https://file.aiquickdraw.com/mathbot/file/173065342167354a9omfw.jpg

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?

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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.

Determine whether the following functions are even, odd, or neither: 1) f(x) = x^4 - 3x^2 + 7, 2) f(x) = (x^2 + 4) / (x^3 - x), 3) f(x) = 4x^3 - 9.

Learn how to determine whether the functions f(x) = x^4 - 3x^2 + 7, f(x) = (x^2 + 4) / (x^3 - x), and f(x) = 4x^3 - 9 are even, odd, or neither. This problem covers concepts of symmetry and polynomial functions, suitable for students in Grades 10-12.

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