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.