2018 MI Promo Q10
Functions $\text{f}$ and $\text{g}$ are defined by
$\text{f}:x\mapsto \frac{1}{x-2},$ $x\in \mathbb{R},$ $x\ne 2,$
$\text{g}:x\mapsto {{x}^{2}}-4x$, $x\in \mathbb{R}.$
(i)
Determine whether the composite functions ${{\text{f}}^{2}}$, $\text{fg}$ and $\text{gf}$ exist.
[4]
(ii)
Give a definition (including the domain) of the composite function that exists, and find the range of this composite function.
[2]
In the rest of the equation, the domain of $\text{f}$ is $x\in \mathbb{R},$ $x<2$.
(iii)
Define ${{\text{f}}^{-1}}$ in similar form.
[3]
(iv)
Sketch on a single clearly labelled diagram, the graphs of $y=\text{f}\left( x \right)$, $y={{\text{f}}^{-1}}\left( x \right)$ and $y={{\text{f}}^{-1}}\text{f}\left( x \right)$. Your sketch should indicate the position of the graphs in relation to the origin.
[3]
(v)
Without using a graphing calculator, solve the equation $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.
[2]
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