Lösungen

1

a, b

f[x_] := (x + 2) (x - 
            2) (x - 4)/(x^2 + x + 2)

Plot[f[x], {x, -6, 8}];

Solve[f[x] == 0, {x}]

Apart[f[x]]

PolynomialQuotient[Numerator[f[x]], Denominator[f[x]], x]

PolynomialRemainder[Numerator[f[x]], Denominator[f[x]], x]

Solve[
      PolynomialQuotient[Numerator[f[x]], Denominator[f[x]], x] == 
        0, {x}]

c

f[x_] := (x + 2) (x - 
            2) (x - 4)/(x^2 + x + 0)

Plot[f[x], {x, -6, 8}];

Solve[f[x] == 0, {x}]

Apart[f[x]]

PolynomialQuotient[Numerator[f[x]], Denominator[f[x]], x]

PolynomialRemainder[Numerator[f[x]], Denominator[f[x]], x]

Solve[
      PolynomialQuotient[Numerator[f[x]], Denominator[f[x]], x] == 
        0, {x}]

2

Remove[f, g, h]

a,b

g[x_] := x - 2;h[x_] := x^2 - 2;

k = 1; f[k] = g[h[x]]; {f[k] // Expand, f[k]}

k = 2; f[k] = h[g[x]]; {f[k] // Expand, f[k]}

k = 3; f[k] = h[g[g[x]]]; {f[k] // Expand, f[k]}

k = 4; f[k] = g[h[h[x]]]; {f[k] // Expand, f[k]}

k = 5; f[k] = h[g[h[x]]]; {f[k] // Expand, f[k]}

k = 6; f[k] = g[h[g[x]]]; {f[k] // Expand, f[k]}

c,d

h[x_] := x - 2;g[x_] := x^2 - 2;

k = 1; f[k] = g[h[x]]; {f[k] // Expand, f[k]}

k = 2; f[k] = h[g[x]]; {f[k] // Expand, f[k]}

k = 3; f[k] = h[g[g[x]]]; {f[k] // Expand, f[k]}

k = 4; f[k] = g[h[h[x]]]; {f[k] // Expand, f[k]}

k = 5; f[k] = h[g[h[x]]]; {f[k] // Expand, f[k]}

k = 6; f[k] = g[h[g[x]]]; {f[k] // Expand, f[k]}

3

Remove[f1, f2, f3, g, h]

a,b

f1[
          x_] := (x^2 - 2) (-1 + 
              x^2) ;
f2[x_] := Sqrt[f1[x]];
f3[x_] := f2[x]^4;

({{Plot[{f1[x]}, {x, 0, 2}, PlotRange -> {-3, 3}];}, {Plot[{f2[x]}, {x, 0, 2}, 
              PlotRange -> {-3, 4}];

          Plot[{f3[x]}, {x, 0, 2}, PlotRange -> {-3, 7}];}})

Solve[f1[x] == 0, {x}]

Solve[f2[x] == 0, {x}]

Solve[f3[x] == 0, {x}]

c,d

f1[
          x_] := (-x^2 + 2) (1 - 
              x^2) ;
f2[x_] := Sqrt[f1[x]];
f3[x_] := f2[x]^4;

({{Plot[{f1[x]}, {x, 0, 2}, PlotRange -> {-3, 3}];}, {Plot[{f2[x]}, {x, 0, 2}, 
              PlotRange -> {-3, 4}];

          Plot[{f3[x]}, {x, 0, 2}, PlotRange -> {-3, 7}];}})

Solve[f1[x] == 0, {x}]

Solve[f2[x] == 0, {x}]

Solve[f3[x] == 0, {x}]

4

?Sign

Remove[f]

f[x_ /; x ≥ 2] := 6 - x^2;
f[x_ /; (x ≥ 0 And x < 2)] := 2; 
    f[x_ /; x < 0] := Sign[Floor[Sin[x Pi/2]]]^2;
p1 = Plot[f[x], {x, -3, 5}];

p2 = Plot[f[x], {x, -5, 5}];

Show[p1, p2, AspectRatio -> Automatic];

Remove[f]

f[x_ /; x ≥ 2] := 6 - x^2;
f[x_ /; (x ≥ 0 And x < 2)] := 2; 
    f[x_ /; x < 0] := -Sign[
            Floor[Sin[x Pi/2]]]^2;
p1 = Plot[f[x], {x, -1, 3.5}];

p2 = Plot[f[x], {x, -4.5, 0}];

Show[p2, p1, AspectRatio -> Automatic];

f1[x_ /; 2 ≤  x] := 6 - x^2;
f2[x_ /; (0 ≤ x   &&  x < 2)] := 2; 
    f3[x_ /; x < 0] := -Sign[
            Floor[Sin[x Pi/2]]]^2;
p1 = 
        Plot[ f1[x] , {x, 2, 3.5}, 
          PlotRange -> {-6, 2}];
p2 = 
        Plot[ f2[x] , {x, 0, 1.99999}, 
          PlotRange -> {-6, 2}];
p3 = 
        Plot[ f3[x] , {x, -5, -0.00001}, 
          PlotRange -> {-6, 2}];
s1 = Show[p2, p1];
s2 = Show[p2, p3];
Show[s1, s2];

Remove[f];

-Sign[Floor[Sin[x Pi/2]]]^2 // TeXForm

f[x_ /; 2 ≤  x] := 6 - x^2;
f[x_ /; (0 ≤ x   &&  x < 2)] := 2; 
    f[x_ /; x < 0] := -Sign[
            Floor[Sin[x Pi/2]]]^2;
p1 = 
        Plot[ f[x] , {x, 2, 3.5}, 
          PlotRange -> {-6, 2}];
p2 = 
        Plot[ f[x] , {x, 0, 1.99999}, 
          PlotRange -> {-6, 2}];
p3 = 
        Plot[ f[x] , {x, -4, -0.00001}, 
          PlotRange -> {-6, 2}];
s1 = Show[p2, p1];
s2 = Show[p3, p2];
Show[s1, s2];

Options[Plot]

?PlotRegion

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

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

Remove[f];

f[x_ /; 2 ≤  x] := 6 - x^2;
f[x_ /; (0 ≤ x   &&  x < 2)] := 2; 
    f[x_ /; x < 0] := -Sign[
            Floor[Sin[x Pi/2]]]^2;
p1 = 
        Plot[ f[x] , {x, 2, 3.5}, 
          PlotRange -> {-6, 2}];
p2 = 
        Plot[ f[x] , {x, 0, 1.99999}, 
          PlotRange -> {-6, 2}];
p3 = 
        Plot[ f[x] , {x, -4, -0.00001}, 
          PlotRange -> {-6, 2}];
s1 = Show[p2, p1];
s2 = Show[p2, p3];

5

Remove[f]

c := Abs[Floor[x + 1] Sqrt[Abs[x]]];
Plot[f[x], {x, -6, 6}];

f[0]

f[0] f[-1]

f[1] f[-1]