Number of integers in domain of function `f(x) =log_(|x^2|)(4-|x|)+log_2{sqrt(x)}` is
a) 0
b) 1
c) 2
d) 3

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To find the number of integers in the domain of the function \( f(x) = \log_{|x^2|}(4 - |x|) + \log_2(\sqrt{x}) \), we need to determine the conditions under which this function is defined.

### Step 1: Analyze the first logarithm
The first term is \( \log_{|x^2|}(4 - |x|) \). For this logarithm to be defined, we need two conditions to be satisfied:
1. The base \( |x^2| \) must be greater than 1.
2. The argument \( 4 - |x| \) must be greater than 0.

**Condition 1**: \( |x^2| > 1 \)
- Since \( |x^2| = x^2 \) (as \( x^2 \) is always non-negative), we need \( x^2 > 1 \).
- This implies \( x > 1 \) or \( x < -1 \).

**Condition 2**: \( 4 - |x| > 0 \)
- This simplifies to \( |x| < 4 \).
- Therefore, \( -4 < x < 4 \).

### Step 2: Combine the conditions
Now we combine the two conditions:
1. From \( |x^2| > 1 \), we have \( x > 1 \) or \( x < -1 \).
2. From \( 4 - |x| > 0 \), we have \( -4 < x < 4 \).

Now, we will analyze these intervals:
- For \( x > 1 \): The intersection with \( -4 < x < 4 \) gives us \( 1 < x < 4 \).
- For \( x < -1 \): The intersection with \( -4 < x < 4 \) gives us \( -4 < x < -1 \).

### Step 3: Determine the integer values in the intervals
Now we need to find the integers in the intervals \( (1, 4) \) and \( (-4, -1) \).

1. In the interval \( (1, 4) \), the integers are \( 2, 3 \).
2. In the interval \( (-4, -1) \), the only integer is \( -2, -3 \).

### Step 4: Count the integers
Now we count the integers:
- From \( (1, 4) \): \( 2, 3 \) (2 integers)
- From \( (-4, -1) \): \( -2, -3 \) (2 integers)

Thus, the total number of integers in the domain of the function is \( 2 + 2 = 4 \).

### Final Answer
The number of integers in the domain of the function \( f(x) \) is **4**.