The domain of `f(x) = log |x| + 1/sqrt|x| + 1/log|x|` is R - A where A is set

Options

A. {-1, 0, 1}
B. {-1, 1}
C. {2, 3, 4}
D. {0, 1, 2}

Answer

To find the domain of the function \( f(x) = \log |x| + \frac{1}{\sqrt{|x|}} + \frac{1}{\log |x|} \), we need to analyze each term in the function and determine the values of \( x \) for which the function is defined.

### Step 1: Analyze the first term \( \log |x| \)

The logarithm function is defined only for positive arguments. Therefore, we require:
\[ |x| > 0 \]
This means \( x \) cannot be zero. Thus, we have:
\[ x \neq 0 \]

### Step 2: Analyze the second term \( \frac{1}{\sqrt{|x|}} \)

The square root function is defined for non-negative arguments, and since we are dealing with a fraction, the denominator must not be zero. Therefore, we require:
\[ |x| > 0 \]
This condition is already satisfied since we found that \( x \neq 0 \).

### Step 3: Analyze the third term \( \frac{1}{\log |x|} \)

For this term to be defined, the logarithm must not equal zero. We know that:
\[ \log |x| = 0 \implies |x| = 1 \]
Thus, we must exclude the values where \( |x| = 1 \), which gives us:
\[ x \neq 1 \quad \text{and} \quad x \neq -1 \]

### Step 4: Combine the conditions

From the analysis above, we have the following restrictions:
1. \( x \neq 0 \)
2. \( x \neq 1 \)
3. \( x \neq -1 \)

Thus, the domain of \( f(x) \) is all real numbers except for \( 0, 1, -1 \). In set notation, we can express this as:
\[ \text{Domain of } f = \mathbb{R} - \{ -1, 0, 1 \} \]

### Conclusion

The set \( A \) is \( \{-1, 0, 1\} \).

### Final Answer

The domain of \( f(x) = \log |x| + \frac{1}{\sqrt{|x|}} + \frac{1}{\log |x|} \) is \( \mathbb{R} - \{-1, 0, 1\} \).

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