To determine if a function is even or odd, we need to evaluate the function for \(-x\) and compare it to the original function.

Let the function be:

\[
f(x) = \frac{x^3}{x - x^3}
\]

### Step 1: Check if \( f(-x) \) is equal to \( f(x) \) (for evenness) or if \( f(-x) = -f(x) \) (for oddness).

We calculate \( f(-x) \):

\[
f(-x) = \frac{(-x)^3}{(-x) - (-x)^3} = \frac{-x^3}{-x - (-x)^3} = \frac{-x^3}{-x + x^3}
\]

Thus, \( f(-x) \) becomes:

\[
f(-x) = \frac{-x^3}{-x + x^3}
\]

### Step 2: Compare \( f(-x) \) with \( f(x) \).

For \( f(x) = \frac{x^3}{x - x^3} \) and \( f(-x) = \frac{-x^3}{-x + x^3} \):

- Clearly, \( f(-x) \neq f(x) \), so the function is not **even**.
- Also, \( f(-x) \neq -f(x) \), so the function is not **odd** either.

### Conclusion:
The function \( f(x) = \frac{x^3}{x - x^3} \) is **neither even nor odd**.

Would you like a detailed breakdown or have any further questions?

Here are 5 follow-up questions related to this topic:
1. How do you determine if a rational function is even or odd?
2. Can a function be neither even nor odd, and under what circumstances?
3. How do you test for even or odd properties using symmetry?
4. What are the implications of a function being even or odd in calculus?
5. Are there specific transformations that preserve the evenness or oddness of functions?

**Tip:** When testing if a function is even or odd, always evaluate \( f(-x) \) and compare it to both \( f(x) \) and \(-f(x) \).

Explore whether the function f(x) = (x^3)/(x - x^3) is even or odd. This detailed solution walks you through the step-by-step process, testing f(-x) for symmetry properties. Ideal for high school students in Grades 10-12 studying algebra and rational functions.

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