The function $\mathrm{f}$ is defined by
$$
\mathrm{f}(x)=\frac{1}{x^2+1}, x \in \mathbb{R}, x \geq k .
$$
(i) 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).
(ii) Sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram, showing clearly the relationship between them.

[3]

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

[4]