Final answer:
To simplify the function (gf)(x), substitute f(x) into g(x), resulting in g(f(x)) = (x-1)² - 1. When expanded and simplified, it becomes (gf)(x) = x² - 2x.
Step-by-step explanation:
To find and simplify the function (gf)(x), where f(x) = x - 1 and g(x) = x² - 1, we need to perform the composition of the two functions; that is, we substitute f(x) into every x in g(x).
Firstly, let's do the substitution to find (gf)(x):
g(f(x)) = g(x - 1) = (x - 1)² - 1
Now, let's simplify the expression:
(x - 1)² - 1 = (x² - 2x + 1) - 1
Simplifying further gives us:
x² - 2x
So the simplified form of (gf)(x) is x² - 2x.