Instructions

Directions for the next 2 questions:

For real numbers x, y, let

f(x, y) = Positive square-root of (x + y), if $$(x + y)^{0.5}$$ is real
f(x, y) = $$(x + y)^2$$; otherwise

g(x, y) = $$(x + y)^2$$, if $$\sqrt{(x + y)}$$ is real
g(x, y) = $$- (x + y)$$ otherwise

Question 113

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

Solution

f(x,y) is always non-negative because $$(x+y)^2$$ is always positive

g(x, y) = $$(x + y)^2$$, if $$\sqrt{(x + y)}$$ is real = Always positive
g(x, y) = $$- (x + y)$$ otherwise = Always positive because this happen when (x+y)<0 and -(x+y) is always greater than zero.

f(x,y)+g(x,y) = Always positive


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