Question

𝑓⁡(𝑥) =𝑥2 +6⁢𝑥 −2⁢𝑔⁡(𝑥) =𝑥 −6

a. Find [𝑓∘𝑔]⁡(𝑥) and [𝑔∘𝑓]⁡(𝑥).
Drag and drop application.
[𝑓∘𝑔]⁡(𝑥)

drop zone empty.
[𝑔∘𝑓]⁡(𝑥)


𝑥2 +7⁢𝑥 −8

𝑥3 −38⁢𝑥 +12

𝑥2 −6⁢𝑥 −2

𝑥2 +6⁢𝑥 −8

state the domain and range for each

Answer 1

Interpreting f(x) = x^2 + 6x − 2 and g(x) = x − 6:

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

Domain and range:

- f(x) = x^2 + 6x − 2
- Domain: (−∞, ∞)
- Vertex at x = −3, f(−3) = −11 → Range: [−11, ∞)

- g(x) = x − 6
- Domain: (−∞, ∞)
- Range: (−∞, ∞)

- f∘g (x^2 − 6x − 2)
- Domain: (−∞, ∞)
- Vertex at x = 3, value = −11 → Range: [−11, ∞)

- g∘f (x^2 + 6x − 8)
- Domain: (−∞, ∞)
- Vertex at x = −3, value = −17 → Range: [−17, ∞)