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Let's analyze each function \( f(x) \), \( g(x) \), and \( h(x) \) to determine whether they are odd, even, or neither.

### Steps to classify functions:
1. **Even functions**: A function is even if \( f(-x) = f(x) \) for all \( x \). Graphically, the function is symmetric about the y-axis.
2. **Odd functions**: A function is odd if \( f(-x) = -f(x) \) for all \( x \). Graphically, the function is symmetric about the origin.
3. **Neither**: If a function is neither even nor odd, it doesn't satisfy either condition.

### 1. \( f(x) = 3x^3 + x \)
- Find \( f(-x) \):
  \[
  f(-x) = 3(-x)^3 + (-x) = 3(-x^3) - x = -3x^3 - x
  \]
- Check if it's odd or even:
  - \( f(-x) = -(3x^3 + x) = -f(x) \), so \( f(x) \) is **odd**.

### 2. \( g(x) = \sqrt{x} \)
- Find \( g(-x) \):
  \[
  g(-x) = \sqrt{-x}
  \]
  The square root of a negative number is undefined in the set of real numbers. So \( g(x) \) doesn't satisfy either condition for being even or odd.
  - Therefore, \( g(x) \) is **neither**.

### 3. \( h(x) = 3 - x^2 \)
- Find \( h(-x) \):
  \[
  h(-x) = 3 - (-x)^2 = 3 - x^2
  \]
  - Since \( h(-x) = h(x) \), the function is **even**.

### Summary:
- \( f(x) = 3x^3 + x \) is **odd**.
- \( g(x) = \sqrt{x} \) is **neither**.
- \( h(x) = 3 - x^2 \) is **even**.

Would you like a more detailed breakdown of any function, or do you have any questions?

### Related Questions:
1. How can you graphically identify if a function is odd or even?
2. Why does the function \( g(x) = \sqrt{x} \) not fit into either category?
3. Can a function be both even and odd? Why or why not?
4. How does the degree of a polynomial influence whether it's odd or even?
5. What happens if you try to apply these properties to piecewise functions?

### Tip:
If you are ever unsure whether a function is odd, even, or neither, always test \( f(-x) \) and compare it to both \( f(x) \) and \( -f(x) \).

For each equation below, determine if the function is Odd, Even, or Neither: f(x) = 3x^3 + x, g(x) = sqrt(x), h(x) = 3 - x^2.

Explore how to determine if a function is odd, even, or neither for equations such as f(x) = 3x^3 + x, g(x) = sqrt(x), and h(x) = 3 - x^2. Learn key algebraic concepts and symmetry rules. Suitable for high school students.

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