plotting - How can I transpose x and y axis on a Plot?

This question has appeared in various forms online, but I have not yet seen a complete answer, so I am posting it here.

More specifically, suppose I have a function

F: X->Y

that is not one-to-one. Mathematica can easily plot this function as follows:

Plot[F[x], {x, 0, 30}, PlotRange -> {{0, 30}, {0, 1}}]

This produces a graph that passes the vertical line test, but does not pass the horizontal line test, because F is not one-to-one. My question is, how do I get a plot of the inverse relation (which is not a function) for all Y?

Edit: I am adding the function F for clarity. I originally omitted it because I figured a generic solution would solve it.

F[x_] := (1000 * x) / 24279 * Sqrt[-1 + x^(2/7)]

Edit 2: I am adding the graph (that I am trying to graph the inverse relation of) I produced using Plot for further clarity.

To be clear, this image is produced by the following three commands:

F[x_] := (1000 * x) / (24279 * Sqrt[-1 + x^(2/7)])
myplot = Plot[F[x], {x,0,30}, PlotRange -> {{0,30}, {0,1}}]
Export["foo.png", myplot]

Answer

Update:

Using the example function provided in op's update:

 ff[x_] := (1000*x)/(24279*Sqrt[-1 + x^(2/7)]);
 prmtrcplt1 = ParametricPlot[{x, ff[x]}, {x, 0, 30},

 PlotRange -> {{0, 30}, {0, 1}}, ImageSize -> 300, AspectRatio -> 1];
 prmtrcplt2 = ParametricPlot[{ff[x], x}, {x, 0, 30},
 PlotRange -> Reverse[PlotRange[prmtrcplt1]], ImageSize -> 300,  AspectRatio -> 1];
 Row[{prmtrcplt1, prmtrcplt2}, Spacer[5]]

 plt = Plot[ff[x], {x, 0, 30}, PlotRange -> {{0, 30}, {0, 1}},ImageSize -> 300,
   AspectRatio -> 1];
 ref1 = MapAt[GeometricTransformation[#, ReflectionTransform[{-1, 1}]] &, plt, {1}];
 ref2 = plt /. line_Line :> GeometricTransformation[line, ReflectionTransform[{-1, 1}]];

 Row[{plt, Graphics[ref1[[1]], PlotRange -> Reverse@PlotRange[ref1], ref1[[2]]],
  Graphics[ref2[[1]], PlotRange -> Reverse@PlotRange[ref2], ref2[[2]]]}, Spacer[5]]

---

original post

ParametricPlot (as suggested by whuber)

 prmtrcplt1 = ParametricPlot[{x, x Sin[2 x]}, {x, -Pi, Pi},
   PlotRange -> {{-Pi, Pi}, {-3, 3}}, ImageSize -> 300];
 prmtrcplt2 = ParametricPlot[{x Sin[2 x], x}, {x, -Pi, Pi},

  PlotRange -> {{-Pi, Pi}, {-3, 3}}, ImageSize -> 300];
 Row[{prmtrcplt1, prmtrcplt2}, Spacer[5]]

Post-process using ReflectionTransform

  reflected = plt /. line_Line :> {Red,  
     GeometricTransformation[line, ReflectionTransform[{-1, 1}]]};
   Row[{plt, reflected, 
   Show[plt, Plot[x, {x, -Pi, Pi}, PlotStyle -> Black], reflected,
   PlotRange -> All, ImageSize -> 300]}, Spacer[5]]

Variations:

 reflected2 = MapAt[GeometricTransformation[#, ReflectionMatrix[{-1, 1}]] &, plt, {1}];
 reflected3 = MapAt[GeometricTransformation[#, ReflectionTransform[{-1, 1}]] &,plt, {1}]