Consider the following functions. f(x) = 2/x , g(x) = 2x + 6

Question

Consider the following functions.

f(x) = 2/x , g(x) = 2x + 6

A)Find (f ∘ g)(x).

B)Find the domain of (f∘ g)(x). (Enter your answer using interval notation.)

C)Find (g ∘ f)(x).

D)Find the domain of (g ∘ f)(x).  (Enter your answer using interval notation.)

E)Find (f ∘ f)(x).

F)Find the domain of (f ∘ f)(x). (Enter your answer using interval notation.)

G)Find (g ∘ g)(x).

H)Find the domain of (g ∘ g)(x). (Enter your answer using interval notation.)

Answer 1

f(x) = 2/x

g(x) = 2x + 6

a) f(g(x)) = 2/(2x + 6) = 1/(x + 3) since we can factor 2 out of the numerator and denominator and divide it out.

b) the domain of this function is all values for which it is defined, that is, where it can return an actual value, which is (-infinity, 3) and (3, infinity) where () denotes non-inclusive as this function is not defined at 3.

c) g(f(x)) = 2(2/x) + 6 = 4/x + 6

d) this function is not defined when x = 0 but returns a value everywhere else, therefore our domain is (-infinity, 0) and (0, infinity).

e) f(f(x)) = 2/(2/x) = 2x/2 = x which is obtained by multiplying the numerator and denominator by x/2.

f) x is defined everywhere so our domain is (-infinity, infinity)

g) g(g(x)) = 2(2x + 6) + 6 = 4x + 12 + 6 = 4x + 18

h) this function is also defined everywhere and has bounds (-infinity, infinity)