Three plots in one plot

Question

• How can we make this comparison more clear,visible and understandable?

• Secondly, I want to add the plot legends to the inside of the plot (to a suitable place).

   f[j_, k_, x_] :=Piecewise[{{1, (j - 1)/4 k <= x && x <= j/4 k}, {0, True}}];
  
    Plot[{f[1, 2, x], f[1, 3, x], f[1, 4, x]}, {x, 0, 1.5},  PlotStyle -> { {Red, Thick}, {Blue, Thick, DotDashed}, {Green,Dashed} } , Frame -> True, GridLines -> Automatic, 
     PlotLegends -> {"f[1,2,x]", "f[1,3,x]", "f[1,4,x]"}]
  

  

Answer 1

"How can we make this comparison more clear, visible and understandable?" Given the nature of the functions, that might be impossible. But here are two approaches:

Plot[{f[1, 2, x], f[1, 3, x], f[1, 4, x]}, {x, 0, 1.5},
 PlotStyle -> {{Red, Thickness[0.03]}, {Blue, 
    Thickness[0.02]}, {Green, Thickness[0.01]}},
 PlotRangeClipping -> False, ImageSize -> Large,
 PlotLegends -> Placed[{"f[1,2,x]", "f[1,3,x]", "f[1,4,x]"}, {.5, .5}],
 PlotRange -> {{-0.1, 1.5}, {-0.1, 1.1}}, AxesOrigin -> {-0.1, -0.1}]

plots = Table[Plot[f[1, i, x], {x, 0, 1.5},
    PlotStyle -> {{Red, Thick}},
    PlotRangeClipping -> False, ImageSize -> Large,
    PlotLabel -> Style["f[1, " <> ToString[i] <> ", x]", Bold, 18],
    PlotRange -> {{-0.1, 1.5}, {-0.1, 1.1}}, 
    AxesOrigin -> {-0.1, -0.1}],
   {i, {2, 3, 4, 3}}];
Export["plots.gif", plots, "DisplayDurations" -> 1, "AnimationRepetitions" -> Infinity]

Answer 2

Clear["Global`*"]

f[j_, k_, x_] := 
  Piecewise[{{1, (j - 1)/4 k <= x && x <= j/4 k}, {0, True}}];

Plot[
 {f[1, 2, x] + 0.025, f[1, 3, x], f[1, 4, x] - 0.025},
 {x, 0, 1.5},
 PlotStyle ->
  {{Red, Thick}, {Blue, Thick, DotDashed}, {Green, Dashed}},
 Frame -> True,
 GridLines -> Automatic,
 PlotLegends ->
  Placed[{"f[1,2,x]", "f[1,3,x]", "f[1,4,x]"}, {.5, .5}],
 FrameTicks -> {Automatic, {0, 1}}]

Answer 3

This may not be suitable for all situations, but here is an attempt to visualize this in a different manner.

n1 = NumberLinePlot[{f[1, 2, x] != 0, f[1, 3, x] != 0, 
   f[1, 4, x] != 0}, {x, 0, 1.5}, 
  PlotLegends -> 
   Placed[{"f[1,2,x]\[NotEqual]0", "f[1,3,x]\[NotEqual]0", 
     "f[1,4,x]\[NotEqual]0"}, {0.9, .5}],
  AspectRatio -> Automatic,
  ImageSize -> Large]


n2 = NumberLinePlot[{f[1, 2, x] == 0, f[1, 3, x] == 0, 
   f[1, 4, x] == 0}, {x, 0, 1.5},
  PlotLegends -> 
   Placed[{"f[1,2,x]\[Equal]0", "f[1,3,x]\[Equal]0", 
     "f[1,4,x]\[Equal]0"}, {0.23, .5}],
  AspectRatio -> Automatic,
  ImageSize -> Large
  ]

GraphicsGrid[{{n1}, {n2}}, ImageSize -> Large]

Answer 4

I like the answers by JimB, Bob Hanlon and Syed as they are both straightforward and visually clear.

The following is an attempt to produce lines that look like multi-colored dashing. For each subset of the input function list, we identify the subdomains where they coincide, and use MeshShading with colors associated with the subset. With the option MeshFunctions -> {"ArcLength"}, we get n equal length dashing pieces using option Mesh -> n.

f[j_, k_, x_] := Piecewise[{{1, (j - 1)/4 k <= x && x <= j/4 k}, {0, True}}];

functions = f[1, #, x] & /@ {2, 3, 4};

colors = {Red, Green, Blue};

legend = LineLegend[Directive[AbsoluteThickness[3], #] & /@ colors,
  {"f[1, 2, x]", "f[1, 3, x]", "f[1, 4, x]"}];

subsetdomains = Join[Thread[{List /@ Range[3], 0 <= x <= 3/2}],
   {#, Reduce[{Equal @@ functions[[#]], 0 <= x <= 3/2}, x]} & /@ 
    Subsets[Range @ 3, {2, 3}]];

layers = Plot[ConditionalExpression[functions[[#[[1, 1]]]], #[[2]]], 
    {x, 0, 1.5}, 
     ImageSize -> 500, 
     MeshFunctions -> {"ArcLength"}, 
     Mesh -> 20, 
     MeshStyle -> None, 
     MeshShading -> (Directive[CapForm["Butt"], AbsoluteThickness[3], #] & /@ 
       colors[[#[[1]]]])] & /@ subsetdomains;

show = Show[layers, 
  GridLines -> {MapThread[{#, Directive[#2, Dashed]} &, {{.5, .75, 1}, colors}], None},
  Frame -> True, Axes -> False, PlotRange -> All];

Using LocatorPane with Appearance -> legend we can interactively control the position of the legend inside the plot frame:

LocatorPane[{1.25, .6}, show, Appearance -> legend]

With

f[j_, k_, x_] := Piecewise[{{Sin[5 x], (j - 1)/4 k <= x && x <= j/4 k}, 
     {Sin[20 x]/2, True}}];

and Mesh -> 30, LocatorPane[{.2, -.5}, show, Appearance -> legend] gives