The domain of definition of `f(x)=log_(2)( -log_(1//2) (1+(1)/(x^(1//4)))-1)` is :

Options

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

Answer

To find the domain of the function \( f(x) = \log_2(-\log_{1/2}(1 + \frac{1}{x^{1/4}}) - 1) \), we need to ensure that the expression inside the logarithm is positive. Here are the steps to find the domain:

### Step 1: Set up the inequality
We need the argument of the outer logarithm to be greater than zero:
\[ -\log_{1/2}(1 + \frac{1}{x^{1/4}}) - 1 > 0 \]
This simplifies to:
\[ -\log_{1/2}(1 + \frac{1}{x^{1/4}}) > 1 \]

### Step 2: Rearranging the inequality
Rearranging gives:
\[ \log_{1/2}(1 + \frac{1}{x^{1/4}}) < -1 \]

### Step 3: Change the base of the logarithm
Since the base of the logarithm is \( \frac{1}{2} \) (which is less than 1), we can change the inequality:
\[ 1 + \frac{1}{x^{1/4}} > 2^{-1} \]
This simplifies to:
\[ 1 + \frac{1}{x^{1/4}} > \frac{1}{2} \]

### Step 4: Isolate the term
Subtracting 1 from both sides gives:
\[ \frac{1}{x^{1/4}} > \frac{1}{2} - 1 \]
\[ \frac{1}{x^{1/4}} > -\frac{1}{2} \]
Since \( \frac{1}{x^{1/4}} \) is always positive for \( x > 0 \), this inequality is always satisfied for positive \( x \).

### Step 5: Solve for \( x \)
Next, we need to ensure that \( 1 + \frac{1}{x^{1/4}} > 0 \):
\[ 1 + \frac{1}{x^{1/4}} > 0 \]
This is true for all \( x > 0 \).

### Step 6: Find the upper limit
Now we need to ensure that \( 1 + \frac{1}{x^{1/4}} < 2 \):
\[ \frac{1}{x^{1/4}} < 1 \]
This implies:
\[ x^{1/4} > 1 \implies x > 1 \]

### Step 7: Combine the results
Thus, we have:
\[ x > 1 \]
Combining the conditions, we find that the domain of \( f(x) \) is:
\[ (1, \infty) \]

### Final Answer
The domain of the function \( f(x) \) is \( (1, \infty) \).