# FIND COMPOSITION OF TWO FUNCTIONS

The composition of a function g with a function f is :

h(x) = g(f (x))

The domain of h is the set of all x-values such that x is in the domain of f and f (x) is in the domain of g.

Problem 1 :

Given f(x) = -9x + 3 and g(x) = x4, find (f g)(x)

Solution :

(f g)(x) = f(g(x))

= f(x4)

= -9(x4) + 3

(f g)(x) = -9x4 + 3

Problem 2 :

Given f(x) = 2x – 5 and g(x) = x + 2, find (f g)(x)

Solution :

(f g)(x) = f(g(x))

= f(x + 2)

= 2(x + 2) – 5

= 2x + 4 – 5

= 2x - 1

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

Problem 3 :

Given f(x) = x2 + 7 and g(x) = x - 3, find (f g)(x)

Solution :

(f g)(x) = f(g(x))

= f(x - 3)

= (x - 3)2 + 7

= x2 + (3)2 – 2(x)(3) + 7

= x2 + 9 - 6x + 7

(f g)(x) = x2 - 6x + 16

Problem 4 :

Given f(x) = 4x + 3 and g(x) = x2, find (g f)(x)

Solution :

(g f)(x) = g(f(x))

= g(4x + 3)

= (4x + 3)2

= (4x)2 + (3)2 + 2(4x)(3)

= 16x2 + 9 + 24x

(g f)(x) = 16x2 + 24x + 9

Problem 5 :

Given f(x) = x – 1 and g(x) = x2 + 2x - 8, find (g f)(x)

Solution :

(g f)(x) = g(f(x))

= g(x - 1)

= (x – 1)2 + 2(x – 1) - 8

= (x)2 + (1)2 – 2(x)(1) + 2x – 2 - 8

= x2 + 1 - 2x + 2x - 10

(g f)(x) = x2 - 9

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