Write the domain of the following real functions
`(1)/(sqrt(x^(2)-7))`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{x^2 - 7}} \), we need to ensure that the expression under the square root is positive, as the square root of a negative number is not defined in the real number system. Additionally, the denominator cannot be zero.

### Step-by-Step Solution:

1. **Identify the expression under the square root**:
We start with the expression \( x^2 - 7 \).

2. **Set the expression greater than zero**:
To ensure the function is defined, we need:
\[ x^2 - 7 > 0 \]

3. **Solve the inequality**:
Rearranging the inequality gives:
\[ x^2 > 7 \]
Now, we can take the square root of both sides. Remember that taking the square root introduces both positive and negative solutions:
\[ |x| > \sqrt{7} \]
This means:
\[ x < -\sqrt{7} \quad \text{or} \quad x > \sqrt{7} \]

4. **Write the domain in interval notation**:
The values of \( x \) that satisfy this inequality are:
\[ (-\infty, -\sqrt{7}) \cup (\sqrt{7}, \infty) \]

### Final Answer:
The domain of the function \( f(x) = \frac{1}{\sqrt{x^2 - 7}} \) is:
\[ (-\infty, -\sqrt{7}) \cup (\sqrt{7}, \infty) \]