Question

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

𝑓 ={(5,13),(−4,−2),(−8,−11),(3,1)}⁢
𝑔 ={(−8,2),(−4,1),(3,−3),(5,7)}

Answer 1

We compute f ∘ g = f(g(x)) and g ∘ f = g(f(x)) by checking whether g(x) (resp. f(x)) is in the domain of f (resp. g).

f = {(5,13), (−4,−2), (−8,−11), (3,1)} so Dom(f) = {5,−4,−8,3}.
g = {(−8,2), (−4,1), (3,−3), (5,7)} so Dom(g) = {−8,−4,3,5}.

For f ∘ g: evaluate g at each x in Dom(g):
- g(−8)=2, 2 ∉ Dom(f)
- g(−4)=1, 1 ∉ Dom(f)
- g(3)=−3, −3 ∉ Dom(f)
- g(5)=7, 7 ∉ Dom(f)
No g(x) lies in Dom(f), so f ∘ g = ∅. Domain(f ∘ g) = ∅, Range(f ∘ g) = ∅.

For g ∘ f: evaluate f at each x in Dom(f):
- f(5)=13, 13 ∉ Dom(g)
- f(−4)=−2, −2 ∉ Dom(g)
- f(−8)=−11, −11 ∉ Dom(g)
- f(3)=1, 1 ∉ Dom(g)
No f(x) lies in Dom(g), so g ∘ f = ∅. Domain(g ∘ f) = ∅, Range(g ∘ f) = ∅.