2022 YIJC Promo Q9
The function $\text{f}$ is defined by
$\text{f}:x\mapsto {{x}^{2}}-4x-5$, for $x\in \mathbb{R}$, $x\le 2$.
(a)
Find ${{\text{f}}^{-1}}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}$.
[3]
(b)
On the same diagram, sketch the graphs of $\text{f}$ and ${{\text{f}}^{-1}}$.
[3]
(c)
Find the exact solution of the equation $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.
[3]
The function $\text{g}$ is defined by
$\text{g}:x\mapsto 1+{{\text{e}}^{2x}}$, for $x\in \mathbb{R}$, $x<0$.
(d)
Explain why the composite function $\text{fg}$ exists and find the range of $\text{fg}$.
[3]
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