| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2008 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Challenging +1.8 This AEA question requires finding inverse functions, solving for self-inverse conditions, sketching transformed rational functions with asymptotes, and using geometric transformations to relate normals on different curves. Part (d) demands significant insight to connect the transformation f(x-2)+2 with the relationship between normals at P and Q, requiring multi-step reasoning beyond standard A-level techniques. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)1.07m Tangents and normals: gradient and equations |
$$f(x) = \frac{ax + b}{x + 2}; \quad x \in \mathbb{R}, x \neq -2,$$
where $a$ and $b$ are constants and $b > 0$.
\begin{enumerate}[label=(\alph*)]
\item Find $f^{-1}(x)$. [2]
\item Hence, or otherwise, find the value of $a$ so that $f(x) = x$. [2]
\end{enumerate}
The curve $C$ has equation $y = f(x)$ and $f(x)$ satisfies $f(x) = x$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item On separate axes sketch
\begin{enumerate}[label=(\roman*)]
\item $y = f(x)$, [3]
\item $y = f(x - 2) + 2$. [3]
\end{enumerate}
\end{enumerate}
On each sketch you should indicate the equations of any asymptotes and the coordinates, in terms of $b$, of any intersections with the axes.
The normal to $C$ at the point $P$ has equation $y = 4x - 39$. The normal to $C$ at the point $Q$ has equation $y = 4x + k$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item By considering the images of the normals to $C$ on the curve with equation $y = f(x - 2) + 2$, or otherwise, find the value of $k$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2008 Q6 [15]}}