3.24: Inverse of an exponential quadratic and finding gg

0606/22/M/J/24 — Question 10 · 8 marks

The functions f\mathrm{f} and fg\mathrm{fg} are defined by
f(x)=ex2+3for x⩽0,\mathrm{f}(x) = \mathrm{e}^{x^{2} + 3} \quad\text{for } x \leqslant 0,fg(x)=e2xfor x⩾32.\mathrm{fg}(x) = \mathrm{e}^{2x} \quad\text{for } x \geqslant \tfrac{3}{2}.
(a)
Explain why f−1\mathrm{f}^{-1} exists.
[1]
(b)
Find an expression for f−1(x)\mathrm{f}^{-1}(x) and state the domain and range of f−1\mathrm{f}^{-1}.
[5]
(c)
Hence find and simplify an expression for g(x)\mathrm{g}(x).
[2]