Sigmoid functions are used to model many natural processes such as population growth of virus. One e

Question

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

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

Sketch the graph of $y=\mathrm{f}\left( x \right)$, indicating clearly the equation(s) of any asymptote(s) and the coordinates of any points where the curve crosses the axes.[2]
Find ${{\mathrm{f}}^{-1}}\left( x \right)$ in similar form.[3]

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

Show that $\mathrm{fg}$ exists and find the range of $\mathrm{fg}$, expressing your answer in terms of $\mathrm{e}$.[4]
Describe a sequence of transformations which transform the graph of $y=\mathrm{f}\left( x \right)$ onto the graph of $y=\mathrm{fg}\left( x \right)$.[2]

Accepted Answer

(ii) ${{\mathrm{f}}^{-1}}:x\to -\ln \left( \frac{1}{x}-1 \right)$, $x\in \left( 0,1 \right)$ (iv) Scale parallel to the $x$-axis with scale factor $\frac{1}{3}$ followed by a translation of $\frac{1}{3}$units in the positive $x$-axis direction.

2021 CJC Promo Q11

2021 CJC Promo Q11

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