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Question Number 47394 by gunawan last updated on 09/Nov/18

f(z)=((3z+1)/(2−4z))  f(f(z))=...

$${f}\left({z}\right)=\frac{\mathrm{3}{z}+\mathrm{1}}{\mathrm{2}−\mathrm{4}{z}} \\ $$$${f}\left({f}\left({z}\right)\right)=... \\ $$

Commented by maxmathsup by imad last updated on 10/Nov/18

fof(z) =((3f(z)+1)/(2−4f(z)))= ((3((3z+1)/(2−4z)) +1)/(2−4 ((3z+1)/(2−4z)))) =((9z+3+2−4z)/(4−8z−12z−4)) =((5z+5)/(−20z))  ⇒fof(z) =−((z+1)/(4z)) .

$${fof}\left({z}\right)\:=\frac{\mathrm{3}{f}\left({z}\right)+\mathrm{1}}{\mathrm{2}−\mathrm{4}{f}\left({z}\right)}=\:\frac{\mathrm{3}\frac{\mathrm{3}{z}+\mathrm{1}}{\mathrm{2}−\mathrm{4}{z}}\:+\mathrm{1}}{\mathrm{2}−\mathrm{4}\:\frac{\mathrm{3}{z}+\mathrm{1}}{\mathrm{2}−\mathrm{4}{z}}}\:=\frac{\mathrm{9}{z}+\mathrm{3}+\mathrm{2}−\mathrm{4}{z}}{\mathrm{4}−\mathrm{8}{z}−\mathrm{12}{z}−\mathrm{4}}\:=\frac{\mathrm{5}{z}+\mathrm{5}}{−\mathrm{20}{z}} \\ $$$$\Rightarrow{fof}\left({z}\right)\:=−\frac{{z}+\mathrm{1}}{\mathrm{4}{z}}\:. \\ $$

Answered by ajfour last updated on 09/Nov/18

f(f(z))=((3(((3z+1)/(2−4z)))+1)/(2−4(((3z+1)/(2−4z)))))         = ((5z+5)/(−20z)) = −(((z+1)/(4z))) .

$${f}\left({f}\left({z}\right)\right)=\frac{\mathrm{3}\left(\frac{\mathrm{3}{z}+\mathrm{1}}{\mathrm{2}−\mathrm{4}{z}}\right)+\mathrm{1}}{\mathrm{2}−\mathrm{4}\left(\frac{\mathrm{3}{z}+\mathrm{1}}{\mathrm{2}−\mathrm{4}{z}}\right)} \\ $$$$\:\:\:\:\:\:\:=\:\frac{\mathrm{5}{z}+\mathrm{5}}{−\mathrm{20}{z}}\:=\:−\left(\frac{{z}+\mathrm{1}}{\mathrm{4}{z}}\right)\:. \\ $$

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