<ol class="i-list"><li><div class="list-item-paragraph">A student defines a function $\mathrm{f}$ and its inverse ${{\mathrm{f}}^{-1}}$ by the following:</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{f}:x\mapsto \sin x,$  $\qquad\,\,\,$  $x\in \mathbb{R},$  $\quad$  $-\pi \le x\le \pi $</div><div class="list-item-paragraph" style="text-align: center">${{\mathrm{f}}^{-1}}:x\mapsto {{\sin }^{-1}}x,$  $\;\;$  $x\in \mathbb{R},$  $\;\;$  $-1\le x\le 1$.</div><div class="list-item-paragraph">With the aid of a sketch, explain what is wrong with the definition of $\mathrm{f}$, and suggest a suitable restriction to its domain such that the definition of ${{\mathrm{f}}^{-1}}$ is correct.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{e}^{-{{\left( x+1 \right)}^{2}}}}$, $x\in \mathbb{R}$, $-1\le x\le 0$. Find the inverse function ${{\mathrm{g}}^{-1}}$ in similar form.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Explain if the composite function ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}$ exists.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Write down ${{\mathrm{f}}^{-1}}{{\mathrm{g}}^{-1}}\left( x \right)$and its domain, and find its range. Hence, or otherwise, find the value of $x$ such that&nbsp;$\mathrm{gf}\left( x \right)=\frac{1}{e}$.<span class="inline-marks">[4]</span></div></li></ol>

A student defines a function $\mathrm{f}$ and its inverse ${{\mathrm{f}}^{-1}}$ by the following:

$\mathrm{f}:x\mapsto \sin x,$  $\qquad\,\,\,$  $x\in \mathbb{R},$  $\quad$  $-\pi \le x\le \pi $

${{\mathrm{f}}^{-1}}:x\mapsto {{\sin }^{-1}}x,$  $\;\;$  $x\in \mathbb{R},$  $\;\;$  $-1\le x\le 1$.

With the aid of a sketch, explain what is wrong with the definition of $\mathrm{f}$, and suggest a suitable restriction to its domain such that the definition of ${{\mathrm{f}}^{-1}}$ is correct.

The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{e}^{-{{\left( x+1 \right)}^{2}}}}$, $x\in \mathbb{R}$, $-1\le x\le 0$. Find the inverse function ${{\mathrm{g}}^{-1}}$ in similar form.

Explain if the composite function ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}$ exists.

Write down ${{\mathrm{f}}^{-1}}{{\mathrm{g}}^{-1}}\left( x \right)$and its domain, and find its range. Hence, or otherwise, find the value of $x$ such that $\mathrm{gf}\left( x \right)=\frac{1}{e}$.

2016 ACJC Promo Q8

2016 ACJC Promo Q8

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