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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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