Sigmoid functions are used to model many natural processes such as population growth of virus. One example of a Sigmoid function $\mathrm{f}$ is given by

$\mathrm{f}: x \mapsto \frac{1}{1+\mathrm{e}^x}, x \in \mathbb{R} .$$

(i) Sketch the graph of $y=\mathrm{f}(x)$, indicating clearly the equation(s) of any asymptote(s) and the coordinates of any points where the curve crosses the axes.

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(ii) Find $\mathrm{f}^{-1}(x)$ in similar form.

Another function $\mathrm{g}$ is given by $\mathrm{g}: x \mapsto 3 x-1, x \in \mathbb{R}, 0 \leq x \leq 2$.

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(iii) Show that fg exists and find the range of $f g$, expressing your answer in terms of e.

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(iv) Describe a sequence of transformations which transform the graph of $y=\mathrm{f}(x)$ onto the graph of $y=f g(x)$.

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