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)
```

The inverse function of `Abs(f(x))`

is only defined in subintervalls `x<-2.17172,-2.17172<x<1.94146,x>1.94146`

.