2022 ASRJC J2 MYCT P1 Q8
The function $\text{f}$ is defined by
$\text{f}:x\mapsto 1+ax-{{x}^{2}}$, $x\in \mathbb{R}$, $x\le k$
where $a$ is a positive constant.
(i)
Explain with a sketch why the largest value of $k$ for which ${{\text{f}}^{-1}}$ exists is $\frac{a}{2}$.
With this value of $k$, find ${{\text{f}}^{-1}}$ in a similar form.
[5]
(ii)
If $k=1$ and $a=2$, find the set of values of $x$ for $\text{f }{{\text{f}}^{-1}}\left( x \right)={{\text{f}}^{-1}}\text{f}\left( x \right)$.
[1]
(iii)
The function $\text{g}$ is defined by
$\text{g}:x\mapsto 2+2x-{{x}^{2}},\text{ }x\in \mathbb{R},\text{ }0\le x\le 1$.
It is also given that $\text{g}\left( x \right)=\text{g}\left( x+2 \right)$ for all real values of $x$.
Sketch $y=\text{g}\left( x \right)$ for $0\le x\le 5$.
[2]
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