f(x+y)+f(x-y)=2f(x)f(y), where  , x,y =R , find whether the function is even or odd.

The given fun. is even.Is it right,I will explain.

 f(0)=1

as per question substitute y=0 to get above result

now when u substitue y=-x

f(2x)=f(x)f(-x)

Put x=x/2

f(x)=f(x/2)f(-x/2)

and f(-x)=f(-x/2)f(x/2)

HENCE EVEN FUNCTION

GOOD QUESTION BOSS!

Here's my method.put x=y=0,we get f(0)=0 or 1.But f(0)=1 only.

Now,put y=x,f(2x)+1=2f(x)f(x)

Put y=-x,1+f(2x)=2f(x)f(-x).Subtract the eqns. and u get f(x)=0 or f(x)=f(-x).Since,f(x) is non-zero,we get that the fun. is even.

hey when u substitute y=-x u get f(0) + f(2x) ryt? u get 1 + f(2x) how did u get f(2x)= 2f(x)f(-x)? cuz u explain again?