Given the functions $f(x) = x^2 - 1$ and $g(x) = 2(x - 1)$, we are tasked with finding $g(f(x))$.

Step 1: Substitute $f(x)$ into $g(x)$.

We start by substituting $f(x)$ into $g(x)$. Since $f(x) = x^2 - 1$, we substitute this into $g(x)$:

$g(f(x)) = 2(f(x) - 1)$

Step 2: Replace $f(x)$ with $x^2 - 1$.

Now, we substitute $f(x) = x^2 - 1$ into the expression:

$g(f(x)) = 2((x^2 - 1) - 1)$

Step 3: Simplify the expression.

Simplify the terms inside the parentheses:

$g(f(x)) = 2(x^2 - 1 - 1) = 2(x^2 - 2)$

Step 4: Final expression.

Distribute the $2$ to get the final result:

$g(f(x)) = 2x^2 - 4$

Thus, the composition $g(f(x))$ is:

$g(f(x)) = 2x^2 - 4$

Would you like further details on the process?

Here are 5 related questions:

1. What is the domain of the composite function $g(f(x))$?
2. How would you find $f(g(x))$?
3. Can you find the inverse of $g(x)$?
4. What would happen if the function $g(x)$ was different, e.g. $g(x) = 3(x-2)$?
5. How do you graph the composite function $g(f(x))$?

Tip: When working with function composition, always substitute carefully and simplify step by step to avoid mistakes!