3.19: Composite functions and inverse form

0606/21/M/J/24 — Question 9 · 7 marks

3.19 diagram3.19 diagram
The functions ff and gg are defined by f(x)=3x24x−1f(x) = \dfrac{3x^{2}}{4x - 1} for x>0x > 0 and g(x)=1x2g(x) = \dfrac{1}{x^{2}} for x>0x > 0.
(a)
Explain why the function fgfg does not exist.
[1]
(b)
Given that gfgf does exist, find and simplify an expression for gf(x)gf(x).
[2]
(c)
Show that f−1(x)f^{-1}(x) can be written as px−x(qx+r)3\dfrac{px - \sqrt{x(qx + r)}}{3} where pp, qq and rr are integers.
[4]