Domain of the function `f(x) = sin(ln(sqrt(4-x^2)/(1-x)))` is

Options

A. [0,1]
B. (-1,0)
C. [-1,1]
D. (-2,1)

Answer

To find the domain of the function \( f(x) = \sin\left(\ln\left(\frac{\sqrt{4 - x^2}}{1 - x}\right)\right) \), we need to ensure that the argument of the logarithm is positive and that the square root is defined.

### Step-by-Step Solution:

1. **Identify the Conditions for the Function**:
The function \( f(x) \) involves a logarithm, which is defined only for positive arguments. Therefore, we need:
\[ \frac{\sqrt{4 - x^2}}{1 - x} > 0 \]

2. **Analyze the Numerator**:
The numerator \( \sqrt{4 - x^2} \) is defined and positive when:
\[ 4 - x^2 > 0 \implies x^2 < 4 \implies -2 < x < 2 \]

3. **Analyze the Denominator**:
The denominator \( 1 - x \) is positive when:
\[ 1 - x > 0 \implies x < 1 \]

4. **Combine the Conditions**:
Now we need to combine the two conditions:
- From the numerator: \( -2 < x < 2 \)
- From the denominator: \( x < 1 \)

The combined condition is:
\[ -2 < x < 1 \]

5. **Conclusion**:
Therefore, the domain of the function \( f(x) \) is:
\[ (-2, 1) \]

### Final Answer:
The domain of the function \( f(x) = \sin\left(\ln\left(\frac{\sqrt{4 - x^2}}{1 - x}\right)\right) \) is \( (-2, 1) \).