F[x_] = 1-(x^2);

F[1]

Solve[F[x]==x,x]

Plot1 = Plot[{F[x],x},{x,-2,2},PlotStyle->{{Blue,Thickness[0.01]},{Red,Thickness[0.01]}},PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

N[Solve[F[x]==x,x]]

Solve[F[x]==x,x]

Table[Nest[F[#] &,-1.6,k],{k,0,20}]

Table[Nest[F[#] &,-1.62,k],{k,0,8}]

Abs[D[F[x],x]]/.x->-1.618

Table[Nest[F[#] &,0.61,k],{k,0,35}]

Table[Nest[F[#] &,0.63,k],{k,0,35}]

N[Abs[D[F[x],x]]/.{x->(1/2) (-1+(Sqrt[5]))}]

Table[Nest[F[#] &,1.1,k],{k,0,35}]

N[(1/2) (1+(Sqrt[5]))]

Table[Nest[F[#] &,1.618,k],{k,0,150}]

Table[Nest[F[#] &,1.61805,k],{k,0,15}]

F[F[x]]

Solve[x==F[F[x]],x]

Plot[{F[F[x]],x},{x,-2,2},PlotStyle->{{Blue,Thickness[0.01]},{Red,Thickness[0.01]}},PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

Plot[{F[F[F[x]]],x},{x,-2,2},PlotStyle->{{Blue,Thickness[0.01]},{Red,Thickness[0.01]}},PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

Plot[{F[F[F[F[x]]]],x},{x,-2,2},PlotStyle->{{Blue,Thickness[0.01]},{Red,Thickness[0.01]}},PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

Plot[{F[F[F[F[F[x]]]]],x},{x,-2,2},PlotStyle->{{Blue,Thickness[0.01]},{Red,Thickness[0.01]}},PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

Plot[{F[F[F[F[F[F[x]]]]]],x},{x,-2,2},PlotStyle->{{Blue,Thickness[0.01]},{Red,Thickness[0.01]}},PlotRange->{{-2,2},{-2,2}}, AspectRatio->1]

D[F[F[x]]]/.x->0

D[F[F[x]]]/.x->1

N[D[F[F[x]]]/.x->-1/2+Sqrt[5]]

N[D[F[F[x]]]/.x->-1/2-Sqrt[5]]

Table[Nest[F[F[#]] &,1.618,k],{k,0,150}]

Table[Nest[F[F[#]] &,1.1,k],{k,0,150}]