Work with Funktions 1

Gauss [x],  Floor[x]

?? Floor

Plot[Floor[x], {x, -5, 5}];

Plot[Floor[10 Sin[Cos[x]]], {x, -5, 5}];

Signum, sgn[x], Sign[x], Sin[...]

?Sign

Plot[Sign[x], {x, -5, 5}];

Options[Plot]

Plot[Sign[Sin[x^2] x^2], {x, -20, 20}];

Plot[Sign[Sin[x^2] x^2], {x, -20, 20}, 
        PlotPoints -> 100];

Plot[Sign[Sin[x^2] x^2], {x, -20, 20}, 
        PlotPoints -> 1000];

Plot[Sin[x^2] x^2, {x, -20, 20}, 
        PlotPoints -> 1000];

Plot[Sin[x^2] x^2, {x, -5, 5}];

Plot[Sin[x^2] x^2, {x, -9, 9}];

Absolute, Abs[x]

?Abs

Plot[Abs[x], {x, -3, 3}];

Plot[x* Sign[x], {x, -3, 3}];

Table,  ListPlot,  Df=N

Table[Tan[n], {n, 1, 20}]

t = Table[Tan[n], {n, 1, 30}] // N

?ListPlot

Options[ListPlot]

ListPlot[t];

c = 2.4

Constant Function

f[x_] := c

Plot[f[x], {x, -3, 3}, PlotLabel -> "Hallo"];

f[x_] := 3;
Plot[f[x], {x, -3, 3}, PlotLabel -> "Hallo"];

Linear Function,   Plot  and  Evaluate

f[x_, a_, b_] := a x  +  b

Table[f[x, a, 2], {a, -3, 3}]

Plot[Table[f[x, a, 2], {a, -3, 3}], {x, -3, 3}];

Evaluate[Table[f[x, a, 2], {a, -3, 3}]]

Plot[
        Evaluate[Table[f[x, a, 2], {a, -3, 3}]], {x, -3, 
          3}];

Animate  (Click on  the Graphics to enimate, then click on "Cell, Animate Selected Graphics" or Ctrl+Y)

Table[Plot[f[x, a, 2], {x, -3, 3}], {a, -3, 3}];

Table[
        Plot[x^2 + b x + 2, {x, -7, 3}], {b, -3, 10}];

Table[Plot[Sin[b x], {x, -7, 3}], {b, -3, 10}];

Nest and so on

Remove["Global`*"]

Nest[f, x, 2]

Nest[f, x, 5]

Nest[f, x, 1]

Nest[f, x, 0]

Nest[Sin, x, 4]

Nest[Sin, 2, 4]

Nest[Sin, 2.5, 4]

Table[Nest[Sin, 2.5, n], {n, 1, 20}]

Options[Plot]

ListPlot[Table[Nest[Sin, 2.5, n], {n, 1, 20}], 
        AxesOrigin -> {0, 0}, 
        PlotStyle -> PointSize[0.04]];

NestList[Sin, 2.5, 20]

ListPlot[NestList[Sin, 2.5, 20], AxesOrigin -> {0, 0}, 
        PlotStyle -> PointSize[0.04]];

Animate!! (In "Cell": Animate selected graphics....)

Do[Plot[Nest[Sin, x, n]/Nest[Sin, Pi/2, n], {x, -2 Pi, 2  Pi}], {n, 
        1, 800, 12}]

3D and so

\(\(Plot3D[
        Nest[Sin, x, Round[y]], {x, \(-2\) Pi, 2  Pi}, {y, 1, 
          15}, 
AxesLabel -> {"x", "y"}];\)\)

Plot3D[
        Nest[Sin, x, Round[y]], {x, -2 Pi, 2  Pi}, {y, 1, 
          15}, 
AxesLabel -> {"x", "y"}, 
        PlotPoints -> {50, 30}];

Plot3D[
        Nest[Sin, x, Round[y]], {x, -2 Pi, 2  Pi}, {y, 1, 
          15}, 
AxesLabel -> {"x", "y"}, 
        PlotPoints -> {30, 15}, Shading -> False];

Options[Plot3D]

Plot3D[
        Nest[Sin, x, Round[y]], {x, -2 Pi, 2  Pi}, {y, 1, 
          15}, 
AxesLabel -> {"x", "y"}, 
        PlotPoints -> {30, 15}, Shading -> False, 
        ViewPoint -> {0, -5, 0}];

Plot3D[
        Nest[Sin, x, Round[y]], {x, -2 Pi, 2  Pi}, {y, 1, 
          15}, 
AxesLabel -> {"x", "y"}, 
        PlotPoints -> {30, 15}, Shading -> False, 
        ViewPoint -> {5, 0, 0}];

Animate!! (In "Cell": Animate selected graphics....)

Do[
        Plot3D[Nest[Sin, x, Round[y]], {x, -2 Pi, 2  Pi}, {y, 1, 
            15}, 
PlotPoints -> {30, 15}, 
          Shading -> False, 
          ViewPoint -> {5/n, 5 - 5/n, -5/n}], {n, 1, 
          50}];