FunctionRange giving "false"

Question

I was trying to find the range of a function I obtained  in some context. The code is as follows
FunctionRange[{1/2 (1 - Sqrt[-1 + 4 Abs[x]] Sqrt[-1 + 4 Abs[z]]),  0 < Abs[x] < 1/4 && 0 < Abs[z] < 1/4}, {Abs[x], Abs[z]}, y] 
The above code when executed gives False. What is the meaning of false and how to find the range of function in this case?

Nasser · Answer

What is the meaning of false and how to find the range of function in
this case?

The default for domain is Reals

When it returns False it means there is no range in the reals.
FunctionRange[{1/2 (1 - Sqrt[-1 + 4 Abs[x]] Sqrt[-1 + 4 Abs[z]]), 
  0 < Abs[x] < 1/4 && 0 < Abs[z] < 1/4}, {Abs[x], 
  Abs[z]}, y]

(*False*)

Change to complex
FunctionRange[{1/2 (1 - Sqrt[-1 + 4 Abs[x]] Sqrt[-1 + 4 Abs[z]]), 
  0 < Abs[x] < 1/4 && 0 < Abs[z] < 1/4}, {Abs[x], 
  Abs[z]}, y, Complexes]

FunctionRange giving "false"

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