Grade 12Algebra
Let f:R-R be a function such that f (x+y)+f (x-y)=f (xy) for all x,y €R.Then f is:
Let f:R-R be a function such that f (x+y)+f (x-y)=f (xy) for all x,y €R.Then f is:
2 Answers
Arun
Put y=0
=> f(x)+f(x)=2f(x).f(0)
=> f(0)=1
Now put x=0 and y=x
=> f(x)+f(-x)=2f(0)f(x)
=> f(x)+f(-x)=2f(x)
=> f(-x)=f(x)
So, even function.
=> f(x)+f(x)=2f(x).f(0)
=> f(0)=1
Now put x=0 and y=x
=> f(x)+f(-x)=2f(0)f(x)
=> f(x)+f(-x)=2f(x)
=> f(-x)=f(x)
So, even function.
Aditya Gupta
aruns ans is wrong.
we are given that f(x+y) + f(x – y)= f(xy).... (1)
put y=0
then f(x)+f(x)= f(0)
or f(x)= f(0)/2= C (a constant)
so from (1),
C+C=C
or C=0
so that f(x)= 0 is the only possibility.
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