We can draw y=f[x]
by
ParametricPlot[{x, f[x]}, {x, 0, 2}]
and draw its inverse x=f[y]
by
ParametricPlot[{f[y], y}, {y, 0, 2}]
f[x_] := 2 A ArcTanh[(# A)/Sqrt[-1 + #^2 B]] +
A Log[1 + #^2 (A^2 - B)] - 2 Log[# B + Sqrt[-1 + #^2 B]] &[
x] /. {A -> 0.2, B -> 0.3} // Abs;
ParametricPlot[{{x, f[x]}, {f[x], x}}, {x, 0, 2}]

We can also use ContourPlot
f[x_] := 2 A ArcTanh[(# A)/Sqrt[-1 + #^2 B]] +
A Log[1 + #^2 (A^2 - B)] - 2 Log[# B + Sqrt[-1 + #^2 B]] &[
x] /. {A -> 0.2, B -> 0.3} // Abs;
ContourPlot[{y == f[x], x == f[y]}, {x, 0, 2}, {y, 0, 2}]