<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f: }x\mapsto \frac{4}{{{\left( 2x-5 \right)}^{2}}+1}\mathrm{ for }x\in \mathbb{R},x\le \frac{5}{2}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y=\mathrm{f}\left( x \right)$ and show that ${{\mathrm{f}}^{-1}}$ exists.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Find ${\mathrm{f}^{-1} \left( x \right)}$ and write down that domain of ${\mathrm{f}^{-1}}$.</span></div><p style="text-align: right;">[3]</p></li></ol>
<p>The solutions to the equation ${\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)}$ are $x=a$ and $x=2$ where $a<2$.</p>
<p>Find, in terms of $a$, the area of the region bounded by the curves $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.</p><p style="text-align: right;">[3]</p>

$\mathrm{f: }x\mapsto \frac{4}{{{\left( 2x-5 \right)}^{2}}+1}\mathrm{ for }x\in \mathbb{R},x\le \frac{5}{2}$.

Sketch the graph of $y=\mathrm{f}\left( x \right)$ and show that ${{\mathrm{f}}^{-1}}$ exists.

Find ${\mathrm{f}^{-1} \left( x \right)}$ and write down that domain of ${\mathrm{f}^{-1}}$.

The solutions to the equation ${\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)}$ are $x=a$ and $x=2$ where $a<2$.

Find, in terms of $a$, the area of the region bounded by the curves $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.

2013 AJC P2 Q1

2013 AJC P2 Q1

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