| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Solve equation with inverses |
| Difficulty | Moderate -0.3 This is a straightforward composite and inverse functions question requiring standard techniques: composing functions, solving a quadratic, finding inverses by swapping and rearranging, and showing no solutions exist. The sketching part is routine. All parts are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
10 Functions $f$ and $g$ are defined by
$$\begin{aligned}
& \mathrm { f } : x \mapsto 2 x + 1 , \quad x \in \mathbb { R } , \quad x > 0 \\
& \mathrm {~g} : x \mapsto \frac { 2 x - 1 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3
\end{aligned}$$
(i) Solve the equation $\operatorname { gf } ( x ) = x$.\\
(ii) Express $\mathrm { f } ^ { - 1 } ( x )$ and $\mathrm { g } ^ { - 1 } ( x )$ in terms of $x$.\\
(iii) Show that the equation $\mathrm { g } ^ { - 1 } ( x ) = x$ has no solutions.\\
(iv) Sketch in a single diagram the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$, making clear the relationship between the graphs.
\hfill \mbox{\textit{CAIE P1 2009 Q10 [13]}}