Find the domain of the following functions.
`f(x)=(sin^(-1)(3-x))/(ln(|x|-2))`

Answer

To find the domain of the function \( f(x) = \frac{\sin^{-1}(3 - x)}{\ln(|x| - 2)} \), we need to consider the conditions imposed by both the sine inverse function and the logarithmic function.

### Step 1: Determine the condition for the sine inverse function
The sine inverse function, \( \sin^{-1}(y) \), is defined for \( y \) in the interval \([-1, 1]\). Therefore, we need:

\[ -1 \leq 3 - x \leq 1 \]

### Step 2: Solve the inequalities for \( x \)
We will break this compound inequality into two parts.

1. **First part:**
\[ 3 - x \geq -1 \]
Rearranging gives:
\[ x \leq 4 \]

2. **Second part:**
\[ 3 - x \leq 1 \]
Rearranging gives:
\[ x \geq 2 \]

Combining these results, we find:
\[ 2 \leq x \leq 4 \]

### Step 3: Determine the condition for the logarithmic function
The logarithmic function \( \ln(|x| - 2) \) is defined for \( |x| - 2 > 0 \), which simplifies to:
\[ |x| > 2 \]
This means:
1. \( x > 2 \)
2. \( x < -2 \)

### Step 4: Combine the conditions
Now we have two conditions:
1. From the sine inverse: \( 2 \leq x \leq 4 \)
2. From the logarithm: \( x > 2 \) or \( x < -2 \)

Since \( x < -2 \) does not overlap with \( 2 \leq x \leq 4 \), we only consider the condition \( x > 2 \).

### Step 5: Finalize the domain
Combining \( 2 \leq x \leq 4 \) with \( x > 2 \), we find:
\[ 2 < x \leq 4 \]

Thus, the domain of the function \( f(x) \) is:
\[ (2, 4] \]

### Summary of the Domain
The domain of the function \( f(x) = \frac{\sin^{-1}(3 - x)}{\ln(|x| - 2)} \) is:
\[ (2, 4] \]