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Question Number 2000 by Yozzi last updated on 29/Oct/15
Find a non−constant function f   satisfying f(0)=1,f(−2)=0 and  f(x−y)=f(x)f(y)−f(−2−x)f(y−2).
$${Find}\:{a}\:{non}−{constant}\:{function}\:{f}\: \\ $$$${satisfying}\:{f}\left(\mathrm{0}\right)=\mathrm{1},{f}\left(−\mathrm{2}\right)=\mathrm{0}\:{and} \\ $$$${f}\left({x}−{y}\right)={f}\left({x}\right){f}\left({y}\right)−{f}\left(−\mathrm{2}−{x}\right){f}\left({y}−\mathrm{2}\right). \\ $$
Commented by prakash jain last updated on 29/Oct/15
x=0  f(−y)=f(0)f(y)−f(−2)f(y−2)  ∵f(−2)=0 for the above relation to be  true ∀y.  f(−y)=f(y)  ⇒even function  y=x  f(0)=[f(x)]^2 −f(−2−x)f(x−2)  [f(x)]^2 =1+f(x−2)f(x+2)
$${x}=\mathrm{0} \\ $$$${f}\left(−{y}\right)={f}\left(\mathrm{0}\right){f}\left({y}\right)−{f}\left(−\mathrm{2}\right){f}\left({y}−\mathrm{2}\right) \\ $$$$\because{f}\left(−\mathrm{2}\right)=\mathrm{0}\:\mathrm{for}\:\mathrm{the}\:\mathrm{above}\:\mathrm{relation}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{true}\:\forall{y}. \\ $$$${f}\left(−{y}\right)={f}\left({y}\right) \\ $$$$\Rightarrow{even}\:{function} \\ $$$${y}={x} \\ $$$${f}\left(\mathrm{0}\right)=\left[{f}\left({x}\right)\right]^{\mathrm{2}} −{f}\left(−\mathrm{2}−{x}\right){f}\left({x}−\mathrm{2}\right) \\ $$$$\left[{f}\left({x}\right)\right]^{\mathrm{2}} =\mathrm{1}+{f}\left({x}−\mathrm{2}\right){f}\left({x}+\mathrm{2}\right) \\ $$
Commented by prakash jain last updated on 30/Oct/15
y=2  f(x−2)=f(x)f(2)−f(x+2)f(0)  f(x−2)=−f(x+2)  f(x)=−f(x+4)  f(x+8)=[f(x+4+4)]=−f(x+4)=f(x)  f is periodic with period 8.  cos [(π/4)(x−y)]=cos (π/4)x∙cos (π/4)y+sin( (π/4)x)sin ((π/4)y)  sin ((πy)/4)=cos (((π(y−2))/4))  sin ((πx)/4)=−cos ((π/2)+(π/4)x)=−cos (((π(−x−2))/4))
$${y}=\mathrm{2} \\ $$$${f}\left({x}−\mathrm{2}\right)={f}\left({x}\right){f}\left(\mathrm{2}\right)−{f}\left({x}+\mathrm{2}\right){f}\left(\mathrm{0}\right) \\ $$$${f}\left({x}−\mathrm{2}\right)=−{f}\left({x}+\mathrm{2}\right) \\ $$$${f}\left({x}\right)=−{f}\left({x}+\mathrm{4}\right) \\ $$$${f}\left({x}+\mathrm{8}\right)=\left[{f}\left({x}+\mathrm{4}+\mathrm{4}\right)\right]=−{f}\left({x}+\mathrm{4}\right)={f}\left({x}\right) \\ $$$${f}\:{is}\:{periodic}\:{with}\:{period}\:\mathrm{8}. \\ $$$$\mathrm{cos}\:\left[\frac{\pi}{\mathrm{4}}\left({x}−{y}\right)\right]=\mathrm{cos}\:\frac{\pi}{\mathrm{4}}{x}\centerdot\mathrm{cos}\:\frac{\pi}{\mathrm{4}}{y}+\mathrm{sin}\left(\:\frac{\pi}{\mathrm{4}}{x}\right)\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}{y}\right) \\ $$$$\mathrm{sin}\:\frac{\pi{y}}{\mathrm{4}}=\mathrm{cos}\:\left(\frac{\pi\left({y}−\mathrm{2}\right)}{\mathrm{4}}\right) \\ $$$$\mathrm{sin}\:\frac{\pi{x}}{\mathrm{4}}=−\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}{x}\right)=−\mathrm{cos}\:\left(\frac{\pi\left(−{x}−\mathrm{2}\right)}{\mathrm{4}}\right) \\ $$
Answered by Rasheed Soomro last updated on 02/Nov/15
Find a non−constant function f   satisfying f(0)=1,f(−2)=0 and  f(x−y)=f(x)f(y)−f(−2−x)f(y−2)...........(1)  −−−−−−×××−−−−−−−  x=−2,y=x  f(−2−x)=f(−2)f(x)−f(−2−{−2})f(x−2)  f(−2−x)=0.f(x)−f(0)f(x−2)=−f(x−2)  f(−2−x)=−f(x−2)=−{f(x)f(2)−f(−2−x)f(2−2)}                =−{f(x)f(2)−f(−2−x)f(0)}=f(−x−2)−f(2)f(x)  f(−x−2)−f(−x−2)=−f(2)f(x)  f(2)=0 ∣ f(x)=0  [But this not non−constant]  ∗∗∗  Substituting y=0  f(x)=f(x)f(0)−f(−2−x)f(−2)           =f(x)  This means f(x)  is arbitrary if y=0.  x=−2−x  f(−2−x−y)=f(−2−x)f(y)−f(−2−{})  continue
$${Find}\:{a}\:{non}−{constant}\:{function}\:{f}\: \\ $$$${satisfying}\:{f}\left(\mathrm{0}\right)=\mathrm{1},{f}\left(−\mathrm{2}\right)=\mathrm{0}\:{and} \\ $$$${f}\left({x}−{y}\right)={f}\left({x}\right){f}\left({y}\right)−{f}\left(−\mathrm{2}−{x}\right){f}\left({y}−\mathrm{2}\right)………..\left(\mathrm{1}\right) \\ $$$$−−−−−−×××−−−−−−− \\ $$$${x}=−\mathrm{2},{y}={x} \\ $$$${f}\left(−\mathrm{2}−{x}\right)={f}\left(−\mathrm{2}\right){f}\left({x}\right)−{f}\left(−\mathrm{2}−\left\{−\mathrm{2}\right\}\right){f}\left({x}−\mathrm{2}\right) \\ $$$${f}\left(−\mathrm{2}−{x}\right)=\mathrm{0}.{f}\left({x}\right)−{f}\left(\mathrm{0}\right){f}\left({x}−\mathrm{2}\right)=−{f}\left({x}−\mathrm{2}\right) \\ $$$${f}\left(−\mathrm{2}−{x}\right)=−{f}\left({x}−\mathrm{2}\right)=−\left\{{f}\left({x}\right){f}\left(\mathrm{2}\right)−{f}\left(−\mathrm{2}−{x}\right){f}\left(\mathrm{2}−\mathrm{2}\right)\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−\left\{{f}\left({x}\right){f}\left(\mathrm{2}\right)−{f}\left(−\mathrm{2}−{x}\right){f}\left(\mathrm{0}\right)\right\}={f}\left(−{x}−\mathrm{2}\right)−{f}\left(\mathrm{2}\right){f}\left({x}\right) \\ $$$${f}\left(−{x}−\mathrm{2}\right)−{f}\left(−{x}−\mathrm{2}\right)=−{f}\left(\mathrm{2}\right){f}\left({x}\right) \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{0}\:\mid\:{f}\left({x}\right)=\mathrm{0}\:\:\left[{But}\:{this}\:{not}\:{non}−{constant}\right] \\ $$$$\ast\ast\ast \\ $$$${Substituting}\:{y}=\mathrm{0} \\ $$$${f}\left({x}\right)={f}\left({x}\right){f}\left(\mathrm{0}\right)−{f}\left(−\mathrm{2}−{x}\right){f}\left(−\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:={f}\left({x}\right) \\ $$$${This}\:{means}\:{f}\left({x}\right)\:\:{is}\:{arbitrary}\:{if}\:{y}=\mathrm{0}. \\ $$$${x}=−\mathrm{2}−{x} \\ $$$${f}\left(−\mathrm{2}−{x}−{y}\right)={f}\left(−\mathrm{2}−{x}\right){f}\left({y}\right)−{f}\left(−\mathrm{2}−\left\{\right\}\right) \\ $$$${continue} \\ $$
Answered by prakash jain last updated on 30/Oct/15
f(x)=cos ((πx)/4)
$${f}\left({x}\right)=\mathrm{cos}\:\frac{\pi{x}}{\mathrm{4}} \\ $$
Commented by prakash jain last updated on 30/Oct/15
see comments in question for explaination.
$${see}\:{comments}\:{in}\:{question}\:{for}\:{explaination}. \\ $$

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