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

I want to plot f(x), inverse f(x) and f'(x) and then f(x)/f'(x) and inversef'(x)/inversef(x)

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};

I don't know from where I can start?

• Your function f[x] is complex, is this feature intended? If yes your question cann#t be answered. Dec 30 '20 at 13:32
• Yes but I like to know inverse f(x) and its differential and their plots mostly. Dec 30 '20 at 13:35
• Please clarify your question: You want to plot parts of a complex function: Re, Im, Abs,... Dec 30 '20 at 13:47
• Abs I would like to know Abs Dec 30 '20 at 13:49

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.

• then I have this problem that how to introduce this f for plotting inversef'(x)/inversef(x) Dec 30 '20 at 17:15
Clear["Global*"]

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[Abs[f[x]], {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"x", "|f(x)|"}]


Use ParametricPlot to plot the inverse

ParametricPlot[{Abs[f[x]], x}, {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"|f(x)|", "x"}]


Use "Log" scaling (ScalingFunctions) for the derivative

Plot[Abs[f'[x]], {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"x", "|f'(x)|"},
ScalingFunctions -> "Log"]


as well as for the ratio

Plot[Abs[f[x]/f'[x]], {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"x", "|f(x)/f'(x)|"},
ScalingFunctions -> "Log"]


• for plotting inverse f'(x) / inverse f(x) ?? Dec 31 '20 at 2:54
• The inverses are not well-defined since they are multi-valued. You could use ParametricPlot to plot {Abs[f'[x]/f[x]], x} or {Log10[Abs[f'[x]/f[x]]], x}` Dec 31 '20 at 3:06