2016 ACJC P2 Q3
The functions $\text{h}$ and $\text{g}$ are defined by
$\text{h}:x\mapsto {{e}^{\left| 2x+1 \right|}}+1,\text{ }x\in \mathbb{R},\text{ }x\le k$
$\text{g}:x\mapsto $ $\left\{ \begin{matrix}
2x\text{ for }0\le x\le \frac{1}{2} \\
2-2x\text{ for }\frac{1}{2}\le x\le 1 \\
\end{matrix} \right.$
(i)
Given that the function $\text{h}$ has an inverse, state the greatest value of $k$. Find ${{\text{h}}^{-1}}(x)$ and write down the domain of ${{\text{h}}^{-1}}$.
[4]
(ii)
Explain why the composite function $\text{gg}$ exists.
[1]
(iii)
Sketch the graph of $y=\text{gg}\left( x \right)$ .
[2]
(iv)
Given that $\text{f}:x\mapsto {{e}^{\left| 2x+1 \right|}}+1,\text{ }x\in \mathbb{R}$ , find the range of $\text{fg}$ exactly.
[1]
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