The domain of definition of the function `f(x) = sqrt(log_(x^(2)-1)) x` is

Options

A. `( sqrt2, oo)`
B. `(0, oo)`
C. `(1, oo)`
D. none of these

Explanation

To find the domain of the function \( f(x) = \sqrt{\log(x^2 - 1)} \cdot x \), we need to ensure that the expression under the square root is non-negative and that the logarithm is defined.

### Step-by-step Solution:

1. **Identify the conditions for the logarithm:**
The logarithm \( \log(x^2 - 1) \) is defined when \( x^2 - 1 > 0 \). This leads to the inequality:
\[ x^2 - 1 > 0 \]
which simplifies to:
\[ x^2 > 1 \]
This implies:
\[ x > 1 \quad \text{or} \quad x < -1 \]

2. **Consider the square root condition:**
Since we have \( \sqrt{\log(x^2 - 1)} \), we need:
\[ \log(x^2 - 1) \geq 0 \]
This means:
\[ x^2 - 1 \geq 1 \]
which simplifies to:
\[ x^2 \geq 2 \]
This implies:
\[ x \geq \sqrt{2} \quad \text{or} \quad x \leq -\sqrt{2} \]

3. **Combine the conditions:**
From the first condition, we have \( x > 1 \) or \( x < -1 \). From the second condition, we have \( x \geq \sqrt{2} \) or \( x \leq -\sqrt{2} \).

- For \( x > 1 \), we need \( x \geq \sqrt{2} \), which is satisfied since \( \sqrt{2} \approx 1.414 \).
- For \( x < -1 \), we need \( x \leq -\sqrt{2} \), which is also satisfied since \( -\sqrt{2} \approx -1.414 \).

4. **Final domain:**
Thus, the domain of the function \( f(x) \) is:
\[ x \in [\sqrt{2}, \infty) \cup (-\infty, -\sqrt{2}] \]

### Conclusion:
The domain of the function \( f(x) = \sqrt{\log(x^2 - 1)} \cdot x \) is:
\[ \boxed{[\sqrt{2}, \infty) \cup (-\infty, -\sqrt{2}]} \]