2013 AJC P2 Q1
The function $\text{f}$ is defined by
$\text{f: }x\mapsto \frac{4}{{{\left( 2x-5 \right)}^{2}}+1}\text{ for }x\in \mathbb{R},x\le \frac{5}{2}$.
(i)
Sketch the graph of $y=\text{f}\left( x \right)$ and show that ${{\text{f}}^{-1}}$ exists.
[3]
(ii)
find ${{\text{f}}^{-1}}\left( x \right)$ and write down that domain of ${{\text{f}}^{-1}}$.
[3]
The solutions to the equation $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$ are $x=a$ and $x=2$ where $a<2$.
Find, in terms of $a$, the area of the region bounded by the curves $y=\text{f}\left( x \right)$ and $y={{\text{f}}^{-1}}\left( x \right)$.
[3]
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