region[z_]:=Norm[z-1]<1;f[z_]:=(3+4I)*z-2+I;g=InverseFunction[f];RegionPlot[region[g[x+I y]],{x,-4,6},{y,0,10}]

region[z_]:=Im[z]+2Re[z]>1;f[z_]:=(3+4I)*z-2+I;g=InverseFunction[f];RegionPlot[region[g[x+I y]],{x,-10,16},{y,0,10}]

region[z_]:=Im[z]>1;f[z_]:=(3+4I)*z-2+I;g=InverseFunction[f];RegionPlot[region[g[x+I y]],{x,-10,10},{y,-10,10}]