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Question Number 134506 by bemath last updated on 04/Mar/21

Given g(x) = ((x+11)/(x−1)) ; x≠1 If f(x)+8(f○g)(x)= 15−x , find the value of f(13).

$$\mathrm{Given}\:\mathrm{g}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}+\mathrm{11}}{\mathrm{x}−\mathrm{1}}\:;\:\mathrm{x}\neq\mathrm{1}\: \\ $$$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)+\mathrm{8}\left(\mathrm{f}\circ\mathrm{g}\right)\left(\mathrm{x}\right)=\:\mathrm{15}−\mathrm{x}\:,\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{13}\right). \\ $$

Answered by EDWIN88 last updated on 04/Mar/21

first step , we must find the value of { ((g(2))),((g(13))) :} ⇔ { ((g(2)=((13)/1)=13)),((g(13)=((24)/(12))=2)) :} second step put x=2 ⇒ f(2)+8f(g(2))= 13 f(2)+8f(13) = 13...(i) put x=13⇒f(13)+8f(g(13))= 2 f(13)+8f(2) = 2...(ii) 8×(i)−(ii)⇒ 63 f(13) = 102 f(13) = ((102)/(63)) ≈1.62

$$\mathrm{first}\:\mathrm{step}\:,\:\mathrm{we}\:\mathrm{must}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\begin{cases}{\mathrm{g}\left(\mathrm{2}\right)}\\{\mathrm{g}\left(\mathrm{13}\right)}\end{cases} \\ $$$$\Leftrightarrow\begin{cases}{\mathrm{g}\left(\mathrm{2}\right)=\frac{\mathrm{13}}{\mathrm{1}}=\mathrm{13}}\\{\mathrm{g}\left(\mathrm{13}\right)=\frac{\mathrm{24}}{\mathrm{12}}=\mathrm{2}}\end{cases} \\ $$$$\mathrm{second}\:\mathrm{step}\: \\ $$$$\mathrm{put}\:\mathrm{x}=\mathrm{2}\:\Rightarrow\:\mathrm{f}\left(\mathrm{2}\right)+\mathrm{8f}\left(\mathrm{g}\left(\mathrm{2}\right)\right)=\:\mathrm{13} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\left(\mathrm{2}\right)+\mathrm{8f}\left(\mathrm{13}\right)\:=\:\mathrm{13}...\left(\mathrm{i}\right) \\ $$$$\mathrm{put}\:\mathrm{x}=\mathrm{13}\Rightarrow\mathrm{f}\left(\mathrm{13}\right)+\mathrm{8f}\left(\mathrm{g}\left(\mathrm{13}\right)\right)=\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\left(\mathrm{13}\right)+\mathrm{8f}\left(\mathrm{2}\right)\:=\:\mathrm{2}...\left(\mathrm{ii}\right) \\ $$$$\mathrm{8}×\left(\mathrm{i}\right)−\left(\mathrm{ii}\right)\Rightarrow\:\mathrm{63}\:\mathrm{f}\left(\mathrm{13}\right)\:=\:\mathrm{102}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\left(\mathrm{13}\right)\:=\:\frac{\mathrm{102}}{\mathrm{63}}\:\approx\mathrm{1}.\mathrm{62}\: \\ $$

Answered by Ñï= last updated on 04/Mar/21

f(13)+8f(g(13))=15−13=2 g(13)=((24)/(12))=2 1)::f(13)+8f(2)=2 f(2)+8f(g(2))=15−2=13 g(2)=13 2)::f(2)+8f(13)=13 1)∧2) ⇒f(13)=((34)/(21))

$${f}\left(\mathrm{13}\right)+\mathrm{8}{f}\left({g}\left(\mathrm{13}\right)\right)=\mathrm{15}−\mathrm{13}=\mathrm{2} \\ $$$${g}\left(\mathrm{13}\right)=\frac{\mathrm{24}}{\mathrm{12}}=\mathrm{2} \\ $$$$\left.\mathrm{1}\right)::{f}\left(\mathrm{13}\right)+\mathrm{8}{f}\left(\mathrm{2}\right)=\mathrm{2} \\ $$$${f}\left(\mathrm{2}\right)+\mathrm{8}{f}\left({g}\left(\mathrm{2}\right)\right)=\mathrm{15}−\mathrm{2}=\mathrm{13} \\ $$$${g}\left(\mathrm{2}\right)=\mathrm{13} \\ $$$$\left.\mathrm{2}\right)::{f}\left(\mathrm{2}\right)+\mathrm{8}{f}\left(\mathrm{13}\right)=\mathrm{13} \\ $$$$\left.\mathrm{1}\left.\right)\wedge\mathrm{2}\right) \\ $$$$\Rightarrow{f}\left(\mathrm{13}\right)=\frac{\mathrm{34}}{\mathrm{21}} \\ $$

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