The domain of definition of the function` f(x) = sin log {(sqrt""(4-x^2))/(1-x))` is

Options

A. `[-2,1-1]`
B. `]-2,1[`
C. `[-2,0]`
D. None

Answer

To find the domain of the function \( f(x) = \sin \left( \log \left( \frac{\sqrt{4 - x^2}}{1 - x} \right) \right) \), we need to ensure that the expression inside the logarithm is positive and that the denominator is not zero.

### Step-by-Step Solution:

1. **Identify the logarithmic condition**:
The argument of the logarithm must be greater than zero:
\[ \frac{\sqrt{4 - x^2}}{1 - x} > 0 \]

2. **Set conditions for the numerator and denominator**:
- The numerator \( \sqrt{4 - x^2} \) must be greater than zero:
\[ \sqrt{4 - x^2} > 0 \implies 4 - x^2 > 0 \implies x^2 < 4 \implies -2 < x < 2 \]
- The denominator \( 1 - x \) must be greater than zero:
\[ 1 - x > 0 \implies x < 1 \]

3. **Combine the conditions**:
We now have two conditions:
- From \( \sqrt{4 - x^2} > 0 \): \( -2 < x < 2 \)
- From \( 1 - x > 0 \): \( x < 1 \)

We need to find the intersection of these intervals:
- The interval from the square root condition is \( (-2, 2) \).
- The interval from the denominator condition is \( (-\infty, 1) \).

4. **Determine the intersection**:
The intersection of \( (-2, 2) \) and \( (-\infty, 1) \) is:
\[ (-2, 1) \]

5. **Final domain**:
The domain of the function \( f(x) \) is:
\[ \text{Domain of } f(x) = (-2, 1) \]