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

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

b)
- Domain(f) = ℝ. Range(f): parabola opens up, vertex at x = −3, f(−3) = −11 ⇒ range = [−11, ∞).
- Domain(g) = ℝ. Range(g): linear with slope 1 ⇒ range = (−∞, ∞).
- Domain(f∘g) = ℝ. (f∘g)(x) = x^2 − 6x − 2 has vertex at x = 3, value −11 ⇒ range = [−11, ∞).
- Domain(g∘f) = ℝ. (g∘f)(x) = x^2 + 6x − 8 has vertex at x = −3, value −17 ⇒ range = [−17, ∞).