The domain of definition of f(x) = `sqrt(1-|x|)/(2-|x|)` is

Options

A. `(-infty,-1)cup (2,infty)`
B. `[-1,1] cup (2,infty)cup(-infty,-2)`
C. `[-1 ,1]`
D. `[-1,1]cup (2,infty)`

Answer

To find the domain of the function \( f(x) = \frac{\sqrt{1 - |x|}}{2 - |x|} \), we need to ensure that the expression is defined. This involves two conditions:

1. The expression under the square root must be non-negative.
2. The denominator must not be zero.

### Step 1: Ensure the expression under the square root is non-negative

The expression under the square root is \( 1 - |x| \). For this to be non-negative, we need:

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

This simplifies to:

\[ |x| \leq 1 \]

This means that \( x \) must lie within the interval:

\[ -1 \leq x \leq 1 \]

### Step 2: Ensure the denominator is not zero

Next, we need to check the denominator \( 2 - |x| \). We want to ensure that this does not equal zero:

\[ 2 - |x| \neq 0 \]

This simplifies to:

\[ |x| \neq 2 \]

However, since \( |x| \leq 1 \) from the previous step, this condition is automatically satisfied because \( |x| \) cannot equal 2 when \( |x| \) is constrained to be at most 1.

### Step 3: Combine the conditions

From the first step, we found that \( -1 \leq x \leq 1 \). The second condition does not impose any additional restrictions since \( |x| \) cannot be 2 in this interval.

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

\[ [-1, 1] \]

### Final Answer

The domain of definition of \( f(x) = \frac{\sqrt{1 - |x|}}{2 - |x|} \) is:

\[ [-1, 1] \]

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