Find the domain and range of
`f(x) = sin{ln((sqrt(4-x^2))/(1-x))}`

Answer

To find the domain and range of the function \( f(x) = \sin\left(\ln\left(\frac{\sqrt{4-x^2}}{1-x}\right)\right) \), we will follow these steps:

### Step 1: Determine the domain of the function

1. **Identify the conditions for the logarithm**: The argument of the logarithm must be positive:
\[ \frac{\sqrt{4-x^2}}{1-x} > 0 \]

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

3. **Analyze the denominator**: The term \( 1-x \) must also be positive:
\[ 1 - x > 0 \implies x < 1 \]

4. **Combine the conditions**: We need to consider both conditions together:
- From \( -2 < x < 2 \)
- From \( x < 1 \)

The intersection of these conditions gives us:
\[ -2 < x < 1 \]

5. **Check for points where the function is not defined**: The function is not defined at \( x = 1 \) (as it makes the denominator zero). Therefore, we exclude this point.

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

### Step 2: Determine the range of the function

1. **Understand the behavior of the logarithm**: The logarithmic function \( \ln\left(\frac{\sqrt{4-x^2}}{1-x}\right) \) can take any real value depending on the value of \( x \) in the domain.

2. **Analyze the limits**:
- As \( x \) approaches \( -2 \), \( \sqrt{4-x^2} \) approaches \( \sqrt{0} = 0 \) and \( 1-x \) approaches \( 3 \). Thus, \( \frac{\sqrt{4-x^2}}{1-x} \) approaches \( 0 \), and \( \ln(0) \) approaches \( -\infty \).
- As \( x \) approaches \( 1 \), \( \sqrt{4-x^2} \) approaches \( \sqrt{3} \) and \( 1-x \) approaches \( 0 \) from the positive side. Thus, \( \frac{\sqrt{4-x^2}}{1-x} \) approaches \( +\infty \), and \( \ln(+\infty) \) approaches \( +\infty \).

3. **Behavior of the sine function**: The sine function oscillates between -1 and 1 for all real values. Therefore, the range of \( f(x) = \sin(k) \) where \( k \) can take any real value is:
\[ \text{Range: } [-1, 1] \]

### Final Answer

- **Domain**: \( (-2, 1) \)
- **Range**: \( [-1, 1] \)