2009 TJC P1 Q8
The functions $\text{f}$ and $\text{g}$ are defined by
$\begin{matrix}
& \text{f}:x\mapsto -{{(x-2)}^{2}}+4,x\in \mathbb{R},x\le 2, \\
& \text{g}:x\mapsto \frac{1}{{{(x-7)}^{2}}-4},x\in \mathbb{R},x\le k,x\ne 5,x\ne 9 \\
\end{matrix}$
(a)
(i) Sketch the graphs of $\text{f}$ and ${{\text{f}}^{-1}}$ on the same axes, showing the relationship between the two graphs.
(ii) If $\text{f}(\alpha )={{\text{f}}^{-1}}(\alpha )$, show that ${{\alpha }^{2}}-3\alpha =0$.
(b)
State the maximum value of $k$such that the inverse of $\text{g}$ exists. With this value of $k$, find
(i) the inverse of $\text{g}$, stating its domain, and
(ii) the range of $\text{gf}$.
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