F[x_] = c x (1-x)

ContourPlot[{F[x]==x},{c,0,4},{x,-1,1},ContourStyle-> {{Red, Thickness[0.01]}}, Axes->True, Frame->False, AspectRatio->1, AxesLabel->{"c","x"}, PlotRange->{{-0.1,4},{-1,1}}]

Plot[{x,0.5 x (1-x)}, {x,-1,1}, PlotRange->{{-1,1},{-1,1}}]

Plot[{x, x (1-x)}, {x,-1,1}, PlotRange->{{-1,1},{-1,1}}]

Plot[{x, 1.5 x (1-x)}, {x,-1,1}, PlotRange->{{-1,1},{-1,1}}]

ContourPlot[{F[F[x]]==x},{c,0,4},{x,-1,1},ContourStyle-> {{Red, Thickness[0.01]}}, Axes->True, Frame->False, AspectRatio->1, AxesLabel->{"c","x"}, PlotRange->{{-0.1,4},{-1,1}}]