Question Number 148035 by mathdanisur last updated on 25/Jul/21
$${if}\:\:\:{f}\left({g}\left({x}\right)\right)\:=\:\mathrm{2}{x}\:-\:\mathrm{5}\:\:{and}\:\:{f}\left({x}\right)\:=\:{x}\:+\:\mathrm{2} \\ $$$${find}\:\:\:{g}\left({x}\right)\:=\:? \\ $$
Commented by prakash jain last updated on 25/Jul/21
$${f}\left({x}\right)={x}+\mathrm{2} \\ $$$${f}\left({g}\left({x}\right)\right)={g}\left({x}\right)+\mathrm{2} \\ $$$${g}\left({x}\right)+\mathrm{2}=\mathrm{2}{x}−\mathrm{5}\Rightarrow{g}\left({x}\right)=\mathrm{2}{x}−\mathrm{7} \\ $$
Commented by mathdanisur last updated on 25/Jul/21
$${Thank}\:{you}\:{Ser} \\ $$
Answered by Olaf_Thorendsen last updated on 25/Jul/21
$${f}\left({x}\right)\:=\:{x}+\mathrm{2}\:\Rightarrow\:{f}'\left({x}\right)\:=\:\mathrm{1} \\ $$$$ \\ $$$${f}\mathrm{o}{g}\left({x}\right)\:=\:\mathrm{2}{x}−\mathrm{5} \\ $$$$\Rightarrow\:{g}'\left({x}\right)×{f}'\mathrm{o}{g}\left({x}\right)\:=\:\mathrm{2} \\ $$$${g}'\left({x}\right)×\mathrm{1}\:=\:\mathrm{2} \\ $$$${g}'\left({x}\right)\:=\:\mathrm{2} \\ $$$${g}\left({x}\right)\:=\:\mathrm{2}{x}+\mathrm{C} \\ $$$${f}\mathrm{o}{g}\left({x}\right)\:=\:\left(\mathrm{2}{x}+\mathrm{C}\right)+\mathrm{2}\:=\:\mathrm{2}{x}−\mathrm{5}\:\Rightarrow\:\mathrm{C}\:=\:−\mathrm{7} \\ $$$$ \\ $$$${g}\left({x}\right)\:=\:\mathrm{2}{x}−\mathrm{7} \\ $$
Commented by mathdanisur last updated on 25/Jul/21
$${Thank}\:{you}\:{Ser} \\ $$