2010 ACJC CT2
(a)
Given the functions
$\text{f}:x\mapsto 3{{x}^{3}}-4{{x}^{2}}-1$, $x\ge 1$,
$\text{g}:x\mapsto {{\text{e}}^{2-0.25{{x}^{2}}}}$, $x\ge -3$.
(i) Explain why the functions ${{\text{f}}^{-1}}$ and $\text{gf}$ exist,
[2]
(ii) Find the exact value of ${{\text{f}}^{-1}}\left( -1 \right)$,
[2]
(iii) Find the range of $\text{gf}$, giving your answer in exact form.
[2]
(b)
The functions $\text{f}$ and $\text{h}$ are defined by
$\text{f}:x\mapsto {{x}^{2}}-{{\lambda }^{2}}$, $x\le -\lambda $,
$\text{h}:x\mapsto \sqrt{x+{{\lambda }^{2}}}$, $x\ge 0$.
where $\lambda >0$. Define the following functions
(i) ${{\text{f}}^{-1}}$.
(ii) $\text{hf}$.
[5]
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