The function f has domain (-∞, ∞) and is defined by f(x) = 3e2x.
The function g has domain (0, ∞) and is defined by g(x) = ln 4x.
(a) Write down the domain and range of f∘g.
(b) Solve the equation (f∘g)(x) = 12
2. The attempt at a solution
Is it correct to think that the domain of f∘g will be all those x in the domain of g which produce g(x) in the domain of f? The domain of f is ℝ, so there's no restrictions on the g(x), so I think the domain of f∘g is the same as the domain of g, which is (0, ∞).
Taking the limit of 3e2x as x→-∞ gives me 0, and x→∞ gives ∞. I got that the range of f is (0, ∞). I, uh, think the range of g is (-∞, ∞).
I notice that the range of g and the domain of f are the same: (-∞, ∞). I therefore conclude that the range of f∘g will be the same as the range of f.
Domain of f∘g: (0, ∞)
Range of f∘g: (0, ∞)
f∘g(x) = 3e2ln4x
(6ln16x)/6 = 12/6
ln16x = 2
16x = e2
x = e2/16 = 0.4618... ≈ 0.5
I really don't know if any of this is right, and these questions always make me scratch my head. Especially any question on the domain and range of a composite function.