To determine if the function $f(x) = 3x^3 - x^2 + 1$ is odd, even, or neither, we need to check the following:

1. For evenness: A function is even if $f(-x) = f(x)$.
2. For oddness: A function is odd if $f(-x) = -f(x)$.

Let's calculate $f(-x)$ first:

$f(-x) = 3(-x)^3 - (-x)^2 + 1$

Simplifying:

$f(-x) = 3(-x^3) - (x^2) + 1 = -3x^3 - x^2 + 1$

Now, compare $f(-x)$ with both $f(x)$ and $-f(x)$:

• $f(x) = 3x^3 - x^2 + 1$
• $-f(x) = -(3x^3 - x^2 + 1) = -3x^3 + x^2 - 1$

Conclusion:

• $f(-x) \neq f(x)$, so the function is not even.
• $f(-x) \neq -f(x)$, so the function is not odd.

Thus, the function is neither odd nor even. The correct answer is (b) neither.

Do you want more details or have any questions?

Here are some related questions:

1. How do you check if a function is odd or even in general?
2. What are the real-life applications of odd and even functions?
3. Can a function be both odd and even? If yes, which one?
4. What is the symmetry of an even function?
5. How does the degree of a polynomial influence whether it's odd or even?

Tip: Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.