Given: f(x) = x - 2, g(x) = x^2 + 1
Find: (f + g)(2), (f • g)(-4), (f o g)(2), (f o f)(3), (f o g o f)(x)
(f + g)(2) = f(2) + g(2) = (2 - 2) + (2^2 + 1) = 0 + 5 = 5
(f • g)(-4) = ( f(-4) ) ( g(-4) ) = (-4 - 2) ( (-4)^2 + 1 ) = -6(17) = -102
(f o g)(2) = f( g(2) ) = f( 2^2 + 1 ) = f(5) = 5 - 2 = 3
(f o f)(3) = f( f(3) ) = f( 3 - 2 ) = f(1) = 1 - 2 = -1
(f o g o f)(x) = f( g( f(x) ) ) = f( g( x - 2 ) ) = f( (x - 2)^2 + 1 )
= ( (x - 2)^2 + 1 ) - 2 = (x - 2)^2 - 1 = x^2 - 4x + 4 - 1 = x^2 - 4x + 3
This is an exercise in Functional Notation; learn it well. You will use all these operations in calculus. Composition of functions is used in the very important and key Chain Rule of Derivatives.