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Let's analyze the symmetry of each given function to determine whether they are even, odd, or neither.

### 1. \( f(x) = \frac{4}{x^5} \)

- **Test for even symmetry:** A function is even if \( f(-x) = f(x) \).
  \[
  f(-x) = \frac{4}{(-x)^5} = \frac{4}{-x^5} = -\frac{4}{x^5}
  \]
  Since \( f(-x) \neq f(x) \), the function is not even.

- **Test for odd symmetry:** A function is odd if \( f(-x) = -f(x) \).
  \[
  f(-x) = -\frac{4}{x^5} \quad \text{and} \quad -f(x) = -\frac{4}{x^5}
  \]
  Since \( f(-x) = -f(x) \), the function is odd.

Thus, \( f(x) \) is **odd**.

### 2. \( g(x) = x^2 + 2x + 2 \)

- **Test for even symmetry:**
  \[
  g(-x) = (-x)^2 + 2(-x) + 2 = x^2 - 2x + 2
  \]
  Since \( g(-x) \neq g(x) \), the function is not even.

- **Test for odd symmetry:**
  \[
  g(-x) = x^2 - 2x + 2 \quad \text{and} \quad -g(x) = -(x^2 + 2x + 2) = -x^2 - 2x - 2
  \]
  Since \( g(-x) \neq -g(x) \), the function is neither odd.

Thus, \( g(x) \) is **neither even nor odd**.

### 3. \( h(x) = 3x^3 - 5x \)

- **Test for even symmetry:**
  \[
  h(-x) = 3(-x)^3 - 5(-x) = -3x^3 + 5x
  \]
  Since \( h(-x) \neq h(x) \), the function is not even.

- **Test for odd symmetry:**
  \[
  h(-x) = -3x^3 + 5x \quad \text{and} \quad -h(x) = -(3x^3 - 5x) = -3x^3 + 5x
  \]
  Since \( h(-x) = -h(x) \), the function is odd.

Thus, \( h(x) \) is **odd**.

### Conclusion:
- \( f(x) = \frac{4}{x^5} \) is **odd**.
- \( g(x) = x^2 + 2x + 2 \) is **neither even nor odd**.
- \( h(x) = 3x^3 - 5x \) is **odd**.

Would you like more details on any of these? Here are some related questions:

1. How do we determine if a function is neither even nor odd?
2. What are some examples of functions that are both even and odd?
3. Can a polynomial function be odd without being a single term?
4. What happens when we combine even and odd functions?
5. How do symmetries of a function affect its graph?

**Tip:** Odd functions exhibit rotational symmetry around the origin, while even functions exhibit symmetry about the y-axis.

Select the symmetry that best describes the function.

Understand how to determine the symmetry of functions with examples like f(x) = 4/x^5, g(x) = x^2 + 2x + 2, and h(x) = 3x^3 - 5x. Learn about even, odd, and neither function types, their properties, and how to test for symmetry. Suitable for students in Grades 10-12.

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