3.4: Quadratic and exponential ranges with composite equation

0606/11/M/J/21 — Question 5 · 7 marks

The functions ff and gg are defined as follows.
f(x)=x2+4xfor x∈Rf(x) = x^{2} + 4x \quad\text{for } x \in \mathbb{R}g(x)=1+e2xfor x∈Rg(x) = 1 + \mathrm{e}^{2x} \quad\text{for } x \in \mathbb{R}
(a)
Find the range of ff.
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(b)
Write down the range of gg.
[1]
(c)
Find the exact solution of the equation fg(x)=21fg(x) = 21, giving your answer as a single logarithm.
[4]