The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \left| \frac{6-2x}{x-2} \right|$, for $x\

Question

The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \left| \frac{6-2x}{x-2} ight|$, for $x\in \mathbb{R}$, $x
e 2$. Sketch the graph of $y=\mathrm{f}(x)$, indicating clearly the equation(s) of any asymptotes and coordinates of any axial intercepts. [3] Show that ${{\mathrm{f}}^{-1}}$ does not exist. [1] If the domain of $\mathrm{f}$ is restricted to $[0,a)$, find the largest value of $a$ for which ${{\mathrm{f}}^{-1}}$ exist. [1] Using the domain in part (a)(iii), find ${{\mathrm{f}}^{-1}}$ in similar form and the solution set for $\mathrm{f}(x)={{\mathrm{f}}^{-1}}(x)$. [4] 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 $\mathrm{gh}$. [3] Given instead that the range of $\mathrm{gh}$ is $(0,\ln 5]$. Find the exact value of $b$. [2]

Accepted Answer

$b = \ln 4$

2021 HCI J1 CT2 Q4

2021 HCI J1 CT2 Q4

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