If `f(x+1/2)+f(x-1/2)=f(x) for all x in R ,` then the period of `f(x)` is 1 (b) 2 (c) 3 (d) 4

Options

A. 1
B. 2
C. 3
D. 4

Answer

C

Explanation

`f(x+(1)/(2))+f(x-(1)/(2))=f(x) " (1)" `
Replacing x by `(x+(1)/(2)),` we get
`f(x+1)+f(x)=f(x+(1)/(2)) " (2)" `
` or f(x+1)+f(x-(1)/(2))=0 " (from (1) and (2))" `
` or f(x+(3)/(2))= - f(x) " "("Replcing x by "(x+(1)/(2)))`
` or f(x+3) = -f(x+(3)/(2))=f(x) " " ("Replacing x by "(x+(3)/(2)))`
Therefore, f(x) is periodic with period 3.