To determine whether each function is even, odd, or neither, let's analyze each case.

Definitions Recap

• Even function: $f(x) = f(-x)$. The function is symmetric about the y-axis.
• Odd function: $f(-x) = -f(x)$. The function has rotational symmetry about the origin.
• Neither: If neither of the above conditions is satisfied.

Analysis of Each Function

1. Function 1: $f(x) = x^4 - 3x^2 + 7$

   • Substitute $-x$ for $x$: $f(-x) = (-x)^4 - 3(-x)^2 + 7 = x^4 - 3x^2 + 7 = f(x)$
   • Since $f(x) = f(-x)$, this function is even.

2. Function 2: $f(x) = \frac{x^2 + 4}{x^3 - x}$

   • Substitute $-x$ for $x$: $f(-x) = \frac{(-x)^2 + 4}{(-x)^3 - (-x)} = \frac{x^2 + 4}{-x^3 + x} = -f(x)$
   • Since $f(-x) = -f(x)$, this function is odd.

3. Function 3: $f(x) = 4x^3 - 9$

   • Substitute $-x$ for $x$: $f(-x) = 4(-x)^3 - 9 = -4x^3 - 9$
   • Here, $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, so this function is neither.

Final Answers

1. Even
2. Odd
3. Neither

Your answers appear to be correct based on this analysis.

Would you like more details on any of the steps, or do you have further questions?

Here are 5 follow-up questions related to this topic:

1. How can you visually identify an even or odd function from its graph?
2. Why is the symmetry property essential for even and odd functions?
3. Can a function be both even and odd? If so, under what conditions?
4. How do transformations (like translations) affect the evenness or oddness of a function?
5. Can composite functions be even or odd?

Tip: For polynomials, check each term: even powers of $x$ suggest even symmetry, and odd powers suggest odd symmetry.