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If the function / satisfies the relation...

If the function / satisfies the relation `f(x+y)+f(x-y)=2f(x),f(y)AAx , y in R and f(0)!=0` , then

A

`f(x)` is an even function

B

`f(x)` is an odd function

C

If `f(2)=a, " then " f(-2)=a`

D

If `f(4)=b, " then " f(-4)= -b`

Text Solution

Verified by Experts

The correct Answer is:
A, C
`f(x+y)+f(x-y) =2f(x)*f(y) " (1)" `
Put `x=0." Then " f(y)+f(-y)=2f(0)f(y) " (2)" `
Put `x=y=0. " Then " f(0) +f(0)=2f(0) f(0)`
` :. f(0) =1 ["as "f(0) ne 0]`
` :. f(-y) =f(y) " [from (2)]" `
Hence, the function is even, Then `f(-2)=f(2) =a.`
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