If ` f:R to R` is a function satisfying the equation
`f(2x+3)+f(2x+7)=2 AA x in R,` then find the fundamental period of `f(x)`.

Options

A. 2
B. 4
C. 8
D. 12

Answer

C

Explanation

We have ,
`f(2x+3)+f(2x+7)=2 ` for all ` x in R `.
`implies f(y)+f(y+4)=2` for all `y in R " "[ :. Y=2x+3]`
`implies f(y-4)+f(y)=2` for all ` y in R ` [ Replacing y by(y-4)]
Thus , we have
`f(y)+f(y+4)=2 and f(y-4) +f(y)=` for all ` y in R `.
On subtracting , we get
`f(y+4)-f(y-4)=0` for all `y in R `.
` implies f(y+4)-f(y-4)=0` for all ` y in R `.
`implies f(y)=f(y+8) ` for all ` y in R ` [ Replacing y by ( y+4)]
`implies f(x)` is periodic with period 8 units.