(* Functions f(x) and g(x). *)
	f[x_]:=x^3;
	g[x_]:= ((2x-1)^3+1)/2;
	h[x_]:=(x+1)/2;
	inverseh[x_]:=2x-1;

f[x]
g[y]
h[f[inverseh[y]]]

(* Locating the dynamical system within the graph and isolating it. *)
	plotx3=Plot[f[x],{x,-3,5},PlotRange->{{-2,2},{-3,3}},PlotStyle->{Red,Thickness[.01]},AspectRatio->.9];
	xdiag=Plot[x,{x,-1,1}];
	xdiagbig = Plot[x,{x,-2,2}];
	xfloor=Plot[-1,{x,-1,1}];
	xceiling=Plot[1,{x,-1,1}];
	xlwall=ParametricPlot[{-1,t},{t,-1,1}];
	xrwall=ParametricPlot[{1,t},{t,-1,1}];
	{Show[plotx3,xdiagbig,ImageSize->400],Show[plotx3,xdiag,xfloor,xceiling,xlwall,xrwall,ImageSize->400]}

(* Transforming it to the [0,1] interval.*)
	plotgy=Plot[g[x],{x,-1,2},PlotRange->{{-2,2},{-2,2}},PlotStyle->Thickness[.01],AspectRatio->.9];
	ydiag=Plot[x,{x,0,1}];
	yfloor=Plot[0,{x,0,1}];
	yceiling=Plot[1,{x,0,1}];
	ylwall=ParametricPlot[{0,t},{t,0,1}];
	yrwall=ParametricPlot[{1,t},{t,0,1}];
	{Show[plotgy,xdiagbig,ImageSize->400],Show[plotgy,ydiag,yfloor,yceiling,ylwall,yrwall,ImageSize->400]}

{Show[plotx3,xdiag,xfloor,xceiling,xlwall,xrwall,ImageSize->400],Show[plotgy,ydiag,yfloor,yceiling,ylwall,yrwall,ImageSize->400]}