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Answer the following question based on the information given below.
For real numbers $x, y,$ let

$f(x, y) = \left\{\begin{matrix} \text{Positive square-root of}\; (x + y),\;\text{if}\; (x + y)^{0.5}\;\text{is real} \\  (x + y)^2,\;\text{otherwise} \end{matrix}\right.$
$g(x, y) = \left\{\begin{matrix} (x + y)^2,\;\text{if}\; (x + y)^{0.5}\;\text{is real} \\  –(x + y),\;\text{otherwise} \end{matrix}\right.$

Which of the following expressions yields a positive value for every pair of non-zero real number $(x, y)?$

  1. $f(x, y) – g(x, y)$
  2. $f(x, y) – (g(x, y))^2$
  3. $g(x, y) – (f(x, y))^2$
  4. $f(x, y) + g(x, y)$
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