Find Domain of f(x,y)=sqrt(5xy-2y^2)

[KUphysstudent]

Homework Statement

Find the domain of f(x,y)=sqrt(5xy-2y^2)

Homework Equations

The Attempt at a Solution

As far as I understand it's all about solving inequalities when you have this kind of problem.
The problem doesn't state it is in ℝ² but I'm pretty sure it is.
So no negative numbers under the root.
Therefore 5xy ≥ -2y^2
x ≥ -2y/5 or -y ≥ 5x/2
But how exactly is this useful information?
So I looked up the topic on Youtube, Multivariable Calculus: Finding and Sketching the Domain.
It sounds perfect and the guy has two examples and one even has a root in it.
Sadly I still don't get it :/

 

[mfb]

Therefore 5xy ≥ -2y^2
x ≥ -2y/5 or -y ≥ 5x/2

Be careful with divisions in inequalities. The sign matters.

But how exactly is this useful information?

That tells you where the function has a chance to be defined.

 

[KUphysstudent]

Yea signs in inequalities shift direction if you multiply or divide with a negative number, I knew there was something so I googled it before hand.
If x ≥ -2y/5 or -y ≥ 5x/2 tells me where the function is defined, well then that is its domain. What format is commonly used to denote it?
I think we use {(x,y) ∈ ℝ² ; 5xy ≥ -2y²}
so (x,y) are a plane in ℝ where 5xy ≥ -2y². Does that make sense?

Edit: My quoting skills lack as much as my math.

 

[Mark44]

The problem doesn't state it is in ℝ² but I'm pretty sure it is.

Yes. The function takes as input pairs of real numbers, and outputs a real number, so it's a map from some subset of $\mathbb R^2$ to $\mathbb R$.

Yea signs in inequalities shift direction if you multiply or divide with a negative number, I knew there was something so I googled it before hand.
If x ≥ -2y/5 or -y ≥ 5x/2 tells me where the function is defined, well then that is its domain.

mfb's point was that this is not valid. When you divide by a variable, you don't know what its sign is, so you can't tell whether the inequality switches direction or not.

The square root is defined, in this context, only when $5xy - 2y^2 \ge 0$. Factor that expression to see where the product of the two factors will be nonnegative.

What format is commonly used to denote it?
I think we use {(x,y) ∈ ℝ² ; 5xy ≥ -2y²}
so (x,y) are a plane in ℝ where 5xy ≥ -2y². Does that make sense?

No, in part because the domain is not a plane. The domain of this function is some subset of the real plane.

 

[LCKurtz]

I second @Mark44 suggestion, factor $5xy-2y^2 = y(5x-2y)$. You want the region where both factors are positive or both are negative, or zero. I like to use plots like this, where I have shaded where each factor is positive. Here you should be able to describe where both factors are positive or neither are.

 

[KUphysstudent]

I second @Mark44 suggestion, factor $5xy-2y^2 = y(5x-2y)$. You want the region where both factors are positive or both are negative, or zero. I like to use plots like this, where I have shaded where each factor is positive. Here you should be able to describe where both factors are positive or neither are.
View attachment 231726

No matter how I solve the inequality there will be a negative sign, so I get the complex part of the function if I try would sketch it. So wouldn't sketching it and taking everything the outside the complex part be a thing you could do? Just a thought I'm going to look at the factoring part :)

 

[KUphysstudent]

No matter how I solve the inequality there will be a negative sign, so I get the complex part of the function if I try would sketch it. So wouldn't sketching it and taking everything the outside the complex part be a thing you could do? Just a thought I'm going to look at the factoring part :)

I mean, don't we already assume that the number is positive since it under the radical? meaning you can basically throw the negative symbol away to begin with so we have 5xy≥2y²

 

[mfb]

No matter how I solve the inequality there will be a negative sign, so I get the complex part of the function if I try would sketch it. So wouldn't sketching it and taking everything the outside the complex part be a thing you could do? Just a thought I'm going to look at the factoring part :)

Huh? You can consider two cases separately, e.g. y<0 and y>0.

I mean, don't we already assume that the number is positive since it under the radical?

The overall expression yes, but not the individual variables.

 

[Mark44]

I mean, don't we already assume that the number is positive since it under the radical? meaning you can basically throw the negative symbol away to begin with so we have 5xy≥2y²

You're not "throwing away" the minus sign. It's true that the inequality $5xy - 2y^2 \ge 0$ is equivalent to $5xy \ge 2y^2$, but this step doesn't do you any good. In your work in post #1, you then divided both sides by y, which is really an invalid step, since y could be negative, positive, of zero, so you can't tell which inequality symbol should be used.

Again, what you need to do is to factor $5xy - 2y^2$ into $y(5x - 2y)$, and investigate the x and y intervals that make the product nonnegative.

 

[KUphysstudent]

Huh? You can consider two cases separately, e.g. y<0 and y>0.The overall expression yes, but not the individual variables.

Yea okay that makes sense.

Okay I'm still very confused, there must be something fundamentally wrong with my results.
So if I start out with 5xy-2y² ≥ 0 → 5xy ≤ -2y² → 5x ≤ -2y → -5x/2 ≥ y , so this is wrong but what is the problem? I cannot remember ever learning about inequalities in school. And reading online tells me it's just algebra and flipping the sign whenever you multiply or divide by something negative.
Holy Moly I just realized that I for some reason keep moving -2y² like I have to add it... oh god. -2y²*-2y² = 0 *(-2y²). I don't know what is wrong with me hehe.
ok so with that out of the way I get to add and not multiply. so we have 5xy-2y² ≥ 0 → 5xy ≥ 2y² → 5x ≥ 2y → 5x/2 ≥ y.
I cannot believe I got into uni…
y=5x/2 gives me a line, like the one mark has in his figure. and if I only want the positive or negative x,y-values at once then I can see the domain. But how exactly did you determine that it was only positives or negatives?

Edit:
Holy Moly I just realized that I for some reason keep moving -2y² like I have to add it
Should have been:
Holy Moly I just realized that I for some reason keep moving -2y² like I have to multiply it

 

[Mark44]

Yea okay that makes sense.

Okay I'm still very confused, there must be something fundamentally wrong with my results.
So if I start out with 5xy-2y² ≥ 0 → 5xy ≤ -2y²

No.
$5xy - 2y^2 \ge 0 \Leftrightarrow 5xy \ge 2y^2$
I added $2y^2$ to both sides of the inequality.
You're playing fast and loose with the signs and your algebra.

→ 5x ≤ -2y → -5x/2 ≥ y , so this is wrong but what is the problem? I cannot remember ever learning about inequalities in school. And reading online tells me it's just algebra and flipping the sign whenever you multiply or divide by something negative.
Holy Moly I just realized that I for some reason keep moving -2y² like I have to add it... oh god. -2y²*-2y² = 0 *(-2y²). I don't know what is wrong with me hehe.
ok so with that out of the way I get to add and not multiply. so we have 5xy-2y² ≥ 0 → 5xy ≥ 2y² → 5x ≥ 2y → 5x/2 ≥ y.
I cannot believe I got into uni…
y=5x/2 gives me a line, like the one mark has in his figure. and if I only want the positive or negative x,y-values at once then I can see the domain. But how exactly did you determine that it was only positives or negatives?

 

[mfb]

5xy ≤ -2y² → 5x ≤ -2y

Consider x=1, y=-1. Clearly 5xy=-5 is smaller than y²=1, but 5x=5 is not smaller than -2y=-2.
If you divide an inequality by a negative number (here: y) you have to swap the direction of the inequality.

$a<b \Leftrightarrow -a > -b$

 

[LCKurtz]

@KUphysstudent Did you ever understand which regions answer your question by looking at the graphic I gave you in post #5?

 

[KUphysstudent]

No.
$5xy - 2y^2 \ge 0 \Leftrightarrow 5xy \ge 2y^2$
I added $2y^2$ to both sides of the inequality.
You're playing fast and loose with the signs and your algebra.

Yea I get the algebra part, I just kept making the same mistake.

@KUphysstudent Did you ever understand which regions answer your question by looking at the graphic I gave you in post #5?

I think I do, the bold line is the one you get from the inequality and the green area is to depict the positive x- and y-values. so you put in an x-value and get a y-value and in the first quadrant it is everything below the line. In the third quadrant it is everything above the line.
I cannot explain this very well, but here is how I think of it. if you have 5x/2=y and set x=2 you will get y=5, so if you instead of using 5x/2=y but then use 2y/5=x and plugin y-values you get all the points below the graph.
I think but it is likely wrong :)

 

[KUphysstudent]

Consider x=1, y=-1. Clearly 5xy=-5 is smaller than y²=1, but 5x=5 is not smaller than -2y=-2.
If you divide an inequality by a negative number (here: y) you have to swap the direction of the inequality.

$a<b \Leftrightarrow -a > -b$

That makes sense. so you can basically check every time you have manipulated the equation to see if the equality still is correct, if not you did it wrong.

 

[LCKurtz]

@KUphysstudent Did you ever understand which regions answer your question by looking at the graphic I gave you in post #5?

I think I do, the bold line is the one you get from the inequality and the green area is to depict the positive x- and y-values. so you put in an x-value and get a y-value and in the first quadrant it is everything below the line. In the third quadrant it is everything above the line.
I cannot explain this very well, but here is how I think of it. if you have 5x/2=y and set x=2 you will get y=5, so if you instead of using 5x/2=y but then use 2y/5=x and plugin y-values you get all the points below the graph.
I think but it is likely wrong :)

Yes, but it is easier just to say the green area and the white area. That's where the factors are either both positive or both negative. If you have trouble figuring out the areas, try this. The points on the line $5x-2y = 0$ are the the only places in the plane where $5x-2y=0$. Everyplace else it is either $>0$ or $<0$. You can tell which side is which by simply plugging in a point not on the line. For example, at $(1,0)$ you get $5\cdot 1 -2\cdot 0 = 5 > 0$ so that is the side where $5x-2y>0$, which I colored yellow. It's easy to visualize the regions that way.