For the three-part question that follows, provide your answer to each part in the given workspace. Identify each part with a coordinating response.

Question

Answer Part A, Part B, and Part C using the functions below.

f(x)=2x+2 g(x)=(sqrt)x-6

Part A: What is the domain of f(x) and g(x)?

Part B: Find f(g(x)). What is the domain of f(g(x))? Show all work and explain answer.

Part C: Find g(f(x)). What is the domain of g(f(x))? Show all work and explain answer.

e4.q18

Answer 1

Since f(x) is a simple polynomial, the domain is all reals.

g(x) has a √, so only non-negatives are allowed. If the radical is over just the x, then the domain is x≥0. If the radical is over the (x-6) the domain is x≥6

Case a: f(g(x)) = 2(√(x)-6)+2 -> 2√x-10, and the domain would be x≥0

Case b: f(g(x)) = 2(√(x-6))+2 and the domain would come from x-6≥0, so x≥6

Case a: g(f(x)) = √(2x+2)-6 and the domain would come from 2x+2≥0, so x≥-1

Case b: g(f(x)) = √(2x+2-6) -> √(2x-4) and the domain would come from 2x-4≥0, so x≥2