% Figure 4 shows curves with asymptotic behavior at x = 3

\includegraphics{figure_4}

Figure 4

\begin{enumerate}[label=(\alph*)]
\item Figure 4 shows a sketch of the curve with equation $y = f(x)$, where
$$f(x) = \frac{x^2 - 5}{3-x}, \quad x \in \mathbb{R}, x \neq 3$$

The curve has a minimum at the point $A$, with $x$-coordinate $\alpha$, and a maximum at the point $B$, with $x$-coordinate $\beta$.

Find the value of $\alpha$, the value of $\beta$ and the $y$-coordinates of the points $A$ and $B$.
[5]

\item The functions $g$ and $h$ are defined as follows
$$g: x \to x + p \quad x \in \mathbb{R}$$
$$h: x \to |x| \quad x \in \mathbb{R}$$

where $p$ is a constant.

% Figure 5 shows curve with minimum points at C and D symmetric about y-axis

\includegraphics{figure_5}

Figure 5

Figure 5 shows a sketch of the curve with equation $y = h(fg(x) + q)$, $x \in \mathbb{R}$, $x \neq 0$, where $q$ is a constant. The curve is symmetric about the $y$-axis and has minimum points at $C$ and $D$.

\begin{enumerate}[label=(\roman*)]
\item Find the value of $p$ and the value of $q$.

\item Write down the coordinates of $D$.
\end{enumerate}
[5]

\item The function $\mathrm{m}$ is given by
$$\mathrm{m}(x) = \frac{x^2 - 5}{3-x} \quad x \in \mathbb{R}, x < \alpha$$

where $\alpha$ is the $x$-coordinate of $A$ as found in part (a).

\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm{m}^{-1}$

\item Write down the domain of $\mathrm{m}^{-1}$

\item Find the value of $t$ such that $\mathrm{m}(t) = \mathrm{m}^{-1}(t)$
\end{enumerate}
[10]
\end{enumerate}
[Total 20 marks]

\hfill \mbox{\textit{Edexcel AEA 2011 Q7 [20]}}