Incomplete curves can often be completed by adding the option PlotRange->All. The logic according to which in absence of this option Mathematica considers parts of curves irrelevant (of detracting from the essential features of the curve?) seems to have changed from version to version. At least my Mathematica 10 works perfectly for the second boxed expression of the original question.

And as to labels, you can try Epilog. Check this post out: http://stackoverflow.com/questions/7221315/how-do-i-label-different-curves-in-mathematica

Hi!

It seems, the problem is, that you didn't define f2 while trying to plot the function. And when you manually substituted f2 by (1-f1) you got your plot. But you can also substitute it by a command:

Plot[With[{r1 = 0.055, r2 = 0.003}, F1 /. f2 -> 1 - f1], {f1, 0, 1}, 
 Frame -> True, FrameLabel -> {Subscript[f, 1], Subscript[F, 1]}]

As a function

 F[f1_, r1_, r2_] := ((1 - f1) f1 + f1^2 r1)/( 2 (1 - f1) f1 + f1^2 r1 + (1 - f1)^2 r2)
    Plot[{F[f1, 0.055, 0.003], F[f1, 55, 0.01], F[f1, 0.1, 10]}, {f1, 0, 1}, Frame -> True, FrameLabel -> {Subscript[f, 1], Subscript[F, 1]}]

Or, if you wish to keep both f1 and f2

FF[f1_, f2_, r1_, r2_] := (f1 f2 + r1 f1^2)/(r1 f1^2 + 2 f1 f2 + r2 f2^2)
Plot[{FF[f1, 1 - f1, 0.055, 0.003], FF[f1, 1 - f1, 55, 0.01], FF[f1, 1 - f1, 0.1, 10]}, {f1, 0, 1}, Frame -> True, FrameLabel -> {Subscript[f, 1], Subscript[F, 1]}]

I am not sure whether I understand you question correctly. Can the following code serve as you need?

Plot[{((1 - f1) f1 + 0.055 f1^2)/(0.003 (1 - f1)^2 + 2 (1 - f1) f1 + 
     0.055 f1^2), ((1 - f1) f1 + 55 f1^2)/(0.01 (1 - f1)^2 + 
     2 (1 - f1) f1 + 55 f1^2), ((1 - f1) f1 + 
     0.1 f1^2)/(10 (1 - f1)^2 + 2 (1 - f1) f1 + 0.1 f1^2), f1}, {f1, 
  0, 1}, Frame -> True, 
 FrameLabel -> {Subscript[f, 1], Subscript[F, 1]}, 
 PlotLegends -> "Expressions"]