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 \( f(x) = \sqrt{\log_{(x^2 - 1)}(x)} \), we need to ensure that the expression under the square root is defined and non-negative. This requires us to analyze the logarithmic function and its base.

### Step 1: Conditions for the logarithm
The logarithm \( \log_{(x^2 - 1)}(x) \) is defined under the following conditions:
1. \( x > 0 \) (the argument of the logarithm must be positive).
2. The base \( x^2 - 1 \) must be greater than 1.
3. The base \( x^2 - 1 \) must not be equal to 1.

### Step 2: Analyze the first condition
From the first condition:
- \( x > 0 \)

### Step 3: Analyze the second condition
For the base \( x^2 - 1 > 1 \):
\[ x^2 - 1 > 1 \implies x^2 > 2 \implies x > \sqrt{2} \text{ or } x < -\sqrt{2} \]
However, since \( x > 0 \), we only consider:
\[ x > \sqrt{2} \]

### Step 4: Analyze the third condition
For the base \( x^2 - 1 \neq 1 \):
\[ x^2 - 1 \neq 1 \implies x^2 \neq 2 \implies x \neq \sqrt{2} \text{ and } x \neq -\sqrt{2} \]
Since \( x > 0 \), we only need to exclude \( x = \sqrt{2} \).

### Step 5: Combine the conditions
From the conditions derived:
1. \( x > 0 \)
2. \( x > \sqrt{2} \)
3. \( x \neq \sqrt{2} \)

The combined condition gives us:
\[ x > \sqrt{2} \]

### Conclusion
Thus, the domain of the function \( f(x) = \sqrt{\log_{(x^2 - 1)}(x)} \) is:
\[ \text{Domain: } x \in (\sqrt{2}, \infty) \]

### Final Answer
The domain of definition of the function \( f(x) = \sqrt{\log_{(x^2 - 1)}(x)} \) is \( (\sqrt{2}, \infty) \).

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