F1[x_] = x+(x^2);

Plot[{x,F1[x]},{x,-2,2}, PlotRange->{{-1,1},{-1,1}}]

Table[Nest[F1[#] &,0.01,k],{k,0,15}]

Table[Nest[F1[#] &,0.1,k],{k,0,15}]

Table[Nest[F1[#] &,-1.1,k],{k,0,15}]

Table[Nest[F1[#] &,-0.75,k],{k,0,500}]

Plot[{x,F1[x],F1[x]+0.1},{x,-2,2}, PlotRange->{{-1,1},{-1,1}}]

Plot[{x,F1[x]-1/2},{x,-2,2}, PlotRange->{{-1,1},{-1,1}}]

Solve[F1[x]-1/2==x,x]

N[D[F1[x],x]/.x->-1/Sqrt[2]]

N[D[F1[x],x]/.x->1/Sqrt[2]]

Solve[F1[x]-λ==x,x]

N[D[F1[x],x]/.x->Sqrt[λ]]

N[D[F1[x],x]/.x->-Sqrt[λ]]

((x^2)+x+λ)/.{x-> (x^2)+x+λ}

ContourPlot[{(x^2)+x+λ==x},{λ,-5,5},{x,-5,5},ContourStyle-> {{Red, Thickness[0.01]}},PlotRange-> All, Axes->True, Frame->False, AspectRatio->1]

ContourPlot[{x+(x^2)+2 λ+(((x+(x^2)+λ))^2)==x},{λ,-5,5},{x,-5,5},ContourStyle-> {{Red, Thickness[0.01]}},PlotRange-> All, Axes->True, Frame->False, AspectRatio->1]

(((x^2)+x+λ)/.{x-> (x^2)+x+λ})/.{x-> (x^2)+x+λ}

ContourPlot[{x+(x^2)+3 λ+(((x+(x^2)+λ))^2)+(((x+(x^2)+2 λ+(((x+(x^2)+λ))^2)))^2)==x},{λ,-5,0.1},{x,-3,3},ContourStyle-> {{Red, Thickness[0.005]}},PlotRange-> All, Axes->True, Frame->False, AspectRatio->1]