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Ulrich Neumann
Ulrich Neumann
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Try ( Thanks @SjoerdSmit helpful comment )

FunctionDomain[cond=FunctionDomain[{f[x], g[x] >= 0 }, x, Reals] //FullSimplify
(*1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 && 
4 x^2 + (-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[2 (3 + \[Pi]) x]*)

This condition gives (Thanks @user64494 )

Reduce[N[1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 &&      4 x^2 +(-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[        2 (3 + \[Pi]) x]] && x>= 0 && x <= 10, x, Reals]
(*0 <= x <= 0.386763 || 0.590121 <= x <= 1.02279 || 
1.02333 <= x <= 1.47324 || 1.60461 <= x <= 2.03514 || 
2.06742 <= x <= 2.43253 || 2.67214 <= x <= 3.049 || 
3.11778 <= x <= 3.42438 || 3.70494 <= x <= 4.06349 || ...<= x <= 10.*)

visualization:

Plot[1, {x, 0, 10}, 
RegionFunction ->Function[{x},1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 &&4 x^2 + (-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[2 (3 + \[Pi]) x]]]

enter image description here

Try ( Thanks @SjoerdSmit helpful comment )

FunctionDomain[{f[x], g[x] >= 0 }, x, Reals] //FullSimplify
(*1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 && 
4 x^2 + (-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[2 (3 + \[Pi]) x]*)

Try ( Thanks @SjoerdSmit helpful comment )

cond=FunctionDomain[{f[x], g[x] >= 0 }, x, Reals] //FullSimplify
(*1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 && 
4 x^2 + (-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[2 (3 + \[Pi]) x]*)

This condition gives (Thanks @user64494 )

Reduce[N[1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 &&      4 x^2 +(-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[        2 (3 + \[Pi]) x]] && x>= 0 && x <= 10, x, Reals]
(*0 <= x <= 0.386763 || 0.590121 <= x <= 1.02279 || 
1.02333 <= x <= 1.47324 || 1.60461 <= x <= 2.03514 || 
2.06742 <= x <= 2.43253 || 2.67214 <= x <= 3.049 || 
3.11778 <= x <= 3.42438 || 3.70494 <= x <= 4.06349 || ...<= x <= 10.*)

visualization:

Plot[1, {x, 0, 10}, 
RegionFunction ->Function[{x},1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 &&4 x^2 + (-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[2 (3 + \[Pi]) x]]]

enter image description here

deleted 57 characters in body
Source Link
Ulrich Neumann
Ulrich Neumann
  • 56.8k
  • 2
  • 26
  • 60

Try ( Thanks @SjoerdSmit helpful comment )

FunctionDomain[{f[x], g[x] >= 0 }, x, Reals] //FullSimplify
(*6*1 + 6 x^2 + x^4 + Cos[2 \[Pi] x] (-1 2+ x^2 Cos[2 \[Pi] x] +x^4)^2 Cos[2 \[Pi] x] != -10 && 
4 x^2 + Cos[2 \[Pi] x] (- 2 x^2 Cos[2 \[Pi] x]1 + x^4x^2)^2 Cos[2 \[Pi] x] -
Cos[2>= (31 + \[Pi]) x] - 2 x^2 Cos[2 (3 + \[Pi]) x] - x^4^2 Cos[2 (3 + \[Pi]) x] >= 0*x]*)

Try

FunctionDomain[{f[x], g[x] >= 0 }, x, Reals]
(*6 x^2 + x^4 + Cos[2 \[Pi] x] - 2 x^2 Cos[2 \[Pi] x] +x^4 Cos[2 \[Pi] x] != -1 && 
4 x^2 + Cos[2 \[Pi] x] - 2 x^2 Cos[2 \[Pi] x] + x^4 Cos[2 \[Pi] x] -
Cos[2 (3 + \[Pi]) x] - 2 x^2 Cos[2 (3 + \[Pi]) x] - x^4 Cos[2 (3 + \[Pi]) x] >= 0*)

Try ( Thanks @SjoerdSmit helpful comment )

FunctionDomain[{f[x], g[x] >= 0 }, x, Reals] //FullSimplify
(*1 + 6 x^2 + x^4 + (-1 + x^2)^2 Cos[2 \[Pi] x] != 0 && 
4 x^2 + (-1 + x^2)^2 Cos[2 \[Pi] x] >= (1 + x^2)^2 Cos[2 (3 + \[Pi]) x]*)
Source Link
Ulrich Neumann
Ulrich Neumann
  • 56.8k
  • 2
  • 26
  • 60

Try

FunctionDomain[{f[x], g[x] >= 0 }, x, Reals]
(*6 x^2 + x^4 + Cos[2 \[Pi] x] - 2 x^2 Cos[2 \[Pi] x] +x^4 Cos[2 \[Pi] x] != -1 && 
4 x^2 + Cos[2 \[Pi] x] - 2 x^2 Cos[2 \[Pi] x] + x^4 Cos[2 \[Pi] x] -
Cos[2 (3 + \[Pi]) x] - 2 x^2 Cos[2 (3 + \[Pi]) x] - x^4 Cos[2 (3 + \[Pi]) x] >= 0*)