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Question Number 196621 by universe last updated on 28/Aug/23
$$\mathrm{find}\:\:\mathrm{f}\left(\mathrm{x}\right)\:\:\mathrm{if} \\ $$$$\:\mathrm{f}\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\right)\:=\:\frac{\mathrm{x}}{\mathrm{x}+\mathrm{1}} \\ $$
Answered by MM42 last updated on 28/Aug/23
$${x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}={u}\Rightarrow{x}=\frac{{u}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{u}} \\ $$$$\Rightarrow{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{1}}\:\:\checkmark \\ $$$$ \\ $$