The following two equations are satisfied:

f(-x)=-f(x)

f(0)=0:f(0)=a/b=0

So: a=0

F(2)=2/(4+b)=2/5, then b= 1.

Therefore, the analytical formula is f (x) = x/(x 2+ 1).

Let- 1 =

=[x 1(x2^2+ 1)-x2(x 1^2+ 1)]/(x 1^2+ 1)(x2^2+ 1)

=[x 1x2(x2-x 1)+(x 1-x2)]/(x 1^2+ 1)(x2^2+ 1)

=(x2-x 1)(x 1x2- 1)/(x 1^2+ 1)(x2^2+ 1)

Because x2-x 1 >: 0, x1x2 <; 1, so there is f (x 1)-f (x2).

Therefore, if the function is increasing function on [- 1, 1], the maximum value is f( 1)= 1/2, and the minimum value is f(- 1)=- 1/2.

Therefore, the range is [- 1/2, 1/2]