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Question Number 57010 by 121194 last updated on 28/Mar/19

f(((x+y)/2))=((f(x)f(y))/(f(2)))  f(x)=?

$${f}\left(\frac{{x}+{y}}{\mathrm{2}}\right)=\frac{{f}\left({x}\right){f}\left({y}\right)}{{f}\left(\mathrm{2}\right)} \\ $$$${f}\left({x}\right)=? \\ $$

Commented by maxmathsup by imad last updated on 29/Mar/19

x=y ⇒f(x)=((f^2 (x))/(f(2)))  with f(2)≠0 ⇒f^2 (x)−f(2)(x)=0 ⇒f(x){f(x)−f(2)}=0  ⇒f(x)=0 of f(x)=f(2) for allx .

$${x}={y}\:\Rightarrow{f}\left({x}\right)=\frac{{f}^{\mathrm{2}} \left({x}\right)}{{f}\left(\mathrm{2}\right)}\:\:{with}\:{f}\left(\mathrm{2}\right)\neq\mathrm{0}\:\Rightarrow{f}^{\mathrm{2}} \left({x}\right)−{f}\left(\mathrm{2}\right)\left({x}\right)=\mathrm{0}\:\Rightarrow{f}\left({x}\right)\left\{{f}\left({x}\right)−{f}\left(\mathrm{2}\right)\right\}=\mathrm{0} \\ $$$$\Rightarrow{f}\left({x}\right)=\mathrm{0}\:{of}\:{f}\left({x}\right)={f}\left(\mathrm{2}\right)\:{for}\:{allx}\:. \\ $$

Answered by kaivan.ahmadi last updated on 28/Mar/19

★x=y=0⇒f(0)=((f^2 (0))/2)⇒f(0)=0 or f(0)=2  ★x=0,y=2⇒f(1)=((f(0)f(2))/(f(2)))=f(0)  ★x=1,y=1⇒f(1)=((f(1)f(1))/(f(2)))⇒f(2)=f(1)=f(0)  ★x=y⇒f(x)=((f^2 (x))/(f(2)))⇒f^2 (x)−2f(x)=0⇒  f(x)=2

$$\bigstar{x}={y}=\mathrm{0}\Rightarrow{f}\left(\mathrm{0}\right)=\frac{{f}^{\mathrm{2}} \left(\mathrm{0}\right)}{\mathrm{2}}\Rightarrow{f}\left(\mathrm{0}\right)=\mathrm{0}\:{or}\:{f}\left(\mathrm{0}\right)=\mathrm{2} \\ $$$$\bigstar{x}=\mathrm{0},{y}=\mathrm{2}\Rightarrow{f}\left(\mathrm{1}\right)=\frac{{f}\left(\mathrm{0}\right){f}\left(\mathrm{2}\right)}{{f}\left(\mathrm{2}\right)}={f}\left(\mathrm{0}\right) \\ $$$$\bigstar{x}=\mathrm{1},{y}=\mathrm{1}\Rightarrow{f}\left(\mathrm{1}\right)=\frac{{f}\left(\mathrm{1}\right){f}\left(\mathrm{1}\right)}{{f}\left(\mathrm{2}\right)}\Rightarrow{f}\left(\mathrm{2}\right)={f}\left(\mathrm{1}\right)={f}\left(\mathrm{0}\right) \\ $$$$\bigstar{x}={y}\Rightarrow{f}\left({x}\right)=\frac{{f}^{\mathrm{2}} \left({x}\right)}{{f}\left(\mathrm{2}\right)}\Rightarrow{f}^{\mathrm{2}} \left({x}\right)−\mathrm{2}{f}\left({x}\right)=\mathrm{0}\Rightarrow \\ $$$${f}\left({x}\right)=\mathrm{2} \\ $$

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