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I'm trying to make Mathematica giving me the inverse function of $f$ (below) when restricted to the interval $[v,1]$.

Using ConditionalExpression or Piecewise as suggested in similar cases does not work.

Result should be the red curve in a way that I can apply the numerical derivative.

v=0.75; f[x_]:=v ArcSin[x]+Cos[ArcSin[x]]; F=InverseFunction[f]; Plot[{f[x],F[x]},{x,-1.3,1.3},AxesOrigin->{0,0}] enter image description here

Thanks in advance!

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    $\begingroup$ Desired picture can be obtained by ParametricPlot[{{x, f[x]}, {f[x], x}}, {x, -1.3, 1.3}], but this in not InverseFunction. $\endgroup$ – Alx Aug 26 '19 at 3:08
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A little dowdy but will do for the moment.

v=0.75; f[x_]:=v ArcSin[x]+Cos[ArcSin[x]]; F=InverseFunction[f]; ifun=Interpolation[Table[{f[x],x},{x,v,1,0.001}]]; inverse[y_]:=Piecewise[{{ifun[y],f[1]<y<f[v]},{Indeterminate,True}}] Plot[{f[x],F[x],inverse[x]},{x,-1.3,1.3},AxesOrigin->{0,0}]

enter image description here

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