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To determine whether the function \( f(x) = 3x^4 - x \) is even, odd, or neither, we evaluate the function at \( -x \) and compare it with \( f(x) \).

1. **Calculate \( f(-x) \):**
   \[
   f(-x) = 3(-x)^4 - (-x) = 3x^4 + x
   \]

2. **Check if \( f(x) \) is even or odd:**
   - A function is **even** if \( f(-x) = f(x) \).
   - A function is **odd** if \( f(-x) = -f(x) \).

   Given \( f(x) = 3x^4 - x \), we find:
   \[
   -f(x) = -(3x^4 - x) = -3x^4 + x
   \]

   Comparing \( f(-x) = 3x^4 + x \) with:
   - \( f(x) = 3x^4 - x \) shows that they are not equal, so the function is not even.
   - \( -f(x) = -3x^4 + x \) shows that they are also not equal, so the function is not odd.

Therefore, the function \( f(x) = 3x^4 - x \) is neither even nor odd.

Would you like more details or have any questions about this explanation?

**Related Questions:**
1. What defines a function as even?
2. How can you prove a function is odd using examples?
3. Are there specific types of functions that are always even or always odd?
4. How does the presence of mixed terms (like \( x^4 \) and \( x \)) affect the parity of a function?
5. Can a function be both even and odd?

**Tip:** When determining the parity of a function, substituting \( -x \) into the function and simplifying the expression can often reveal its nature directly through algebraic manipulation.

Determine whether the function f(x) = 3x^4 - x is even, odd or neither.

Learn how to determine whether the function f(x) = 3x^4 - x is even, odd, or neither by checking the function’s symmetry properties. Suitable for high school students learning algebra and function parity.

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