SetDirectory[NotebookDirectory[]]

U=Import["U.wl"]

Get["Figures_style.wl"]

w=Plot[U[x],{x,0,12},PlotRange->{0,1.1},PlotStyle->{Black,AbsoluteThickness[thickness]},
AxesLabel->{"x","U(x)"},
Ticks->{{0,1,{InverseFunction[U][1/GoldenRatio],"(Subscript[x, t])"},3,4,5,10},{0,0.25,0.5,{1/GoldenRatio,"1/(Subscript[η, c])"},0.75,1}},
Prolog->
{Dashed,AbsoluteThickness[thickness],
Line[{{0,1/GoldenRatio},{InverseFunction[U][1/GoldenRatio],1/GoldenRatio}}],
Line[{{InverseFunction[U][1/GoldenRatio],1/GoldenRatio},{InverseFunction[U][1/GoldenRatio],0}}],
Point[{InverseFunction[U][1/GoldenRatio],1/GoldenRatio}]
}
]

inset=LogLogPlot[U[x]-Sqrt[2]/x//Abs,{x,1,2048},PlotRange->{10^-10,All},PlotStyle->{Black,AbsoluteThickness[thickness]},
AxesLabel->{"x","|U(x)-(Sqrt[2])/x|"},Ticks->{{{1,1},{10,10},{100,"(10^2)"},{1000,"(10^3)"}},Automatic}
]

fig1=Show[w,Epilog->Inset[inset,{6,0.9},{0,0},6]]

Export["Fig1_function_U.pdf",fig1]