(*Plot the function*)
R1[t_]:= 800 t^2;
R2[t_]:= 9 - 2 (3-20t)^2;
Show[Plot[R1[t],{t,0,3/40}],Plot[R2[t],{t,3/40,3/20}], PlotRange->{{0,0.15},{0,9}},
	AxesLabel->{"t","y"},PlotLabel->"y = R(t)",AxesOrigin->{-0.006,-0.35}]
(*Plot the inverse function the slow way*)
Solve[R1[t]==y,t]
Q1[y_]:= t/.Solve[R1[t]==y,t][[2]];
Q1[y]
Solve[R2[t]==y,t]
Q2[y_]:= t/.Solve[R2[t]==y,t][[1]];
Q2[y]
Show[Plot[Q1[y],{y,0,9/2}],Plot[Q2[y],{y,9/2,9}], PlotRange->{{0,9},{0,0.15}},
	AxesLabel->{"y","t"},PlotLabel->"t = Q(y)", AxesOrigin->{-0.35,-0.006}]
(*Plot the inverse function the fast way*)
Show[ParametricPlot[{R1[t],t},{t,0,3/40}],ParametricPlot[{R2[t],t},{t,3/40,3/20}], PlotRange->{{0,9},{0,0.15}},
	AxesLabel->{"y","t"},PlotLabel->"t = Q(y)", AxesOrigin->{-0.35,-0.006}]