The domain of the function `f(x)=log_(1//2)(-log_(2)(1+1/(root(4)(x)))-1)` is:

Options

A. `0ltxlt1`
B. `0ltxle1`
C. `xge1`
D. null set

Explanation

To find the domain of the function \( f(x) = \log_{1/2}(-\log_{2}(1 + \frac{1}{\sqrt[4]{x}}) - 1) \), we need to ensure that the expression inside the logarithm is greater than zero. Let's go through the steps:

### Step 1: Set up the inequality
We start with the condition that the argument of the logarithm must be greater than zero:
\[ -\log_{2}(1 + \frac{1}{\sqrt[4]{x}}) - 1 > 0 \]

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

### Step 3: Exponentiating both sides
Using the property of logarithms, we can convert the logarithmic inequality to an exponential form:
\[ 1 + \frac{1}{\sqrt[4]{x}} < 2^{-1} \]
This simplifies to:
\[ 1 + \frac{1}{\sqrt[4]{x}} < \frac{1}{2} \]

### Step 4: Isolate the term involving \( x \)
Now, isolate the term involving \( x \):
\[ \frac{1}{\sqrt[4]{x}} < \frac{1}{2} - 1 \]
This simplifies to:
\[ \frac{1}{\sqrt[4]{x}} < -\frac{1}{2} \]

### Step 5: Analyze the inequality
The left side, \( \frac{1}{\sqrt[4]{x}} \), is always positive for \( x > 0 \). However, the right side is negative. Therefore, there are no values of \( x \) that satisfy this inequality.

### Conclusion
Since there are no values of \( x \) that can satisfy the inequality, the domain of the function \( f(x) \) is the empty set.

Thus, the domain of the function is:
\[ \text{Domain} = \emptyset \]