Question

Find 𝑓 ∘𝑔 and 𝑔 ∘𝑓, if they exist. State the domain and range for each.

𝑓 ={(−4,−14),(0,−6),(−6,−18),(2,−2)}⁢
𝑔 ={(−6,1),(−18,13),(−14,9),(−2,−3)}

Answer 1

First note the domains:
- dom(f) = {−4, 0, −6, 2}
- dom(g) = {−6, −18, −14, −2}

f ∘ g:
For x in dom(g) we need g(x) to be in dom(f). g(x) values are {1, 13, 9, −3}, none of which are in dom(f). So f ∘ g has no ordered pairs:
- f ∘ g = ∅
- domain(f ∘ g) = ∅
- range(f ∘ g) = ∅

g ∘ f:
Compute g(f(x)) for each x in dom(f):
- f(−4) = −14, g(−14) = 9 → (−4, 9)
- f(0) = −6, g(−6) = 1 → (0, 1)
- f(−6) = −18, g(−18) = 13 → (−6, 13)
- f(2) = −2, g(−2) = −3 → (2, −3)

So
- g ∘ f = { (−4, 9), (0, 1), (−6, 13), (2, −3) }
- domain(g ∘ f) = {−4, 0, −6, 2}
- range(g ∘ f) = {9, 1, 13, −3}