The domain of ` sqrt(log_(x^2 -1)(x)` is

Options

A. `( sqrt(2),oo)`
B. `(0,oo)`
C. `(1,oo)`
D. None

Answer

To find the domain of the function \( f(x) = \sqrt{\log_{(x^2 - 1)}(x)} \), we need to ensure that the expression inside the square root is non-negative and that the logarithm is defined. Here are the steps to determine the domain:

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

### Step 2: Solve the conditions
1. **Base greater than 0**:
\[ x^2 - 1 > 0 \implies x^2 > 1 \implies x > 1 \text{ or } x < -1 \]

2. **Base not equal to 1**:
\[ x^2 - 1 \neq 1 \implies x^2 \neq 2 \implies x \neq \sqrt{2} \text{ and } x \neq -\sqrt{2} \]

3. **Argument greater than 0**:
\[ x > 0 \]

### Step 3: Combine the conditions
From the above conditions, we have:
- From \( x^2 - 1 > 0 \), we have \( x > 1 \) or \( x < -1 \).
- From \( x > 0 \), we only consider \( x > 1 \).
- We also need to exclude \( x = \sqrt{2} \).

Thus, the combined conditions give us:
\[ x > 1 \text{ and } x \neq \sqrt{2} \]

### Step 4: Final domain
The domain of the function \( f(x) \) is:
\[ (1, \infty) \text{ excluding } \sqrt{2} \]
In interval notation, this can be expressed as:
\[ (1, \sqrt{2}) \cup (\sqrt{2}, \infty) \]

### Final Answer:
The domain of \( \sqrt{\log_{(x^2 - 1)}(x)} \) is \( (1, \sqrt{2}) \cup (\sqrt{2}, \infty) \).

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