Finding the joint domain for a couple of functions of three variables

Question

I have two functions $f_{1}(x,y,z)$ and $f_{2}(x,y,z)$ defined respectively as

  Sqrt[(2 x)/y + y^2/x + (2 y (-0.04258557948619213` - z)^2)/x] Sqrt[
   x - 17.37121059964452` z + (2 x z^2)/y + (y^2 z^2)/x + (
    y (1 + z^4))/(2 x)]; 

and

 (Sqrt[(y^3 + 2 y^2 (-1 + z)^2)/(x y)] Sqrt[(
 x (1 + 2 y (-1 + z)^2 + (-2 + z)^2 z^2))/y])/Sqrt[2]; 

subject to the restrictions: $x >0,~y\geq 1,~0<z<1.$ Is there a way (using mathematica), to find the domain for the $x,y,z$ variables in order to satsfy $f_{1}(x,y,z)<2$ and $f_{2}(x,y,z)<2$ simultaneously? in other words, how can I find (using mathematica) the allowed values for $x,y,z$ in order to satsfy $f_{1}(x,y,z)<2$ and $f_{2}(x,y,z)<2$?

Answer 1

Clear["Global`*"]

$Version

(* "12.2.0 for Mac OS X x86 (64-bit) (December 12, 2020)" *)

$Assumptions = x > 0 && y >= 1 && 0 < z < 1;

f1[x_, y_, z_] = 
  Sqrt[(2 x)/y + y^2/x + (2 y (-0.04258557948619213` - z)^2)/x] Sqrt[
      x - 17.37121059964452` z + (2 x z^2)/y + (y^2 z^2)/
        x + (y (1 + z^4))/(2 x)] // Rationalize[#, 0] & // Simplify;

f2[x_, y_, 
   z_] = (Sqrt[(y^3 + 
          2 y^2 (-1 + z)^2)/(x y)] Sqrt[(x (1 + 
            2 y (-1 + z)^2 + (-2 + z)^2 z^2))/y])/Sqrt[2] // Simplify;

RegionPlot3D indicates that a description of the region would be extremely complicated.

RegionPlot3D[
  f1[x, y, z] < 2 && f2[x, y, z] < 2 && $Assumptions, {x, 0, 2}, {y, 1, 
   4.5}, {z, 0, 1}, PlotPoints -> 200, MaxRecursion -> 5, 
  WorkingPrecision -> 30,
  AxesLabel -> Automatic] // Quiet