CAIE P1 2022 November — Question 8

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2022
SessionNovember
TopicComposite & Inverse Functions

8 The function f is defined by \(\mathrm { f } ( x ) = 2 - \frac { 3 } { 4 x - p }\) for \(x > \frac { p } { 4 }\), where \(p\) is a constant.
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence determine whether f is an increasing function, a decreasing function or neither.
  2. Express \(\mathrm { f } ^ { - 1 } ( x )\) in the form \(\frac { p } { a } - \frac { b } { c x - d }\), where \(a , b , c\) and \(d\) are integers.
  3. Hence state the value of \(p\) for which \(\mathrm { f } ^ { - 1 } ( x ) \equiv \mathrm { f } ( x )\).