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.

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