You may want to have a look at this
fy[x_, a_, b_, n_] := a (1 - Exp[-x/b] - x/b Exp[-x/b])
fx[y_, a_, b_, n_] := -b - b ProductLog[n, (-a + y)/(a E)]
fx[fy[.5, .1, .5, -1], .1, .5, -1] // FullSimplify
fy[fx[.5, .1, .5, -1], .1, .5, -1] // FullSimplify
(fx[fy[#, .1, .5, -1], .1, .5, -1] // FullSimplify) & /@ Range[20/2]
(fy[fx[#, .1, .5, -1], .1, .5, -1] // FullSimplify) & /@ Range[20/2] // Chop