The domain of the fiction f defined by `f(x)=(l)/(sqrt(9-x^(2)))` is

Options

A. (0,3)
B. (-3,0)
C. [-3,3]
D. (-3,3)

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{9 - x^2}} \), we need to determine the values of \( x \) for which the function is defined.

### Step-by-step solution:

1. **Identify the condition for the square root**:
The expression under the square root, \( 9 - x^2 \), must be greater than zero since the square root of a negative number is not defined in the real number system. Therefore, we set up the inequality:
\[ 9 - x^2 > 0 \]

2. **Rearranging the inequality**:
Rearranging the inequality gives us:
\[ x^2 < 9 \]

3. **Taking the square root**:
To solve \( x^2 < 9 \), we take the square root of both sides. Remember that taking the square root introduces both positive and negative solutions:
\[ -3 < x < 3 \]

4. **Expressing the domain**:
The solution \( -3 < x < 3 \) means that \( x \) can take any value between -3 and 3, but not including -3 and 3 themselves. Therefore, we can express the domain in interval notation:
\[ \text{Domain of } f = (-3, 3) \]

### Final Answer:
The domain of the function \( f(x) = \frac{1}{\sqrt{9 - x^2}} \) is \( (-3, 3) \).