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 $-f(x)$.