Funny functions

Algebra Level 3

f(12009)+f(22009)+f(32009)++f(20082009) \large f \left(\frac{1}{2009} \right)+f\left(\frac{2}{2009}\right) +f\left(\frac{3}{2009}\right) + \ldots +f\left(\frac{2008}{2009}\right)

Let f(x)=4x4x+2\large f(x)=\dfrac{4^{x}}{4^{x}+2}, find the value of the expression above.

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