functions:-subjective

if f(x+y)+f(x-y)=2f(x)f(y) for all x,y Îµ R and f(0)â‰ 0 then prove that f(x) is an even function.

2 Answers

106

Asish Mahapatra ·2009-07-28 09:51:29

Setting x=y=0, we get
2f(0)=2f²(0)
So, f(0) = 1 [given f(0)â‰ 0]

Set x=y=x
f(2x)+1 = 2f²(x) ... (i)

Set y= -x, we have
1+f(2x) = 2f(x) .. (ii)

From 1 and 2
2f(x) = 2f²(x)
=> f(x)=0 OR f(x)=1

Either case f(x) is even

after, f(0)=1,
u can directly have

x=0, so

1*f(-y)+1*f(y)=2*1*f(y)

or f(-y)=f(y)
simple way out.

so f(x) is even

cheers!!

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