2020 ACJC Promo Q7
The functions $\text{f}$ and $\text{g}$ are defined as follows:
$\text{f}:x\mapsto 7{{\left( x-a \right)}^{2}}-1$, $x\in \mathbb{R}$, $x<3$
$\text{g}:x\mapsto 1+b-{{e}^{-x}}$, $x\ge 0$,
where $a$ and $b$ are real constants.
(i)
State the smallest value of $a$ such that ${{\text{f}}^{-1}}$ exists.
For the rest of this question, $a=4$.
(ii)
Explain if ${{\text{f}}^{-1}}$ exists, and if it does, find ${{\text{f}}^{-1}}\left( x \right)$ and state its domain.
(iii)
Find the largest value of $b$ such that the composite function $\text{fg}$ exists.
For this value of $b$, find $\text{fg}\left( x \right)$, state its domain, and find its range.
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