If `a f(x+1)+bf((1)/(x+1))=x,x ne -1, a ne b, " then " f(2)` is

Options

A. a+b
B. `a^(2)-b^(2)`
C. `(1)/(a+b)`
D. none of these

Answer

D

Explanation

We have, `af(x+1)+bf((1)/(+1))=x` for all ` x in R, x ne -1" "`…(i)
Replacing `x+1 by (1)/(x+1)`, we get
`af((1)/(x+1))+bf(x+1)+(1)/(x+1)-1`
`implies af((1)/(x+1))+bf(x+1)=(-x)/(x+1)" "`….(ii)
Multiplying (i) by a , (ii) by (b) and subtracting, we get
`(a^(2)-b^(2))f(x+1)=ax+(bx)/(x+1)`
`implies f(x+1)(x)/(a^(2)-b^(2)){a+(b)/(x+1)}` for all `x in R , x ne -1`
Putting x=0 in this question , we get f(1)=0