The domain of `f(x) = sqrt((2-|x|)/(3-|x|))` is

Options

A. `(-infty, -3) cup )3, infty)`
B. `[-2,3)`
C. `(-infty, -3) cup(3, infty) cup [-2,2]`
D. `(-infty, infty) - {-3,3}`

Answer

To find the domain of the function \( f(x) = \sqrt{\frac{2 - |x|}{3 - |x|}} \), we need to ensure that the expression inside the square root is non-negative. This means that:

1. The numerator \( 2 - |x| \) must be greater than or equal to 0.
2. The denominator \( 3 - |x| \) must be greater than 0 (since division by zero is undefined).

### Step 1: Analyze the numerator
We start with the condition for the numerator:

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

This can be rearranged to:

\[ |x| \leq 2 \]

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

\[ -2 \leq x \leq 2 \]

### Step 2: Analyze the denominator
Next, we analyze the condition for the denominator:

\[ 3 - |x| > 0 \]

This can be rearranged to:

\[ |x| < 3 \]

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

\[ -3 < x < 3 \]

### Step 3: Combine the conditions
Now we need to find the intersection of the two intervals obtained from the numerator and the denominator.

1. From the numerator, we have \( [-2, 2] \).
2. From the denominator, we have \( (-3, 3) \).

The intersection of these two intervals is:

\[ [-2, 2] \]

### Step 4: Exclude points where the denominator is zero
We also need to ensure that the denominator does not equal zero. The denominator \( 3 - |x| = 0 \) when \( |x| = 3 \), which corresponds to \( x = 3 \) and \( x = -3 \). However, these points are already excluded from our interval since they are not included in \( [-2, 2] \).

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

\[ [-2, 2] \]