Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2012-11-03 10:29:42

zetafunc.
Guest

'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 2012-11-03 11:30:00

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,673

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

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#3 2012-11-03 11:32:20

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

Is that a proper function?

Offline

#4 2012-11-03 11:33:20

zetafunc.
Guest

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

Why isn't it a function?

#5 2012-11-03 11:34:37

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,673

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.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#6 2012-11-03 11:35:21

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

Y can only have one value.

Offline

#7 2012-11-03 11:40:41

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 2012-11-03 11:42:35

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

No, it is a function.

Offline

#9 2012-11-03 11:43:05

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,673

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...


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#10 2012-11-03 11:45:02

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

y^2 = 4ax would be a parabola.

Offline

#11 2012-11-03 11:46:13

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 2012-11-03 11:48:53

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

Is it not the form for a parabola?

Offline

#13 2012-11-03 11:51:22

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 2012-11-03 11:53:15

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,673

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

y^2=4ax is not a function.

I will try plotting it like that.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#15 2012-11-03 11:53:16

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

Offline

#16 2012-11-03 11:58:18

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 2012-11-03 11:59:46

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

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

Offline

#18 2012-11-03 12:01:25

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,673

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.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#19 2012-11-03 12:02:31

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 2012-11-03 12:03:13

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

In those terms, yes.

Offline

#21 2012-11-03 12:07:27

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,673

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.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#22 2012-11-03 12:12:49

ShivamS
Member
Registered: 2011-02-07
Posts: 3,630

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

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

Offline

#23 2012-11-03 12:21:43

zetafunc.
Guest

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

I am not taking a maths course.

#24 2012-11-03 20:44:41

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,542

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

Online

Board footer

Powered by FluxBB