3.22: Composite and inverse functions; solve f=gf = g

0606/21/O/N/21 — Question 9 · 9 marks

Do not use a calculator in part (c).

The functions are defined for x>−1x > -1, x≠0x \neq 0 as appropriate:
f(x)=3x−1f(x) = \dfrac{3}{x} - 1,\quad g(x)=2x+1g(x) = \dfrac{2}{x + 1}.
(a)
Find fg(x)f\mathrm{g}(x).
[2]
(b)
Find g−1(x)g^{-1}(x).
[2]
(c)
Solve f(x)=g(x)f(x) = g(x).
[5]