The domain of the function f (x) = ` 3sqrt((x)/(1 - |x|))`

Options

A. `( - infty - 1) cup (- 1, 1) `
B. `(- infty , -1)`
C. `[0, infty)`
D. None of these

Answer

To find the domain of the function \( f(x) = 3\sqrt{\frac{x}{1 - |x|}} \), we need to ensure that the expression inside the square root is defined and non-negative. This involves two main conditions:

### Step 1: Identify the denominator condition
The denominator \( 1 - |x| \) should not be equal to zero because division by zero is undefined. Therefore, we set up the equation:

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

This implies:

\[ |x| \neq 1 \]

Thus, \( x \) cannot be equal to 1 or -1.

### Step 2: Identify the positivity condition
For the square root to be defined, the expression inside it must be positive:

\[ 1 - |x| > 0 \]

This simplifies to:

\[ |x| < 1 \]

### Step 3: Solve the absolute value inequality
The inequality \( |x| < 1 \) can be rewritten as:

\[ -1 < x < 1 \]

### Step 4: Combine the conditions
From Step 1, we know that \( x \neq 1 \) and \( x \neq -1 \). However, since \( |x| < 1 \) already restricts \( x \) to the interval (-1, 1), we do not need to worry about the endpoints being included in the domain.

### Conclusion
Thus, the domain of the function \( f(x) \) is:

\[ \text{Domain of } f(x) = (-1, 1) \]