To determine if a function is odd, even, or neither, we follow these rules:

• Even function: $f(-x) = f(x)$
• Odd function: $f(-x) = -f(x)$
• Neither: If neither of the above conditions hold.

Let's analyze each function:

1. For $f(x) = 2x - x^3$:

   • Compute $f(-x)$: $f(-x) = 2(-x) - (-x)^3 = -2x + x^3$
   • Compare this with $-f(x)$: $-f(x) = -(2x - x^3) = -2x + x^3$
   • Since $f(-x) = -f(x)$, the function is odd.

2. For $g(x) = x$:

   • Compute $g(-x)$: $g(-x) = -x$
   • Compare this with $-g(x)$: $-g(x) = -x$
   • Since $g(-x) = -g(x)$, the function is odd.

3. For $h(x) = 2x^4$:

   • Compute $h(-x)$: $h(-x) = 2(-x)^4 = 2x^4$
   • Since $h(-x) = h(x)$, the function is even.

Final answers:

• $f(x) = 2x - x^3$ is odd.
• $g(x) = x$ is odd.
• $h(x) = 2x^4$ is even.

Would you like more details on any of these calculations?

5 Related Questions:

1. What is the geometric interpretation of odd and even functions?
2. How can you determine if a polynomial function is odd, even, or neither by looking at its degree?
3. How does symmetry about the origin relate to odd functions?
4. Can a function be both odd and even? Why or why not?
5. How does a function’s graph change based on whether it is odd, even, or neither?

Tip: If all powers in a polynomial are odd, the function is likely odd, and if all powers are even, it's likely even.