Domain of definition of the function `f(x) = log_2 (-log_(1/2) (1+x^(-4))-1)` is

Options

A. `(0,1)`
B. `(0,1]`
C. `[1,oo)`
D. `(1,oo)`

Explanation

To find the domain of the function \( f(x) = \log_2(-\log_{1/2}(1 + x^{-4}) - 1) \), we need to ensure that the expression inside the logarithm is positive. Let's break it down step by step.

### Step 1: Set up the inequality
The function \( f(x) \) is defined when:
\[ -\log_{1/2}(1 + x^{-4}) - 1 > 0 \]

### Step 2: Rearrange the inequality
Rearranging the inequality gives:
\[ -\log_{1/2}(1 + x^{-4}) > 1 \]
This can be rewritten as:
\[ \log_{1/2}(1 + x^{-4}) < -1 \]

### Step 3: Convert the logarithmic inequality
Using the property of logarithms, we can convert this to an exponential form:
\[ 1 + x^{-4} < (1/2)^{-1} \]
Since \( (1/2)^{-1} = 2 \), we have:
\[ 1 + x^{-4} < 2 \]

### Step 4: Solve for \( x^{-4} \)
Subtracting 1 from both sides gives:
\[ x^{-4} < 1 \]
This can be rewritten as:
\[ \frac{1}{x^4} < 1 \]

### Step 5: Solve the inequality
Taking the reciprocal (and flipping the inequality since \( x^4 > 0 \)):
\[ x^4 > 1 \]
Taking the fourth root of both sides results in:
\[ x > 1 \quad \text{or} \quad x < -1 \]

### Step 6: Consider the restriction on \( x \)
Since \( x^{-4} \) is undefined for \( x = 0 \), we must also exclude \( x = 0 \) from our domain.

### Step 7: Combine the results
Thus, the domain of the function \( f(x) \) is:
\[ (-\infty, -1) \cup (1, \infty) \]

### Final Answer
The domain of definition of the function \( f(x) = \log_2(-\log_{1/2}(1 + x^{-4}) - 1) \) is:
\[ (-\infty, -1) \cup (1, \infty) \]