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The given function is:

\[
f(x) = \ln(x + 1)
\]

### Domain:
The natural logarithm function, \( \ln(y) \), is only defined when \( y > 0 \). Therefore, for \( f(x) = \ln(x + 1) \) to be defined, we must have:

\[
x + 1 > 0
\]

Solving this inequality:

\[
x > -1
\]

Thus, the domain of the function is:

\[
(-1, \infty)
\]

### Range:
The range of the natural logarithm function \( \ln(y) \) is all real numbers, as it can take any value from \( -\infty \) to \( \infty \). Since there are no restrictions placed by the function \( x + 1 \), the range of \( f(x) \) is:

\[
(-\infty, \infty)
\]

### Summary:
- **Domain**: \( (-1, \infty) \)
- **Range**: \( (-\infty, \infty) \)

Would you like further details on how the logarithmic function behaves or any additional clarifications?

Here are five related questions to expand this topic:
1. What is the derivative of \( f(x) = \ln(x + 1) \)?
2. How does shifting the logarithmic function affect its graph?
3. Can you calculate the inverse of \( f(x) = \ln(x + 1) \)?
4. What is the behavior of \( f(x) = \ln(x + 1) \) as \( x \to -1 \)?
5. How does the range of \( f(x) \) compare to other logarithmic functions, such as \( \ln(x) \)?

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

Domain and range of the function f(x) = ln(x + 1)

Learn how to find the domain and range of the logarithmic function f(x) = ln(x + 1). This problem explores key mathematical concepts related to logarithmic functions and their properties. Suitable for students in Grades 10-12.

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