Given that $a0$, functions $\mathrm{f}$ and $\mathrm{g}$ are defined by \(\mathrm{f}:

Question

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

Accepted Answer

(i) $\mathrm{gf}$ does not exist regardless of the value of $a$. (ii) $\mathrm{fg}$ exists if $\ln a\ge 1\Rightarrow a\ge \mathrm{e}$ (iii) $\mathrm{fg}:x\mapsto \ln \left( x+a \right)+2+\frac{4}{\ln \left( x+a \right)-1}$ for $x\in \mathbb{R}$, $x>0$ (iv) ${{\mathrm{R}}_{\mathrm{fg}}}=\left( \ln a+2+\frac{4}{\ln a-1},\infty \right)$

2021 TMJC Promo Q5

2021 TMJC Promo Q5

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