2008 AJC P1 Q11
The function $\text{f}$ is defined by $\text{f}:x\to \ln \left[ {{\left( x-1 \right)}^{2}}-k \right]$, where $x<-2$ and $k$ is a constant. Find the maximum value of $k$. Another function $\text{g}$ is defined by $\text{g}:x\to \tan x,-\frac{\pi }{2}<x<\frac{\pi }{2}.$ If $k=8$,
(i)
define ${{\text{f}}^{-1}}$, the inverse function of $\text{f}$, in a similar form.
(ii)
show that the composite function ${{\text{g}}^{-1}}{{\text{f}}^{-1}}$ exists and find the range of this composite function.
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