CJC/2021/JC1/Promo/P1/Q11

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. [2] (ii) Find$\mathrm{f}^{-1}(x)$in similar form. Another function$\mathrm{g}$is given by$\mathrm{g}: x \mapsto 3 x-1, x \in \mathbb{R}, 0 \leq x \leq 2$. [3] (iii) Show that fg exists and find the range of$f g$, expressing your answer in terms of e. [4] (iv) Describe a sequence of transformations which transform the graph of$y=\mathrm{f}(x)$onto the graph of$y=f g(x)\$.

[2]