CPI 200, G. Farin

Function types

Recall: one function value y per x-value.

Example

f1[x_]:=Log[x+2]+(3^(-0.05 x))*Sin[x];

Plot[f1[x],{x,0,100 },PlotStyle-> Thick,PlotLabel->Style["f1(x)",FontSize->18,Bold]]

bounded functions

(* discretize function values: *)functable[f_,a_,b_,eps_]:=Table[f[i],{i,a,b,eps}];

f2[x_]:= ArcTan[x];
Plot[f2[x],{x,0,30 },PlotStyle-> Thick,PlotLabel->Style["f2(x)",FontSize->18,Bold]]

table2=functable[f2,0,10,.5];
ListPlot[table2,Filling->Axis,PlotStyle->PointSize[Large
],PlotLabel->Style["f2(x)",FontSize->18,Bold]]

f3[x_]:=Log[x];
Plot[f3[x],{x,0,10 },PlotStyle-> Thick,PlotLabel->Style["f3(x)",FontSize->18,Bold]]

Now let’s check if a function is bounded. Instead of discretizing (being afraid we might miss some important information), we use Mathematica’s built-in functions.

boundedQ[f_]:=If [MaxValue[f,x]<Infinity && MinValue[f,x]>-Infinity,True, False];

boundedQ[Sqrt[x]]
boundedQ[Sin[x]]

periodic functions

monotone  functions

If f(x+ϵ)>f(x) for all x and ϵ, then f is is monotonically increasing. To check if f is monotone, discretize and check differences:

monoQ[f_,a_,b_,eps_]:=(table=functable[f,a,b,eps];
n=Length[table];
deltable=Table[table[[i+1]]-table[[i]],{i,1,n-1}];
If[Min[deltable]>0, True, False]);

monoQ[f1,0,10,0.2]
monoQ[f2,0,10,0.2]

asymptotes

There are two kinds of limits (at least) for functions: as they tend to infinity,
they often times take on simple shapes like parabolas etc. Those are asymptotes. For finite arguments, they may have poles. f7 below has both.

f7[x_]:= (x-2)^3/(1-x)^4;
f8[x_]:= (x-2)^4/(1-x)^3;
Plot[{f7[x],f8[x]},{x,-10,10 },PlotStyle-> Thick,PlotLabel->Style["f7(x), f8(x)",FontSize->18,Bold]]