2021 TMJC Promo Q5
Given that $a>0$, functions $\text{f}$ and $\text{g}$ are defined by
$\text{f}:x\mapsto x+2+\frac{4}{x-1}$ for $x\in \mathbb{R}$, $x\ne 1$,
$\text{g}:x\mapsto \ln \left( x+a \right)$ for $x\in \mathbb{R}$, $x>0$.
(i)
Explain clearly why $\text{gf}$ does not exist.
[2]
(ii)
Find the range of values of $a$, in exact form, such that $\text{fg}$ exists.
[2]
For the rest of the question, assume that the function $\text{fg}$ exists.
(iii)
Define $\text{fg}$ in a similar form, in terms of $a$.
[2]
(iv)
Given that $a>{{\text{e}}^{3}}$, find the range of $\text{fg}$, in terms of $a$.
[2]
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