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

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: $$x >0,~y\geq 1,~0 Is there a way (using mathematica), to find the domain for the $$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$$?

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
`