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To find the domain of the function \( f(x) = \ln(x^2 + 1) \), we need to determine the values of \( x \) for which the argument of the logarithm, \( x^2 + 1 \), is positive. The logarithm function is defined only when its argument is greater than 0:

\[
x^2 + 1 > 0
\]

Since \( x^2 \geq 0 \) for all real \( x \), adding 1 ensures that \( x^2 + 1 \) is always positive. Therefore, there are no restrictions on \( x \) based on this inequality.

### Domain:
The domain of \( f(x) = \ln(x^2 + 1) \) is all real numbers:

\[
\text{Domain: } (-\infty, \infty)
\]

Would you like more details or have any other questions?

Here are some related questions to further explore the topic:
1. How would the domain change if the function was \( \ln(x^2 - 1) \)?
2. What is the range of the function \( f(x) = \ln(x^2 + 1) \)?
3. How does the derivative of \( f(x) = \ln(x^2 + 1) \) look?
4. What is the behavior of \( f(x) = \ln(x^2 + 1) \) as \( x \) approaches infinity?
5. How does the function behave for negative values of \( x \)?

**Tip:** When determining the domain of logarithmic functions, always ensure the argument of the logarithm is strictly positive.

Find the domain of the function f(x) = ln(x^2 + 1).

Discover how to find the domain of the logarithmic function f(x) = ln(x^2 + 1). This problem involves understanding the properties of logarithms and the behavior of quadratic functions, making it ideal for students in Grades 10-12.

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