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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**.

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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.

Is the function f(x) = 3x^3 - x^2 + 1 odd, even, or neither?

Learn how to determine if the polynomial function f(x) = 3x^3 - x^2 + 1 is odd, even, or neither. This problem covers core algebraic concepts such as the properties of odd and even functions and their symmetries. Suitable for students in Grades 9-12.

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