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## Homework Statement

The function f has domain (-∞, ∞) and is defined by f(x) = 3e

^{2x}.

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**

(a)

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 3e

^{2x}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, ∞)

(b)

f∘g(x) = 3e

^{2ln4x}

= 3e

^{ln(4x)2}= 3ln16x

^{2}= 6ln16x

(6ln16x)/6 = 12/6

ln16x = 2

16x = e

^{2}

x = e

^{2}/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.