Question

𝑓⁡(𝑥) =−3⁢𝑥⁢𝑔⁡(𝑥) =−𝑥 +8



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

Answer 1

Given f(x) = −3x and g(x) = −x + 8:

a)
- (f ∘ g)(x) = f(g(x)) = −3(−x + 8) = 3x − 24
- (g ∘ f)(x) = g(f(x)) = −(−3x) + 8 = 3x + 8

b) Domains and ranges:
- f: domain = (−∞, ∞), range = (−∞, ∞)
- g: domain = (−∞, ∞), range = (−∞, ∞)
- f ∘ g: domain = (−∞, ∞), range = (−∞, ∞)
- g ∘ f: domain = (−∞, ∞), range = (−∞, ∞)