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To determine whether the given functions are **odd**, **even**, or **neither**, we can apply the following tests:

- **Even function**: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).
- **Odd function**: A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).

Let's test both functions using these criteria.

### a) \( f(x) = x^3 - x + 3 \)

1. Calculate \( f(-x) \):
   \[
   f(-x) = (-x)^3 - (-x) + 3 = -x^3 + x + 3
   \]
   
2. Compare \( f(-x) \) with \( f(x) \):
   \[
   f(-x) = -x^3 + x + 3 \quad \text{and} \quad f(x) = x^3 - x + 3
   \]
   
   Clearly, \( f(-x) \neq f(x) \) (so it's not even), and \( f(-x) \neq -f(x) \) (so it's not odd).

**Conclusion**: The function \( f(x) = x^3 - x + 3 \) is **neither** even nor odd.

---

### b) \( f(x) = x^2 + 6 \)

1. Calculate \( f(-x) \):
   \[
   f(-x) = (-x)^2 + 6 = x^2 + 6
   \]
   
2. Compare \( f(-x) \) with \( f(x) \):
   \[
   f(-x) = x^2 + 6 \quad \text{and} \quad f(x) = x^2 + 6
   \]
   
   Since \( f(-x) = f(x) \), the function is even.

**Conclusion**: The function \( f(x) = x^2 + 6 \) is **even**.

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Would you like further details or explanations on any step?

Here are 5 related questions to expand your understanding:
1. What are the implications of a function being neither odd nor even?
2. Can a function be both odd and even? If yes, under what conditions?
3. How do symmetry properties of functions relate to their odd/even nature?
4. What transformations of a function could change its odd/even nature?
5. How does the domain of a function affect the determination of odd or

State whether each is odd, even, or neither for the functions f(x) = x^3 - x + 3 and f(x) = x^2 + 6.

Learn how to determine if functions are odd, even, or neither using the examples f(x) = x^3 - x + 3 and f(x) = x^2 + 6. This step-by-step solution explains how to test for function symmetry and determine the nature of the functions.

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