Find the domain of the function `log|4-x^(2)|.`

Options

A. `(-infty ,infty)`
B. `(-infty ,2)`
C. `(-2 ,infty)`
D. `(-infty ,infty)` `- ``{-2,2}`

Answer

To find the domain of the function \( f(x) = \log |4 - x^2| \), we need to determine the values of \( x \) for which the expression inside the logarithm is positive. The logarithm function is defined only for positive values.

### Step-by-step Solution:

1. **Identify the condition for the logarithm**:
The logarithm function \( \log |4 - x^2| \) is defined when \( |4 - x^2| > 0 \).

2. **Set up the inequality**:
We need to solve the inequality:
\[ |4 - x^2| > 0 \]

3. **Consider the cases for the absolute value**:
The absolute value \( |A| > 0 \) implies that \( A \neq 0 \). Therefore, we need:
\[ 4 - x^2 \neq 0 \]

4. **Solve for \( x \)**:
Set the equation \( 4 - x^2 = 0 \):
\[ 4 = x^2 \]
Taking the square root of both sides, we find:
\[ x = \pm 2 \]

5. **Determine the values excluded from the domain**:
The values \( x = 2 \) and \( x = -2 \) make the expression inside the logarithm zero, which is not allowed. Thus, these values must be excluded from the domain.

6. **Write the domain**:
The domain of the function is all real numbers except \( x = 2 \) and \( x = -2 \). In interval notation, this can be expressed as:
\[ \text{Domain} = (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \]

### Final Answer:
The domain of the function \( \log |4 - x^2| \) is:
\[ (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \]