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#1 2012-11-04 09:29:42

zetafunc.
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'Gap' in Graph of x^y = y^x

I was looking at the graph of x^y = y^x today and I was curious about the point where the two lines meet (you'll know what I mean if you look at the graph). Initially I guessed that that might be the point (x,y) = (e,e) but according to W|A it seems like a range of values for which this function is discontinuous. What is this range of values, and why is this the case?

#2 2012-11-04 10:30:00

anonimnystefy
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Re: 'Gap' in Graph of x^y = y^x

That isn't even a function, let alone a continuous one...


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#3 2012-11-04 10:32:20

ShivamS
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Re: 'Gap' in Graph of x^y = y^x

Is that a proper function?


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#4 2012-11-04 10:33:20

zetafunc.
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Re: 'Gap' in Graph of x^y = y^x

Why isn't it a function?

#5 2012-11-04 10:34:37

anonimnystefy
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Re: 'Gap' in Graph of x^y = y^x

A function has exactly one value of y for any given x in the domain of the function.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#6 2012-11-04 10:35:21

ShivamS
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Re: 'Gap' in Graph of x^y = y^x

Y can only have one value.


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#7 2012-11-04 10:40:41

zetafunc.
Guest

Re: 'Gap' in Graph of x^y = y^x

Oh I see, so y^2 = 4ax are also not functions, then? I did think that functions had to be one-to-one, but that sort of graph (or something like (x^2 - 2)^2 + (y^2 - 2)^2 = 2) is many-to-one.

So, can you tell me what it happening in the gap?

#8 2012-11-04 10:42:35

ShivamS
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Re: 'Gap' in Graph of x^y = y^x

No, it is a function.


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#9 2012-11-04 10:43:05

anonimnystefy
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Re: 'Gap' in Graph of x^y = y^x

It doesn't have to be one-to-one.

W|A doesn't show any gap to me...


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#10 2012-11-04 10:45:02

ShivamS
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Re: 'Gap' in Graph of x^y = y^x

y^2 = 4ax would be a parabola.


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#11 2012-11-04 10:46:13

zetafunc.
Guest

Re: 'Gap' in Graph of x^y = y^x

Shivamcoder3013 wrote:

No, it is a function.

Why?

Plot it with W|A, but tell it to show the graph in the range x = 2 to x = 3, or x = 2.5 to x = 3.

#12 2012-11-04 10:48:53

ShivamS
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Re: 'Gap' in Graph of x^y = y^x

Is it not the form for a parabola?


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#13 2012-11-04 10:51:22

zetafunc.
Guest

Re: 'Gap' in Graph of x^y = y^x

Yes, it is a parabola, but I'm not understanding why that would be a function (you can have two values for y given a value for x), but x^y = y^x is not a function, even though there are two values for y you can have given an x-value.

#14 2012-11-04 10:53:15

anonimnystefy
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Re: 'Gap' in Graph of x^y = y^x

y^2=4ax is not a function.

I will try plotting it like that.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#15 2012-11-04 10:53:16

ShivamS
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Re: 'Gap' in Graph of x^y = y^x

Well, you would consider it as a function. For example:
y2 = 2x to y2 = 4ax form, y2 = 4 (2/4) x


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#16 2012-11-04 10:58:18

zetafunc.
Guest

Re: 'Gap' in Graph of x^y = y^x

Shivamcoder3013 wrote:

Well, you would consider it as a function. For example:
y2 = 2x to y2 = 4ax form, y2 = 4 (2/4) x

I know that y^2 = 4ax is the general form of a parabola, I was just asking why it was not a function...

So, what is an example of a function that is not one-to-one?

#17 2012-11-04 10:59:46

ShivamS
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Re: 'Gap' in Graph of x^y = y^x

(0,1), (1,0), (2,0), (3,2)


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#18 2012-11-04 11:01:25

anonimnystefy
Real Member

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Re: 'Gap' in Graph of x^y = y^x

y=x^2. At every value of x has only one value of y, but -1 and 1 yield the same value of y.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#19 2012-11-04 11:02:31

zetafunc.
Guest

Re: 'Gap' in Graph of x^y = y^x

Oh I see, so the distinction is that it can be many to one, but not one to many.

#20 2012-11-04 11:03:13

ShivamS
Super Member

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Re: 'Gap' in Graph of x^y = y^x

In those terms, yes.


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#21 2012-11-04 11:07:27

anonimnystefy
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Re: 'Gap' in Graph of x^y = y^x

zetafunc. wrote:

Oh I see, so the distinction is that it can be many to one, but not one to many.

Yes. Another thing that must be satisfied by a function is that it must have a value at every element of its domain.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#22 2012-11-04 11:12:49

ShivamS
Super Member

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Re: 'Gap' in Graph of x^y = y^x

Zeta, if you don't mind me questioning, what course are you taking?


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#23 2012-11-04 11:21:43

zetafunc.
Guest

Re: 'Gap' in Graph of x^y = y^x

I am not taking a maths course.

#24 2012-11-04 19:44:41

bob bundy
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Re: 'Gap' in Graph of x^y = y^x

hi zetafunc

For a good explanation of what makes a function see

http://www.mathsisfun.com/sets/function.html

The early stuff on that page you will find trivial, so skip down to about half way.

The problem with using a graph plotter is caused by the way they work.

The domain for x is set and the lowest value calculated using the formula, and a point plotted.

Then x is given a small increment, a new point calculated, and a line is drawn from the first point.

This carries on for a sequence of points throughout the domain.

This is OK for many functions where the curve is continuous throughtout the domain.

If there is an asymptote the process can lead to incorrect plots.

For example, many plotters doing  y = tanx try to connect a point at x just less than pi/2 to a point just over pi/2 and end up with a nearly vertical line at the discontinuity.

There are fixes that improve the plotting  to avoid this.

http://www.mathsisfun.com/data/function-grapher.php

shows y = tanx correctly.

As your equation is not a function (more than one y for each x) you cannot use that plotter .

But MIF has also made a plotter that will handle it at

http://www.mathsisfun.com/data/grapher-equation.html

This shows no gap, even when you zoom in at (e,e).

(e,e) certainly is a valid point for the equation.  (Any point on y=x is valid)

But it does show y = 0 as part of the curve and I'm not sure that is OK.

For me W/A looks ok and doesn't have y = 0.

Moral of this:  Beware when you let a computer do your maths.  Not all results are correct.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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