plotting – Plot Parametric f(x), inverse f(x) and f'(x) and then f(x)/f'(x) and inversef'(x)/inversef(x)

To plot Abs(f(x))==Sqrt(f(x) Conjugate(f(x)) try

f = 2 a ArcTanh((# a)/Sqrt(-1 + #^2 b)) + a Log(1 + #^2 (a^2 - b)) -2 Log(# b + Sqrt(-1 + #^2 b)) & /. {a -> 2/10, b -> 3/10};
Plot(Sqrt(f(x) Conjugate(f(x))) , {x, -5, 5} , PlotRange -> All)

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The inverse function of Abs(f(x)) is only defined in subintervalls x<-2.17172,-2.17172<x<1.94146,x>1.94146.