Here are the cob-web-plots for this function:


As you see there are two solutions:
1/2 (3 \[PlusMinus] Sqrt[5])
One is stable, one is unstable, as you can see from the cob-web-plot above.
Play around with it yourself using this code:
ClearAll[CobwebPlot]
SetAttributes[CobwebPlot,HoldAll]
CobwebPlot[f_,start_?NumericQ,n_,{xrange:{xmin_,xmax_},yrange:{_,_}}]:=Module[{cob,x,g1,coor},
cob=NestList[f,start,n];
coor = Partition[Riffle[cob,cob],2,1];
coor[[1,2]]=0;
g1=Graphics[{Red,Line[coor]}];
Show[{Plot[{x,f[x]},{x,xmin,xmax},PlotStyle->{{Thick,Black},Black },PlotRange->{xrange,yrange}],g1}]
]
Manipulate[CobwebPlot[Sqrt[3 #-1]&,\[Beta],50,{{0,4},{0,4}}],{{\[Beta],1},0,5}]