2021 NYJC Promo Q4
The function $\text{f}$ is defined by
$\text{f}\left( x \right)=\frac{1}{{{x}^{2}}+1}$, $x\in \mathbb{R}$, $x\ge k$.
(i)
State the minimum value of $k$ for which the function ${{\text{f}}^{-1}}$ exists.
[1]
For the rest of the question, use the value of $k$ found in part (i).
(ii)
Sketch the graphs of $y=\text{f}\left( x \right)$ and $y={{\text{f}}^{-1}}\left( x \right)$ on the same diagram, showing clearly the relationship between them.
[3]
The function $\text{g}$ is defined by
$\text{g}\left( x \right)=\frac{{{x}^{2}}+1}{x}$, $x\in \mathbb{R}$, $x>0$.
(iii)
By finding $\text{fg}\left( x \right)$ or otherwise, solve $\text{g}\left( x \right)={{\text{f}}^{-1}}\left( \frac{1}{5} \right)$.
[4]
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