If `f(x)="sin ln" (sqrt(4-x^(2)))/(1-x),` then

Options

A. its domain is (-1,1)
B. its domain is (-2,1)
C. its domain is (-2,5)
D. its domain is (1,2)

Answer

To find the domain of the function \( f(x) = \frac{\sin(\ln(\sqrt{4 - x^2}))}{1 - x} \), we need to determine the values of \( x \) for which the function is defined. This involves checking two main conditions:

1. The expression inside the logarithm must be positive.
2. The denominator must not be zero.

### Step 1: Determine when the logarithm is defined

The logarithm function \( \ln(y) \) is defined for \( y > 0 \). Here, we have:

\[ \sqrt{4 - x^2} > 0 \]

This implies:

\[ 4 - x^2 > 0 \]

### Step 2: Solve the inequality

Rearranging the inequality gives:

\[ x^2 < 4 \]

Taking the square root of both sides, we find:

\[ -2 < x < 2 \]

### Step 3: Determine when the denominator is not zero

The denominator \( 1 - x \) must not be equal to zero:

\[ 1 - x \neq 0 \implies x \neq 1 \]

### Step 4: Combine the conditions

From Step 2, we have \( -2 < x < 2 \), and from Step 3, we have \( x \neq 1 \). Therefore, the domain of the function is:

\[ (-2, 2) \text{ excluding } x = 1 \]

This can be expressed in interval notation as:

\[ (-2, 1) \cup (1, 2) \]

### Conclusion

The domain of the function \( f(x) = \frac{\sin(\ln(\sqrt{4 - x^2}))}{1 - x} \) is:

\[ (-2, 1) \cup (1, 2) \]