Let the function
`f(x)=4 sinx+3 cos x+ log (|x|+sqrt(1+x^(2)))` be defined on the interval [0,1]. The odd extension of f(x) to the inteval `[-1,1]` is

Options

A. `4 sin x+ 3 cos x+log(|x|+sqrt(1+x^(2))), -1 lex lt 0`
B. `4 sin x- 3 cos x-log(|x|+sqrt(1+x^(2))), -1 lex lt 0`
C. `4 sin x+ 3 cos x+log(|x|+sqrt(1+x^(2))), -1 lex lt 0`
D. none of these

Explanation

To find the odd extension of the function \( f(x) = 4 \sin x + 3 \cos x + \log(|x| + \sqrt{1 + x^2}) \) defined on the interval \([0, 1]\), we will follow these steps:

### Step 1: Define the odd extension
The odd extension \( g(x) \) of a function \( f(x) \) defined on an interval can be expressed as:
\[ g(x) = \begin{cases} f(x) & \text{if } x \in [0, 1] \\ -f(-x) & \text{if } x \in [-1, 0] \end{cases} \]

### Step 2: Determine \( g(x) \) for \( x \in [0, 1] \)
For \( x \in [0, 1] \), we have:
\[ g(x) = f(x) = 4 \sin x + 3 \cos x + \log(|x| + \sqrt{1 + x^2}) \]

### Step 3: Calculate \( g(x) \) for \( x \in [-1, 0] \)
For \( x \in [-1, 0] \), we need to compute \( -f(-x) \):
\[ g(x) = -f(-x) = -\left(4 \sin(-x) + 3 \cos(-x) + \log(|-x| + \sqrt{1 + (-x)^2})\right) \]

### Step 4: Simplify \( f(-x) \)
Using the properties of sine and cosine:
\[ \sin(-x) = -\sin x \quad \text{and} \quad \cos(-x) = \cos x \]
Thus,
\[ f(-x) = 4(-\sin x) + 3\cos x + \log(|-x| + \sqrt{1 + x^2}) = -4 \sin x + 3 \cos x + \log(|x| + \sqrt{1 + x^2}) \]

### Step 5: Substitute \( f(-x) \) back into \( g(x) \)
Now substituting \( f(-x) \) into the equation for \( g(x) \):
\[ g(x) = -(-4 \sin x + 3 \cos x + \log(|x| + \sqrt{1 + x^2})) \]
This simplifies to:
\[ g(x) = 4 \sin x - 3 \cos x - \log(|x| + \sqrt{1 + x^2}) \]

### Final Form of \( g(x) \)
Combining both cases, we have:
\[ g(x) = \begin{cases} 4 \sin x + 3 \cos x + \log(|x| + \sqrt{1 + x^2}) & \text{if } x \in [0, 1] \\ 4 \sin x - 3 \cos x - \log(|x| + \sqrt{1 + x^2}) & \text{if } x \in [-1, 0] \end{cases} \]

### Summary
Thus, the odd extension of the function \( f(x) \) to the interval \([-1, 1]\) is:
\[ g(x) = \begin{cases} 4 \sin x + 3 \cos x + \log(|x| + \sqrt{1 + x^2}) & \text{if } x \in [0, 1] \\ 4 \sin x - 3 \cos x - \log(|x| + \sqrt{1 + x^2}) & \text{if } x \in [-1, 0] \end{cases} \]