`f:RrarrR` is a function satisfying `f(x+5)gef(x)+5 and f(x+1)lef(x)+1`.
Given `f(1)=1` and `g(x)=f(x)+1-x, g(2009)` is equal to

Options

A. `2009`
B. `2008`
C. `1`
D. None of these

Answer

C

Explanation

`f(x+5)gef(x)+5 ….(i)`
and `f(x+1)lef(x)+1 …(ii)`
or `f(x)ge f(x+1)-1….(iii)`
`implies f(x+5)gef(x)+5gef(x+1)-1+5`
or `f(x+5)ge f(x+1)+4…..(iv)`
`impliesf(x+5)ge f(x+4)+1`
and `therefore (x+1)ge f(x)+1….(v)`
`("replacing" x by x-4)`
from (ii) and (v) `implies f(x+1)=f(x)+1....(vi)`
`therefore f(1)=1`
from (vi), f(2), f(3)=3 etc.
f(2009) =2009
`therefore g(2009)=f(2009)+1-2009=1`