Plotting values excluded by FunctionRange

The following FunctionRange

(*a=((1-x) y)/(1-y z)*)
FunctionRange[{{((1 - x) y)/(1 - y z)}, 0 <= x <= 1, y > 1,
0 < z < 1}, {x}, a, Reals] // Simplify


returns

y > 1 && z < 1 && z > 0 && y z != 1 && a y z <= a && a <= y + a y z


I'm struggling to see how to illustrate/plot (in 2D or 3D) the values of x, y, z that are excluded and the resulting values for a that are excluded.

Appreciate any hints or tips.

• Maybe result = FunctionRange[{((1 - x) y)/(1 - y z), 0 <= x <= 1, y > 1, 0 < z < 1}, {x}, a, Reals]; Reduce[result, a] Aug 27, 2023 at 21:55
• Apologies for the ambiguity. I've clarified the OP. What I meant by "illustrate the values" is some type of plot, 2D or 3D Aug 27, 2023 at 22:01

a = ((1 - x) y)/(1 - y z);
RegionPlot3D[
y > 1 && z < 1 && z > 0 && y z != 1 && a y z <= a && a <= y + a y z,
{x, 0, 2}, {y, 0, 2}, {z, 0, 2},
PlotPoints -> 100]


• Thanks, that led me to, what I believe is, the end result I was after. Aug 28, 2023 at 3:33

If I understand correctly the following is the program:

1. Calculate the FunctionRange in terms of the function a for one of the variables.
2. Use Reduce to obtain a form suitable for plotting (HT: @cvgmt)
3. Replace a to return to the original variables
4. Again Reduce and then
5. Plot the complement of the constraints using RegionPlot3D (HT: @david-p-stork)
6. Some sort of color scale to show the values of a...
(*step 1*)
result =
FunctionRange[{((1 - x) y)/(1 - y z), 0 <= x <= 1, y > 1,
0 < z < 1}, {x}, a, Reals]
(*step 2, 3 and 4*)
r = Reduce[Reduce[result, a] /. a -> ((1 - x) y)/(1 - y z)]
(*step 5*)
RegionPlot3D[! (0 < z <
1 && ((1 < y < 1/z && 0 <= x <= 1) || (y > 1/z &&
0 <= x <= 1))), {x, 0, 1}, {y, 1, 10}, {z, 0, 1}, ColorFunction -> "TemperatureMap",
PlotPoints -> 50]