T[x_] = Piecewise[{{2x, x≥0 && x≤ (1/2)},{2-2x, x>(1/2) && x≤ 1}}]

Tplot = Plot[T[x],{x,0,1}, PlotStyle->{{Blue,Thickness[0.01]}}]

XPlot = Plot[x,{x,0,1}, PlotStyle->{{Red,Thickness[0.01]}}];

Show[Tplot,XPlot]

T2 = Simplify[T[T[x]]]

Plot[{x,T2},{x,0,1}, PlotStyle->{{Red,Thickness[0.01]},{Blue,Thickness[0.01]}}]

T3 = Simplify[T[T[T[x]]]]

Plot[{x,T3},{x,0,1}, PlotStyle->{{Red,Thickness[0.01]},{Blue,Thickness[0.01]}}]

Solve[2-8 x == x,x]

T[2/9]

T[4/9]

T[8/9]

Solve[-2 + 8 x == x,x]

T[2/7]

T[4/7]

T[6/7]

T2New[x_] = (Piecewise[{{4-4 x, (3/4)≤x≤1}, {4 x, 0≤x≤(1/4)}, {2-4 x, (1/4)<x≤(1/2)}, {-2+4 x, (1/2)<x<(3/4)}, {0, True}}]);

T4 = Simplify[T2New[T2New[x]]]

Plot[{x,T4},{x,0,1}, PlotStyle->{{Red,Thickness[0.01]},{Blue,Thickness[0.01]}}]

Solve[2 - 16 x == x,x]

T[2/17]

T[T[2/17]]

T[T[T[2/17]]]

T[T[T[T[2/17]]]]