Number of integral values of x in the domain of function `f (x)= sqrt(ln(|ln|x||)) + sqrt(7|x|-(|x|)^2-10)` is equal to

Options

A. 5
B. 6
C. 7
D. 8

Answer

To find the number of integral values of \( x \) in the domain of the function \( f(x) = \sqrt{\ln(|\ln|x||)} + \sqrt{7|x| - (|x|)^2 - 10} \), we need to ensure that both components of the function are defined and non-negative.

### Step 1: Analyze the first component \( \sqrt{\ln(|\ln|x||)} \)

For \( \sqrt{\ln(|\ln|x||)} \) to be defined, we need:
\[ \ln(|\ln|x||) \geq 0 \]
This implies:
\[ |\ln|x|| \geq 1 \]
Thus, we have two cases:
1. \( \ln|x| \geq 1 \) which gives \( |x| \geq e \)
2. \( \ln|x| \leq -1 \) which gives \( |x| \leq e^{-1} \)

This leads to the intervals:
\[ |x| \geq e \quad \text{or} \quad |x| \leq \frac{1}{e} \]

### Step 2: Analyze the second component \( \sqrt{7|x| - (|x|)^2 - 10} \)

For \( \sqrt{7|x| - (|x|)^2 - 10} \) to be defined, we need:
\[ 7|x| - (|x|)^2 - 10 \geq 0 \]
Rearranging gives:
\[ (|x|)^2 - 7|x| + 10 \leq 0 \]
Factoring the quadratic:
\[ (|x| - 5)(|x| - 2) \leq 0 \]
This inequality holds when:
\[ 2 \leq |x| \leq 5 \]

### Step 3: Combine the intervals

Now we have two conditions:
1. \( |x| \geq e \) or \( |x| \leq \frac{1}{e} \)
2. \( 2 \leq |x| \leq 5 \)

Since \( e \approx 2.718 \), we can summarize the intervals:
- From \( |x| \geq e \), we have \( |x| \) can take values in \( [e, 5] \).
- From \( |x| \leq \frac{1}{e} \), this interval does not contribute any integral values since \( \frac{1}{e} \approx 0.367 \).

Thus, we focus on the interval \( [e, 5] \).

### Step 4: Identify integral values

The integral values of \( |x| \) in the interval \( [e, 5] \) are:
- The integers between \( 2.718 \) and \( 5 \) are \( 3, 4, 5 \).

Additionally, since \( |x| \) can be negative, we also consider:
- The negative counterparts: \( -3, -4, -5 \).

### Step 5: Count the integral values

The integral values of \( x \) are:
- Positive: \( 3, 4, 5 \)
- Negative: \( -3, -4, -5 \)

Thus, the total number of integral values of \( x \) is \( 3 + 3 = 6 \).

### Final Answer

The number of integral values of \( x \) in the domain of the function is \( \boxed{6} \).

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