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.