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The function given is \( f(x) = \ln(x^2 - 4) \), and you are asked to determine the domain and range.

### Domain:
For the natural logarithm function \( \ln(u) \), the argument \( u \) must be strictly positive, i.e., \( u > 0 \). In this case:
\[
x^2 - 4 > 0
\]
\[
x^2 > 4
\]
\[
|x| > 2
\]
Thus, \( x > 2 \) or \( x < -2 \). The domain is \( (-\infty, -2) \cup (2, \infty) \).

### Range:
The natural logarithm function \( \ln(u) \) can take any real value as long as \( u > 0 \). Since \( x^2 - 4 \) is always positive and greater than zero for the domain we found, the range of \( f(x) \) is \( (-\infty, \infty) \).

### Answer:
The correct option is **b**:
- **Domain**: \( (-\infty, -2) \cup (2, \infty) \)
- **Range**: \( (-\infty, \infty) \)

Would you like a detailed explanation of any specific part? Here are some related questions to explore further:

1. How is the domain of logarithmic functions determined in general?
2. What happens to the function \( f(x) = \ln(x^2 - 4) \) at \( x = \pm 2 \)?
3. Can logarithmic functions have restricted ranges?
4. How do you graph \( f(x) = \ln(x^2 - 4) \)?
5. What transformations can be applied to the function to modify its domain or range?

**Tip**: Always remember that the argument of the logarithm must be strictly positive to define the domain.

Determina el dominio y el rango de la función: f(x) = ln(x^2 - 4)

Learn how to find the domain and range of the logarithmic function f(x) = ln(x^2 - 4). This solution explains the inequality setup for logarithmic functions and shows how to determine valid x-values. Suitable for Grades 10-12.

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