If `log_(4)((2f(x))/(1-f(x)))=x, " then "(f(2010)+f(-2009))` is equal to

Options

A. 0
B. -1
C. 1
D. 2

Answer

C

Explanation

`log_(4)((2f(x))/(1-f(x)))=x`
`implies (2f(x))/(1-f(x))=4^(x) " (1)" `
Replacing x by `1-x`, we get
`(2f(1-x))/(1-f(1-x))=4^(1-x) " (2)" `
Multiplying (1) and (2), we get
`(2f(x))/(1-f(x))((2f(1-x))/(1-f(1-x)))=4^(1-x)`
`implies f(x)f(1-x)=1-f(x)-f(1-x)+f(x)f(1-x)`
`implies f(x)+f(1-x)=1`
Putting `x=2010,` we get
`f(2010)+f(-2009)=1`