Question

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

𝑓 ={(−8,−4),(0,4),(2,6),(−6,−2)}
⁢𝑔 ={(4,−4),(−2,−1),(−4,0),(6,−5)

Answer 1

First list the domains:
- Domain(f) = {-8, 0, 2, -6}
- Domain(g) = {4, -2, -4, 6}

Compute f ∘ g (f(g(x))). Evaluate g(x) for each x in Domain(g) and keep those whose g(x) ∈ Domain(f):
- g(4) = -4 (not in Domain(f)) → no composition
- g(-2) = -1 (not in Domain(f)) → no composition
- g(-4) = 0 (0 ∈ Domain(f)) → (−4) maps to f(0)=4, so (-4,4)
- g(6) = -5 (not in Domain(f)) → no composition

So f ∘ g = {(-4, 4)}.
- Domain(f ∘ g) = {-4}
- Range(f ∘ g) = {4}

Compute g ∘ f (g(f(x))). Evaluate f(x) for each x in Domain(f) and apply g:
- f(-8) = -4 → g(-4)=0, so (-8,0)
- f(0) = 4 → g(4) = -4, so (0,-4)
- f(2) = 6 → g(6) = -5, so (2,-5)
- f(-6) = -2 → g(-2)= -1, so (-6,-1)

So g ∘ f = {(-8,0), (0,-4), (2,-5), (-6,-1)}.
- Domain(g ∘ f) = {-8, 0, 2, -6}
- Range(g ∘ f) = {0, -4, -5, -1}