The domain of function `sqrt( log_(0.75) x)` is

Options

A. `(0,oo)`
B. `[0.75 ,1]`
C. `(0,1]`
D. `[0,1)`

Answer

To find the domain of the function \( f(x) = \sqrt{\log_{0.75}(x)} \), we need to ensure that the expression inside the square root is non-negative. This means we need to solve the inequality:

\[ \log_{0.75}(x) \geq 0 \]

### Step 1: Understanding the Logarithm
The logarithm \( \log_{0.75}(x) \) is defined for \( x > 0 \). Therefore, the first condition for the domain is:

\[ x > 0 \]

### Step 2: Solving the Inequality
Next, we need to solve the inequality \( \log_{0.75}(x) \geq 0 \).

Using the properties of logarithms, we know that:

\[ \log_{b}(a) = 0 \quad \text{when} \quad a = 1 \]

Thus, we set:

\[ \log_{0.75}(x) = 0 \implies x = 1 \]

### Step 3: Analyzing the Base of the Logarithm
Since the base \( 0.75 < 1 \), the logarithmic function \( \log_{0.75}(x) \) is decreasing. Therefore:

- \( \log_{0.75}(x) > 0 \) when \( x < 1 \)
- \( \log_{0.75}(x) = 0 \) when \( x = 1 \)
- \( \log_{0.75}(x) < 0 \) when \( x > 1 \)

### Step 4: Combining Conditions
From our analysis, we have two conditions:
1. \( x > 0 \)
2. \( x \leq 1 \) (since \( \log_{0.75}(x) \) is non-negative when \( x \leq 1 \))

Combining these two inequalities, we find:

\[ 0 < x \leq 1 \]

### Step 5: Writing the Domain
In interval notation, the domain of the function \( f(x) = \sqrt{\log_{0.75}(x)} \) is:

\[ (0, 1] \]

### Final Answer
Thus, the domain of the function is:

\[ (0, 1] \]