Question

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

𝑓 ={(−7,0),(4,5),(8,12),(−3,6)}⁢
𝑔 ={(6,8),(−12,−5),(0,5),(5,1)}

Answer 1

Recall f ∘ g means f(g(x)) and g ∘ f means g(f(x)).

f = {(-7,0),(4,5),(8,12),(-3,6)}, Dom(f) = {-7,4,8,-3}, Range(f) = {0,5,12,6}
g = {(6,8),(-12,-5),(0,5),(5,1)}, Dom(g) = {6,-12,0,5}, Range(g) = {8,-5,5,1}

Compute f ∘ g:
- g(6)=8 and f(8)=12 ⇒ (6,12)
- g(-12)=-5 not in Dom(f) ⇒ no value
- g(0)=5 not in Dom(f) ⇒ no value
- g(5)=1 not in Dom(f) ⇒ no value

So f ∘ g = {(6,12)}. Domain(f ∘ g) = {6}. Range(f ∘ g) = {12}.

Compute g ∘ f:
- f(-7)=0 and g(0)=5 ⇒ (-7,5)
- f(4)=5 and g(5)=1 ⇒ (4,1)
- f(8)=12 not in Dom(g) ⇒ no value
- f(-3)=6 and g(6)=8 ⇒ (-3,8)

So g ∘ f = {(-7,5),(4,1),(-3,8)}. Domain(g ∘ f) = {-7,4,-3}. Range(g ∘ f) = {5,1,8}.