Find the domain and range of the following function:
`f(x)=sqrt(log_(1//2)log_(2)[x^(2)+4x+5])` where [.] denotes the greatest integer function

Explanation

To find the domain and range of the function \( f(x) = \sqrt{\log_{1/2}(\lfloor \log_2(x^2 + 4x + 5) \rfloor)} \), we will follow these steps:

### Step 1: Identify the conditions for the square root
Since \( f(x) \) involves a square root, the expression inside the square root must be non-negative:
\[ \log_{1/2}(\lfloor \log_2(x^2 + 4x + 5) \rfloor) \geq 0 \]
This implies:
\[ \lfloor \log_2(x^2 + 4x + 5) \rfloor \leq 1 \]

### Step 2: Convert the logarithmic inequality
The inequality \( \lfloor \log_2(x^2 + 4x + 5) \rfloor \leq 1 \) means:
\[ \log_2(x^2 + 4x + 5) < 2 \]
This can be rewritten using the properties of logarithms:
\[ x^2 + 4x + 5 < 2^2 \]
\[ x^2 + 4x + 5 < 4 \]
\[ x^2 + 4x + 1 < 0 \]

### Step 3: Factor the quadratic inequality
Now we factor the quadratic:
\[ (x + 2)^2 < 0 \]
The expression \( (x + 2)^2 \) is always non-negative, hence it can never be less than zero. Therefore, there are no values of \( x \) that satisfy this inequality.

### Step 4: Determine the domain
Since there are no values of \( x \) that satisfy the inequality derived from the square root condition, we conclude that:
\[ \text{Domain of } f(x) = \emptyset \]

### Step 5: Determine the range
Since the domain is empty, the function \( f(x) \) is not defined for any \( x \). Consequently, the range of \( f(x) \) is also empty:
\[ \text{Range of } f(x) = \emptyset \]

### Final Answer
- **Domain:** \( \emptyset \)
- **Range:** \( \emptyset \)