<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The function $\mathrm{f}$ is given by</span></div><div class="list-item-paragraph" style="text-align: center">$\mathrm{f}:x\mapsto \left| \frac{6-2x}{x-2} \right|$, for $x\in \mathbb{R}$, $x\ne 2$.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y=\mathrm{f}(x)$, indicating clearly the equation(s) of any asymptotes and coordinates of any axial intercepts.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that ${{\mathrm{f}}^{-1}}$ does not exist.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">If the domain of $\mathrm{f}$ is restricted to $[0,a)$, find the largest value of $a$ for which ${{\mathrm{f}}^{-1}}$ exist.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Using the domain in part (a)(iii), find ${{\mathrm{f}}^{-1}}$ in similar form and the solution set for $\mathrm{f}(x)={{\mathrm{f}}^{-1}}(x)$.</span></div><p style="text-align: right;">[4]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The functions $\mathrm{g}$ and $\mathrm{h}$ are given by</span></div><div class="list-item-paragraph" style="text-align: center">$g:x\mapsto \ln (x-3)$, for $x\in \mathbb{R}$, $x>4$,</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{h}:x\mapsto 4+{{\mathrm{e}}^{-x}}$, for $x\ge -b$, where $b$ is a positive constant.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the composite function $\mathrm{gh}$ exists and find the range of $\mathrm{gh}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Given instead that the range of $\mathrm{gh}$ is $(0,\ln 5]$. Find the exact value of $b$.</span></div><p style="text-align: right;">[2]</p></li></ol></li></ol>

$\mathrm{f}:x\mapsto \left| \frac{6-2x}{x-2} \right|$, for $x\in \mathbb{R}$, $x\ne 2$.

Sketch the graph of $y=\mathrm{f}(x)$, indicating clearly the equation(s) of any asymptotes and coordinates of any axial intercepts.

Show that ${{\mathrm{f}}^{-1}}$ does not exist.

If the domain of $\mathrm{f}$ is restricted to $[0,a)$, find the largest value of $a$ for which ${{\mathrm{f}}^{-1}}$ exist.

Using the domain in part (a)(iii), find ${{\mathrm{f}}^{-1}}$ in similar form and the solution set for $\mathrm{f}(x)={{\mathrm{f}}^{-1}}(x)$.

The functions $\mathrm{g}$ and $\mathrm{h}$ are given by

$g:x\mapsto \ln (x-3)$, for $x\in \mathbb{R}$, $x>4$,

$\mathrm{h}:x\mapsto 4+{{\mathrm{e}}^{-x}}$, for $x\ge -b$, where $b$ is a positive constant.

Show that the composite function $\mathrm{gh}$ exists and find the range of $\mathrm{gh}$.

Given instead that the range of $\mathrm{gh}$ is $(0,\ln 5]$. Find the exact value of $b$.

2021 HCI J1 CT2 Q4

2021 HCI J1 CT2 Q4

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