The functions $\mathrm{h}$ and $\mathrm{g}$ are defined by $\mathrm{h}:x\mapsto {{e}^{\left| 2x+1 \r

Question

The functions $\mathrm{h}$ and $\mathrm{g}$ are defined by $\mathrm{h}:x\mapsto {{e}^{\left| 2x+1 ight|}}+1,\mathrm{ }x\in \mathbb{R},\mathrm{ }x\le k$ $\mathrm{g}:x\mapsto $ $\left\{ \begin{matrix} 2x\mathrm{ for }0\le x\le \frac{1}{2} \ 2-2x\mathrm{ for }\frac{1}{2}\le x\le 1 \ \end{matrix} ight.$ Given that the function $\mathrm{h}$ has an inverse, state the greatest value of $k$. Find ${{\mathrm{h}}^{-1}}(x)$ and write down the domain of ${{\mathrm{h}}^{-1}}$. [4] Explain why the composite function $\mathrm{gg}$ exists. [1] Sketch the graph of $y=\mathrm{gg}\left( x ight)$ . [2] Given that $\mathrm{f}:x\mapsto {{e}^{\left| 2x+1 ight|}}+1,\mathrm{ }x\in \mathbb{R}$ , find the range of $\mathrm{fg}$ exactly. [1]

Accepted Answer

(i) $h^{-1}(x)=-\frac{ \ln(x-1)+1}{2}$, $x\ge2$; (ii) $gg$ exists; (iv) $R_{fg}=[e+1,e^3+1]$

2016 ACJC P2 Q3

2016 ACJC P2 Q3

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