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Given-f-x-y-4f-x-f-y-for-all-real-numbers-x-and-y-If-f-3-32-then-f-1-




Question Number 119815 by bemath last updated on 27/Oct/20
Given f(x+y)=4f(x).f(y) for  all real numbers x and y.  If f(3)=32 then f(1)=_
$${Given}\:{f}\left({x}+{y}\right)=\mathrm{4}{f}\left({x}\right).{f}\left({y}\right)\:{for} \\ $$$${all}\:{real}\:{numbers}\:{x}\:{and}\:{y}. \\ $$$${If}\:{f}\left(\mathrm{3}\right)=\mathrm{32}\:{then}\:{f}\left(\mathrm{1}\right)=\_ \\ $$
Answered by Olaf last updated on 27/Oct/20
f(1+1) = 4f(1)f(1)  f(2) = 4f(1)^2     f(2+1) = 4f(2)f(1)  f(3) = 16f(1)^3  = 32  f(1) = (2)^(1/3)
$${f}\left(\mathrm{1}+\mathrm{1}\right)\:=\:\mathrm{4}{f}\left(\mathrm{1}\right){f}\left(\mathrm{1}\right) \\ $$$${f}\left(\mathrm{2}\right)\:=\:\mathrm{4}{f}\left(\mathrm{1}\right)^{\mathrm{2}} \\ $$$$ \\ $$$${f}\left(\mathrm{2}+\mathrm{1}\right)\:=\:\mathrm{4}{f}\left(\mathrm{2}\right){f}\left(\mathrm{1}\right) \\ $$$${f}\left(\mathrm{3}\right)\:=\:\mathrm{16}{f}\left(\mathrm{1}\right)^{\mathrm{3}} \:=\:\mathrm{32} \\ $$$${f}\left(\mathrm{1}\right)\:=\:\sqrt[{\mathrm{3}}]{\mathrm{2}} \\ $$$$ \\ $$
Commented by bemath last updated on 27/Oct/20
yes....
$${yes}…. \\ $$

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