Which of the following is an odd function?
I. `f(x)=3x^(3)+5`
II. `g(x)=4x^(6)+2x^(4)-3x^(2)`
III. `h(x)=7x^(5)-8x^(3)+12x`

Options

A. only I
B. only II
C. only III
D. only I and II

Explanation

To determine which of the given functions is an odd function, we will use the property of odd functions: a function \( f(x) \) is odd if \( f(-x) = -f(x) \).

### Step-by-Step Solution:

1. **Check the first function: \( f(x) = 3x^3 + 5 \)**

- Calculate \( f(-x) \):
\[ f(-x) = 3(-x)^3 + 5 = 3(-x^3) + 5 = -3x^3 + 5 \]

- Calculate \( -f(x) \):
\[ -f(x) = -(3x^3 + 5) = -3x^3 - 5 \]

- Compare \( f(-x) \) and \( -f(x) \):
\[ f(-x) = -3x^3 + 5 \quad \text{and} \quad -f(x) = -3x^3 - 5 \]
Since \( f(-x) \neq -f(x) \), \( f(x) \) is **not an odd function**.

2. **Check the second function: \( g(x) = 4x^6 + 2x^4 - 3x^2 \)**

- Calculate \( g(-x) \):
\[ g(-x) = 4(-x)^6 + 2(-x)^4 - 3(-x)^2 = 4x^6 + 2x^4 - 3x^2 \]
(All powers are even, so they remain positive.)

- Calculate \( -g(x) \):
\[ -g(x) = -(4x^6 + 2x^4 - 3x^2) = -4x^6 - 2x^4 + 3x^2 \]

- Compare \( g(-x) \) and \( -g(x) \):
\[ g(-x) = 4x^6 + 2x^4 - 3x^2 \quad \text{and} \quad -g(x) = -4x^6 - 2x^4 + 3x^2 \]
Since \( g(-x) \neq -g(x) \), \( g(x) \) is **not an odd function**.

3. **Check the third function: \( h(x) = 7x^5 - 8x^3 + 12x \)**

- Calculate \( h(-x) \):
\[ h(-x) = 7(-x)^5 - 8(-x)^3 + 12(-x) = -7x^5 + 8x^3 - 12x \]

- Calculate \( -h(x) \):
\[ -h(x) = -(7x^5 - 8x^3 + 12x) = -7x^5 + 8x^3 - 12x \]

- Compare \( h(-x) \) and \( -h(x) \):
\[ h(-x) = -7x^5 + 8x^3 - 12x \quad \text{and} \quad -h(x) = -7x^5 + 8x^3 - 12x \]
Since \( h(-x) = -h(x) \), \( h(x) \) is an **odd function**.

### Conclusion:
The only odd function among the given options is **III. \( h(x) = 7x^5 - 8x^3 + 12x \)**.