To determine whether the given functions are even, odd, or neither, we follow these steps:

1. Definitions:
   • A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$.
   • A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$.
   • If neither condition is satisfied, the function is neither.

(a) $f(x) = 2x^4 - 4x^3$

• Compute $f(-x)$: = 2x^4 + 4x^3$$
• Compare $f(-x)$ with $f(x)$:
  • $f(-x) \neq f(x)$ (not even).
  • $f(-x) \neq -f(x)$ (not odd).

Thus, $f(x) = 2x^4 - 4x^3$ is neither.

(b) $g(x) = x - x^3$

• Compute $g(-x)$: = -x + x^3 = -(x - x^3) = -g(x)$$
• Compare $g(-x)$ with $g(x)$:
  • $g(-x) = -g(x)$ (odd condition satisfied).

Thus, $g(x) = x - x^3$ is odd.

Final Results:

1. $f(x) = 2x^4 - 4x^3$: neither.
2. $g(x) = x - x^3$: odd.

Would you like me to explain this process further or provide related examples?

Related Questions:

1. What are the differences between even and odd functions in terms of symmetry?
2. Can a function be both even and odd? If so, under what conditions?
3. How does the degree of polynomial terms affect whether a function is even, odd, or neither?
4. Can you identify symmetry from the graph of a function without computation?
5. How are even and odd functions used in real-world applications like physics?

Tip:

To test evenness or oddness quickly, focus on substituting $-x$ into the function and compare it to the original $f(x)$ and $-f(x)$.