Answer:
g(f(x))=x^2-2x
(g-f)(x)=x^2-x
Explanation:
g(f(x)) and (f-g)(x) are both composite functions.
Given: f(x)=x-1, g(x)=x^2-1, find: g(f(x)) and (f-g)(x)
In order to find g(f(x)), we have to plug in f(x) into the function g(x):
f(x)=x-1
g(x)=x^2-1
g(f(x))=g(x-1)
g(x-1)=(x-1)^2-1
(x-1)(x-1)-x^2-2x+1
Thus, x^2-2x+1-1=x^2-2x=x(x-2)
To find (g-f)(x), you just subtract both functions:
f(x)=x-1
g(x)=x^2-1
(g-f)(x)=(x^2-1)-(x-1)=x^2-1-x+1=x^2-x=x(x-1) [this is factored form]
g(f(x))=x^2-2x or x(x-2) factored
(g-f)(x)=x^2-x or x(x-1) factored