The range of the function `f(x)=sin{log_(10)((sqrt(4-x^(2)))/(1-x))}` ,is

Options

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

Answer

To find the range of the function \( f(x) = \sin\left(\log_{10}\left(\frac{\sqrt{4 - x^2}}{1 - x}\right)\right) \), we will follow these steps:

### Step 1: Determine the Domain of the Function
1. **Identify Restrictions from the Square Root**:
The expression inside the square root, \( 4 - x^2 \), must be non-negative:
\[ 4 - x^2 \geq 0 \implies x^2 \leq 4 \implies -2 \leq x \leq 2 \]

2. **Identify Restrictions from the Logarithm**:
The argument of the logarithm, \( \frac{\sqrt{4 - x^2}}{1 - x} \), must be positive:
\[ \frac{\sqrt{4 - x^2}}{1 - x} > 0 \]
This requires:
- \( \sqrt{4 - x^2} > 0 \) (which is true for \( -2 < x < 2 \))
- \( 1 - x > 0 \implies x < 1 \)

3. **Combine the Restrictions**:
The domain of \( f(x) \) is the intersection of the intervals:
\[ -2 < x < 2 \quad \text{and} \quad x < 1 \implies -2 < x < 1 \]

### Step 2: Analyze the Expression Inside the Logarithm
We need to analyze the behavior of \( \frac{\sqrt{4 - x^2}}{1 - x} \) over the interval \( -2 < x < 1 \).

1. **Evaluate at the Endpoints**:
- As \( x \to -2 \):
\[ \frac{\sqrt{4 - (-2)^2}}{1 - (-2)} = \frac{\sqrt{0}}{3} = 0 \]
- As \( x \to 1 \):
\[ \frac{\sqrt{4 - 1^2}}{1 - 1} \to \text{undefined (approaches } +\infty\text{)} \]

2. **Evaluate at a Point in the Interval**:
- At \( x = 0 \):
\[ \frac{\sqrt{4 - 0^2}}{1 - 0} = \frac{2}{1} = 2 \]

### Step 3: Determine the Range of the Logarithm
The logarithm function \( \log_{10}(y) \) is defined for \( y > 0 \) and ranges from \( -\infty \) to \( +\infty \) as \( y \) moves from \( 0 \) to \( +\infty \).

1. **Behavior of the Logarithm**:
- As \( x \to -2 \), \( \frac{\sqrt{4 - x^2}}{1 - x} \to 0 \) implies \( \log_{10}(0) \to -\infty \).
- As \( x \to 1 \), \( \frac{\sqrt{4 - x^2}}{1 - x} \to +\infty \) implies \( \log_{10}(+\infty) \to +\infty \).

### Step 4: Determine the Range of the Sine Function
The sine function, \( \sin(t) \), where \( t \) ranges from \( -\infty \) to \( +\infty \), oscillates between -1 and 1.

### Conclusion
Thus, the range of \( f(x) = \sin\left(\log_{10}\left(\frac{\sqrt{4 - x^2}}{1 - x}\right)\right) \) is:
\[ \text{Range of } f(x) = [-1, 1] \]