The domain of `y=9-sqrt(9-x^(2))` is :

Options

A. `x lt-3`
B. `-3ltxlt3`
C. `x in [-3, 3]`
D. none of these

Answer

To find the domain of the function \( y = 9 - \sqrt{9 - x^2} \), we need to ensure that the expression under the square root is non-negative, as the square root of a negative number is not defined in the realm of real numbers.

### Step-by-Step Solution:

1. **Identify the expression under the square root**:
\[ 9 - x^2 \]
We need this expression to be greater than or equal to zero:
\[ 9 - x^2 \geq 0 \]

2. **Rearranging the inequality**:
To solve the inequality, we can rearrange it:
\[ -x^2 \geq -9 \]
By multiplying both sides by -1 (remember to flip the inequality sign):
\[ x^2 \leq 9 \]

3. **Taking the square root**:
Now, we take the square root of both sides. Remember that taking the square root introduces both positive and negative solutions:
\[ -3 \leq x \leq 3 \]

4. **Expressing the domain**:
The domain of the function can be expressed in interval notation as:
\[ [-3, 3] \]

### Final Answer:
The domain of \( y = 9 - \sqrt{9 - x^2} \) is:
\[ [-3, 3] \]