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Consider the function $f =\sqrt{xy}$? What is the largest domain for the function?

My first instinct was that x and y both be positive reals. But this is not the largest since we have not accounted for both negative ones. So I believe the answer should be $(R+, R+) \cup (R-, R-)$ is this correct?

Is my way of describing the solution using symbols correct?

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Do R+ and R- include 0? Should your domain? –  Ross Millikan Jul 5 '12 at 14:19
    
@RossMillikan yeah I forgot about 0,0. So there will be another set containing that and the union of these three sets. Correct? –  Inquest Jul 5 '12 at 14:24
    
Isn't the question rather ill-formed as it stands? Are we assuming the function is from $\mathbb{R} \times \mathbb{R}$ to $\mathbb{R}$? If so then I agree with the answers below - otherwise it might be different. –  Old John Jul 5 '12 at 14:40

2 Answers 2

up vote 2 down vote accepted

The largest domain, of course, is $$ \left\{ (x,y) \mid xy \geq 0 \right\}. $$ Therefore you are right, provided that you do not forget the coordinate axes.

Beware: your notation is rather old-fashioned. We write $\mathbb{R}^{+} \times \mathbb{R}^{+}$ instead of $(\mathbb{R}^{+},\mathbb{R}^+)$.

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Yes, you're right. From $xy\geq 0$ we have:

$x\leq 0$

$y\leq 0$

And

$x\geq 0$

$y\geq 0$

i.e. The first and third quadrants of the Cartesian plane.

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