The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\begin{matrix} & \mathrm{f}:x\mapsto

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\begin{matrix} & \mathrm{f}:x\mapsto \left| x-2 ight|,x\in \mathbb{R} \ & \mathrm{g}:x\mapsto {{x}^{2}}-x,x\in \mathbb{R},x\le 0 \ \end{matrix}$ By considering the graphs of $\mathrm{f}$ and $\mathrm{g}$, solve exactly the inequality of $\mathrm{f}(x)>\mathrm{g}(x)$. [4] Explain why the composite function $\mathrm{fg}$ exists. [2] Find $\mathrm{fg}(x)$, giving the domain of $\mathrm{fg}$. [2] Show by means of a graphical argument that ${{(\mathrm{fg})}^{-1}}$ does not exist. [2] If the domain of $\mathrm{fg}$ is restricted to $x\le a$, state the greatest value of $a$ for ${{(\mathrm{fg})}^{-1}}$to exist. With the domain, find ${{(\mathrm{fg})}^{-1}}(x)$. [4]

Accepted Answer

$x_1 = -\sqrt{2}$, $-\sqrt{2} < x \leq 0$, fg exists, $fg: x \mapsto |x^2-x-2|$, $(fg)^{-1}$ doesn't exist, $a = -1$, $(fg)^{-1}: x \mapsto \dfrac{1}{2} - \sqrt{x + \dfrac{9}{4}}$

2016 JJC Promo Q7

2016 JJC Promo Q7

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