CAIE P1 2006 June — Question 11

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2006
SessionJune
TopicComposite & Inverse Functions

11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto k - x & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, }
\mathrm { g } : x \mapsto \frac { 9 } { x + 2 } & \text { for } x \in \mathbb { R } , x \neq - 2 . \end{array}$$
  1. Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has two equal roots and solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) in these cases.
  2. Solve the equation \(\operatorname { fg } ( x ) = 5\) when \(k = 6\).
  3. Express \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).