A function is defined as $\mathrm{f}\left( x \right)=\frac{1-{{\mathrm{e}}^{x}}}{{{\mathrm{e}}^{x-2}

Question

A function is defined as $\mathrm{f}\left( x \right)=\frac{1-{{\mathrm{e}}^{x}}}{{{\mathrm{e}}^{x-2}}}$. Show that $\mathrm{f}\left( x \right)$ can be written in the form ${{\mathrm{e}}^{\mathrm{g}\left( x \right)}}+A$, where $A$ is a constant and $\mathrm{g}\left( x \right)$ is a function in $x$ to be determined. [2] Hence describe a sequence of transformations that transforms the graph of $y={{\mathrm{e}}^{x}}$ onto the graph of $y=\mathrm{f}\left( x \right)$. [3]

Accepted Answer

(i) $A=-\mathrm{e}^2$, $g(x)=2-x$ (ii) Reflect in the $y$-axis, translate $2$ units right, then translate $\mathrm{e}^2$ units down

2021 ACJC P1 Q1

2021 ACJC P1 Q1

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