The domain of the function `sqrt((log_(0.5)x))` is

Options

A. `(1,oo)`
B. `(0,oo)`
C. `(0,1)`
D. `(0*5,1)`

Answer

To find the domain of the function \( f(x) = \sqrt{\log_{0.5}(x)} \), we need to ensure that the expression inside the square root is non-negative. This means we need to solve the inequality:

\[ \log_{0.5}(x) > 0 \]

### Step 1: Understanding the logarithm
The logarithm \(\log_{0.5}(x)\) is defined when \(x > 0\). Therefore, our first condition is:

\[ x > 0 \]

### Step 2: Solve the inequality
Next, we need to solve the inequality \(\log_{0.5}(x) > 0\). Recall that the logarithm \(\log_{b}(a) > 0\) when \(a > b^0\) if \(0 < b < 1\). Since \(0.5 < 1\), we can rewrite the inequality as:

\[ x < 0.5^0 \]

Calculating \(0.5^0\):

\[ 0.5^0 = 1 \]

Thus, we have:

\[ x < 1 \]

### Step 3: Combine the conditions
Now we combine our two conditions:

1. \(x > 0\)
2. \(x < 1\)

This gives us:

\[ 0 < x < 1 \]

### Step 4: Write the domain in interval notation
The domain of the function in interval notation is:

\[ (0, 1) \]

### Conclusion
Therefore, the domain of the function \( f(x) = \sqrt{\log_{0.5}(x)} \) is:

\[ \text{Domain: } (0, 1) \]