Question

𝑓⁡(𝑥) =𝑥^2 +6⁢𝑥 −2⁢𝑔⁡(𝑥) =𝑥 −6

a. Find [𝑓∘𝑔]⁡(𝑥) and [𝑔∘𝑓]⁡(𝑥).
b. State the domain and range for each.

Answer 1

Given f(x) = x^2 + 6x − 2 and g(x) = x − 6.

a) Compositions
- (f ∘ g)(x) = f(g(x)) = f(x − 6) = (x − 6)^2 + 6(x − 6) − 2
= x^2 − 12x + 36 + 6x − 36 − 2
= x^2 − 6x − 2.

- (g ∘ f)(x) = g(f(x)) = f(x) − 6 = (x^2 + 6x − 2) − 6
= x^2 + 6x − 8.

b) Domain and range
- Domain((f ∘ g)) = ℝ (polynomial).
Vertex of x^2 − 6x − 2 at x = 3 gives minimum value 3^2 − 6·3 − 2 = −11.
Range((f ∘ g)) = [−11, ∞).

- Domain((g ∘ f)) = ℝ (polynomial).
Vertex of x^2 + 6x − 8 at x = −3 gives minimum value (−3)^2 + 6(−3) − 8 = −17.
Range((g ∘ f)) = [−17, ∞).