F[x_] = Piecewise[{{2(x-1)+3, x≥1 && x< 2},{-1(x-2)+5, x≥ 2 && x< 3},{-2(x-3)+4, x≥ 3 && x< 4},{-1(x-4)+2, x≥ 4 && x≤  5}}]

Table[Nest[F[#] &,1,k],{k,0,10}]

Plot[{x,F[x]},{x,1,5},PlotRange->{{1,5},{1,5}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,F[F[x]]},{x,1,5},PlotRange->{{1,5},{1,5}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,F[F[F[x]]]},{x,1,5},PlotRange->{{1,5},{1,5}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

G[x_] = Piecewise[{{3(x-1)+4, x≥1 && x< 2},{-1(x-2)+7, x≥ 2 && x< 3},{-1(x-3)+6, x≥ 3 && x< 4},{-2(x-4)+5, x≥ 4 && x<  5},{-1(x-5)+3, x≥ 5 && x<  6},{-1(x-6)+2, x≥ 6 && x≤  7}}]

Table[Nest[G[#] &,1,k],{k,0,14}]

Plot[{x,G[x]},{x,1,7},PlotRange->{{1,7},{1,7}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,G[G[x]]},{x,1,7},PlotRange->{{1,7},{1,7}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,G[G[G[x]]]},{x,1,7},PlotRange->{{1,7},{1,7}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,G[G[G[G[x]]]]},{x,1,7},PlotRange->{{1,7},{1,7}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,G[G[G[G[G[x]]]]]},{x,1,7},PlotRange->{{1,7},{1,7}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,G[G[G[G[G[G[x]]]]]]},{x,1,7},PlotRange->{{1,7},{1,7}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

G3[x_] = Simplify[G[G[G[x]]]];

Plot[{x,G[G[G[G[G[G[G[x]]]]]]]},{x,1,7},PlotRange->{{1,7},{1,7}},PlotStyle->{{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]