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?

Answer 1

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]