Lösungen

1

f[x_] := x;
f[x_ /; x < 0] := 2;
f[x_ /; Integer[x]  &&  Positive[x]] := 
        3; 
Plot[f[x], {x, -4, 4}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
        Epilog -> {PointSize[0.02], Point[{1, 3}], Point[{2, 3}], 
            Point[{3, 3}], Point[{4, 3}]}];

2

Plot[Sqrt[(x - 10) (x - 20)], {x, -40, 40}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

3

Remove[f]

f[x_] := (x - 1) (x - 2);  f[x] // Expand

Solve[{f[x] == a x^2 + b x  + c /. x -> 0, 
        f[x] == a x^2 + b x  + c /. x -> 1, 
        f[x] == a x^2 + b x  + c /. x -> 2}, {a, b, 
        c}]

Reduce[f[x] == a x^2 + b x  + c, {a, b, c}, {x}]

Plot[(x - 1) (x - 2), {x, 0, 3}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

<< Algebra`InequalitySolve`

InequalitySolve[(x - 1) (x - 2) > 0, x]

4

Remove[f]

f[x_] := (x^2 - 3  x + 1)/(x - 1)

Plot[f[x], {x, 0, 3}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

Plot[{f[x], x - 2}, {x, -2, 4}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
        PlotRange -> {-4, 4}, PlotPoints -> 50];

f[x] // Apart

5

Remove[f]

f[z_] := Sin[z];  g[x_] := Cos[x];

Plot[f[g[x]], {x, -10, 10}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

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

Table[f[g[(π/2) + k 2  Pi]], {k, -5, 5}]

6

Remove[f, g]

f[x_] := Sign[x];  g[x_] := Sin[x];

Plot[f[x], {x, -3, 3}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

Plot[f[g[x]], {x, -8, 8}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

Plot[g[Abs[f[x]]], {x, -8, 8}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];