Find the domain of the following functions.
`f(x)=(sin^(-1)(x-3))/(sqrt(9-x^(2)))`

Answer

To find the domain of the function \( f(x) = \frac{\sin^{-1}(x-3)}{\sqrt{9-x^2}} \), we need to consider the conditions under which both the numerator and the denominator are defined.

### Step 1: Analyze the numerator
The numerator is \( \sin^{-1}(x-3) \). The inverse sine function is defined for inputs in the range \([-1, 1]\). Therefore, we need:
\[ -1 \leq x - 3 \leq 1 \]
Adding 3 to all parts of the inequality gives:
\[ 2 \leq x \leq 4 \]

### Step 2: Analyze the denominator
The denominator is \( \sqrt{9 - x^2} \). The square root function is defined for non-negative inputs, so we need:
\[ 9 - x^2 > 0 \]
This can be rearranged to:
\[ x^2 < 9 \]
Taking the square root of both sides gives:
\[ -3 < x < 3 \]

### Step 3: Combine the conditions
Now we have two conditions:
1. \( 2 \leq x \leq 4 \)
2. \( -3 < x < 3 \)

To find the domain of \( f(x) \), we take the intersection of these two intervals:
- The first interval is \([2, 4]\).
- The second interval is \((-3, 3)\).

The intersection of these two intervals is:
\[ [2, 3) \]

### Conclusion
Thus, the domain of the function \( f(x) \) is:
\[ \boxed{[2, 3)} \]