2021 NYJC Promo Q4

The function \(\mathrm{f}\) is defined by

\(\mathrm{f}\left( x \right)=\frac{1}{{{x}^{2}}+1}\), \(x\in \mathbb{R}\), \(x\ge k\).

1. State the minimum value of \(k\) for which the function \({{\mathrm{f}}^{-1}}\) exists.[1]

For the rest of the question, use the value of \(k\) found in part (i).

2. Sketch the graphs of \(y=\mathrm{f}\left( x \right)\) and \(y={{\mathrm{f}}^{-1}}\left( x \right)\) on the same diagram, showing clearly the relationship between them.[3]

The function \(\mathrm{g}\) is defined by

\(\mathrm{g}\left( x \right)=\frac{{{x}^{2}}+1}{x}\), \(x\in \mathbb{R}\), \(x>0\).

3. By finding \(\mathrm{fg}\left( x \right)\) or otherwise, solve \(\mathrm{g}\left( x \right)={{\mathrm{f}}^{-1}}\left( \frac{1}{5} \right)\).[4]