Question Number 173652 by saly last updated on 15/Jul/22
Commented by saly last updated on 15/Jul/22
$$\:\:{Help}\:{me} \\ $$
Commented by JDamian last updated on 15/Jul/22
$${you}\:{really}\:{need}\:{help}\:−{have}\:{you}\:{missed}\:{any}\:{term}? \\ $$
Commented by saly last updated on 15/Jul/22
$$\:\:\:\:{log}\sqrt{\frac{\mathrm{2022}}{{x}^{\mathrm{3}} +{x}+\mathrm{2}}} \\ $$
Answered by okbruh123 last updated on 15/Jul/22
$${first}\:{of}\:{all},\:{theres}\:{no}\:{base}\:{mentioned}. \\ $$$${so}\:{i}\:{am}\:{assuming}\:{its}\:{common}\:{log} \\ $$$${since}\:{base}\:>\:\mathrm{0}\:{and}\:\neq\mathrm{1},\:{we}\:{need}\:{not}\:{worry}\:{about}\:{it}. \\ $$$$\Rightarrow\sqrt{\frac{\mathrm{2022}}{{x}^{\mathrm{3}} +{x}+\mathrm{2}}\:}\:>\:\mathrm{0}\:\cap\:{x}^{\mathrm{3}} +{x}+\mathrm{2}\neq\mathrm{0} \\ $$$$\Rightarrow\:{as}\:{square}\:{root}\:{is}\:{always}\:\geqslant\:\mathrm{0}\:{and}\:{in}\:{this}\:{case}\:{also}\:\neq\:\mathrm{0} \\ $$$$\Rightarrow\:{x}\:\in\:\mathbb{R}\:\cap\:\left({x}+\mathrm{1}\right)+{x}^{\mathrm{3}} +\mathrm{1}\:\neq\:\mathrm{0} \\ $$$$\Rightarrow\:{x}\:\in\:\mathbb{R}\:\cap\:\left({x}+\mathrm{1}\right)+\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}−{x}\right)\neq\mathrm{0} \\ $$$$\Rightarrow\:\mathbb{R}\:\cap\:\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{2}\right)\:\neq\mathrm{0}\Rightarrow{x}\neq−\mathrm{1}\: \\ $$$$\Rightarrow\:{x}\:\in\:\mathbb{R}−\left\{−\mathrm{1}\right\} \\ $$$${this}\:{was}\:{domain}. \\ $$$${ill}\:{answer}\:{range}\:{separately} \\ $$