3.20: Composite gf domain and equation

0606/22/F/M/25 — Question 7 · 5 marks

It is given that f(x)=2ex+af(x) = 2\mathrm{e}^{x} + a for x⩾0x \geqslant 0, where aa is an integer, and g(x)=x−1g(x) = \sqrt{x - 1} for x⩾1x \geqslant 1.
(a)
Find the least value of aa so that the function gfgf exists for all x⩾0x \geqslant 0.
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(b)
In the case where a=5a = 5, solve the equation gf(x)=3gf(x) = 3. Give your answer correct to 33 decimal places.
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