example involving the function f(x) = (x^3 - x) / (x^2 + 1). Learn the key concepts of function symmetry, including definitions of odd and even functions. Suitable for advanced high school students and those interested in mathematical function properties.

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