The problem provides two functions, $f(x) = \sqrt{x-2}$ and $g(x) = x^2 - 1$. The goal is to find the expression for $(g \circ f)(x)$, which represents the composition of the functions $g$ and $f$. Then, we need to choose the correct expression for $(g \circ f)(x)$ from the provided options.

1. Problem Description

The problem provides two functions, $f(x) = \sqrt{x-2}$ and $g(x) = x^2 - 1$. The goal is to find the expression for $(g \circ f)(x)$, which represents the composition of the functions $g$ and $f$. Then, we need to choose the correct expression for $(g \circ f)(x)$ from the provided options.

2. Solution Steps

The composition of functions $g$ and $f$, denoted as $(g \circ f)(x)$, is defined as $g(f(x))$.

First, substitute $f(x)$ into $g(x)$:

$$(g \circ f)(x) = g(f(x)) = g(\sqrt{x-2})$$

Next, replace the variable $x$ in the expression for $g(x)$ with $\sqrt{x-2}$:

$$g(\sqrt{x-2}) = (\sqrt{x-2})^2 - 1$$

Now, simplify the expression:

$$(\sqrt{x-2})^2 - 1 = (x-2) - 1$$

$$= x - 2 - 1$$

$$= x - 3$$

Thus, $(g \circ f)(x) = x - 3$.

3. Final Answer

The final answer is (d) $(g \circ f)(x) = x-3$