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The problem you have shared involves determining the composition of two functions, \( f(x) = \ln x \) and \( g(x) = x^2 - 1 \). We are tasked to find \( (g \circ f)(x) = g(f(x)) \) and the domain (definition set) of this composite function.

### Step 1: Calculate \( g(f(x)) \)
- \( f(x) = \ln x \), so \( f(x) \) is the input for \( g(x) \).
- The function \( g(x) = x^2 - 1 \) will now take \( f(x) = \ln x \) as its argument.

Thus, the composite function \( g(f(x)) \) is:

\[
g(f(x)) = g(\ln x) = (\ln x)^2 - 1
\]

This simplifies to:

\[
g(f(x)) = (\ln x)^2 - 1
\]

### Step 2: Determine the domain of \( g(f(x)) \)
- The domain of \( f(x) = \ln x \) is \( x > 0 \), because the natural logarithm function is only defined for positive real numbers.
- The function \( g(x) = x^2 - 1 \) is defined for all real numbers, so the only restriction comes from the domain of \( f(x) \).

Therefore, the domain of \( g(f(x)) \) is the same as the domain of \( f(x) \), which is:

\[
x > 0
\]

### Final Answer:
\[
g(f(x)) = (\ln x)^2 - 1
\]
with the domain \( x > 0 \).

---

Would you like further clarification or details on any part of this solution?

### Related Questions:
1. What is the range of \( g(f(x)) \) for \( x > 0 \)?
2. How would the domain change if \( g(x) \) was a different function?
3. Can the composition \( f(g(x)) \) also be calculated? What would its domain be?
4. How does the behavior of \( g(f(x)) \) change as \( x \to \infty \)?
5. What happens to \( g(f(x)) \) if we use a base-10 logarithm instead of the natural logarithm?

### Tip:
When composing functions, always check the domain of the inner function, as it can impose restrictions on the entire composite function.

Låt f(x) = ln x och g(x) = x^2 - 1. Bestäm (g ∘ f)(x) = g(f(x)) och dess definitionsmängd.

Learn how to compose the functions f(x) = ln x and g(x) = x^2 - 1 to find (g ∘ f)(x). This problem also covers the calculation of the domain for the composite function. Suitable for students in Grades 10-12.

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