CAIE P1 2012 June — Question 10

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2012
SessionJune
TopicComposite & Inverse Functions

10 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + 5 & \text { for } x \in \mathbb { R } ,
\mathrm {~g} : x \mapsto \frac { 8 } { x - 3 } & \text { for } x \in \mathbb { R } , x \neq 3 \end{array}$$
  1. Obtain expressions, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\), stating the value of \(x\) for which \(\mathrm { g } ^ { - 1 } ( x )\) is not defined.
  2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram, making clear the relationship between the two graphs.
  3. Given that the equation \(\operatorname { fg } ( x ) = 5 - k x\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\).