Domain of $\sqrt[4]{1-xy}$

Question

I'm having hard time understanding the graph of the domain (within $\mathbb{R}^2$) of the function

$$\sqrt[4]{1-xy}$$

So that function exists under the condition $1-xy\ge0$ which leads to $xy\le1$. So the domain of the function is

$$Dom(f)=\{\forall(x,y)\in \mathbb{R}^2 | xy\le1\}$$

Which plots to

I know how to graph a domain of a function, I'm having hard time understanding why the graph of $xy\le1$ is like the picture above.

user · Accepted Answer · 2019-11-16T10:50:09.377

4

The points $xy=1\iff y=\frac1x$ are the boundaries for the region.

Then let consider the cases

for $x,y >0 $

$$xy< 1 \iff y< \frac1x$$

for $x>0$ and $y <0 $

$$xy<0< 1 $$

for $x<0$ and $y >0 $

$$xy<0< 1 $$

for $x,y <0 $

$$xy< 1 \iff y> \frac1x$$

to see that the region corresponds to your plot.

edited Nov 16 '19 at 10:50

answered Nov 16 '19 at 10:13

user

154,566

score 2 · Answer 2 · edited Nov 16 '19 at 10:31

If $x\leqslant 0\leqslant y$ or $y\leqslant 0\leqslant x$ then $xy\leqslant0$. So, that region must contain the second and the fourth quadrants.

Now, if $x,y\geqslant0$ and $xy\leqslant1$, then $y\leqslant\frac1x$. So, the points of the first quadrant that belong to your region are those which are below the graph of $x\mapsto\frac1x$.

And you can apply a similar argument to the third quadrant.

Domain of $\sqrt[4]{1-xy}$

2 Answers2