Question Number 82078 by jagoll last updated on 18/Feb/20
$${f}\left(\mathrm{10}^{{x}} \right)\:=\:\sqrt{{x}}\: \\ $$$${what}\:{is}\:{f}^{−\mathrm{1}} \left({x}\right)=? \\ $$
Commented by john santu last updated on 18/Feb/20
$${hahaha}..{i}\:{think}\:{you}\:{can}\:{solve}\:{this} \\ $$$${problem}\: \\ $$
Answered by Kunal12588 last updated on 18/Feb/20
$$\mathrm{10}^{{x}} ={t} \\ $$$$\Rightarrow{x}\:{log}\:\mathrm{10}\:=\:{log}\:{t} \\ $$$$\Rightarrow{x}=\:\frac{{log}\:{t}}{{log}\:\mathrm{10}}={log}_{\mathrm{10}} {t} \\ $$$${f}\left({t}\right)=\sqrt{{log}_{\mathrm{10}} {t}} \\ $$$$\Rightarrow{t}=\mathrm{10}^{\left({f}\left({t}\right)\right)^{\mathrm{2}} } \\ $$$${f}^{−\mathrm{1}} \left({t}\right)=\mathrm{10}^{{t}^{\mathrm{2}} } \\ $$$$\Rightarrow{f}^{−\mathrm{1}} \left({x}\right)=\mathrm{10}^{{x}^{\mathrm{2}} } \\ $$