CAIE P1 2011 November — Question 9

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2011
SessionNovember
TopicComposite & Inverse Functions

9 Functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + 3 & \text { for } x \leqslant 0
\mathrm {~g} : x \mapsto x ^ { 2 } - 6 x & \text { for } x \leqslant 3 \end{array}$$
  1. Express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) and solve the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\).
  2. On the same diagram sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), showing the coordinates of their point of intersection and the relationship between the graphs.
  3. Find the set of values of \(x\) which satisfy \(\operatorname { gf } ( x ) \leqslant 16\).