ff[x_]=Exp[-(x^2)]

Manipulate[ParametricPlot[{ff[ξ] s+ξ,s},{s,0,10},PlotRange->{{-4,4},{0,6}},AspectRatio->Automatic],{ξ,-4,4}]

Show[Table[ParametricPlot[{ff[ξ] s+ξ,s},{s,0,10},PlotRange->{{-4,4},{0,2}},AspectRatio->Automatic],{ξ,-4,4,0.1}]]

ParametricPlot3D[{ff[ξ] s+ξ,s,ff[ξ]},{s,0,10},{ξ,-4,4},PlotRange->{{-4,4},{0,1},{0,1}}]

Manipulate[ParametricPlot[{ff[ξ] t0+ξ,ff[ξ]},{ξ,-4,4},PlotRange->{{-4,4},{-0.1,1.2}}],{t0,0,2}]

ff[x_]=ArcTan[x]

Manipulate[ParametricPlot[{ff[ξ] s+ξ,s},{s,0,10},PlotRange->{{-4,4},{0,6}},AspectRatio->Automatic],{ξ,-4,4}]

Show[
Table[ParametricPlot[{ff[ξ] s+ξ,s},{s,0,10},PlotRange->{{-4,4},{0,6}},AspectRatio->Automatic],{ξ,-4,4,0.1}]
]

ParametricPlot[Evaluate[Table[{ff[ξ] s+ξ,s},{ξ,-4,4,0.1}]],{s,0,10},PlotRange->{{-4,4},{0,6}},AspectRatio->Automatic]

ff[x]

ParametricPlot3D[{ff[ξ] s+ξ,s,ff[ξ]},{s,0,10},{ξ,-4,4},PlotRange->{{-4,4},{0,10},{-Pi/2,Pi/2}},ImageSize->600]

Manipulate[ParametricPlot[{ff[ξ] t0+ξ,ff[ξ]},{ξ,-4,4},PlotRange->{{-4,4},{-0.1,1.2}}],{t0,0,2}]

Solve[{x==ff[ξ] s+ξ,y==s},{s,ξ}]

Plot[If[x<0,x,Sin[x]],{x,-Pi,2 Pi}]

Plot3D[If[x<0,x,Sin[x]],{x,-Pi,2 Pi},{y,-1,1},Exclusions->None]