Definition of a function: Why do we not define functions as one-to-many?

[apple2357]

I understand functions can only be one-to-one or many-to-one ( like f(x) = x^2), is there a reason why we choose to not define functions as one-to-many , for example f(x) = +-sqrt(x). Equations that fail the vertical line test are not functions?

My own thinking on this is that we don't want to get into the habit of saying an input can result in two different outputs, but i don't exactly understand why? Is it some vague idea of being well-defined? lack of ambiguity? Or we can't undo a function that is one-to-many ?

Thoughts would be helpful!

 

[JeffM]

I understand functions can only be one-to-one or many-to-one ( like f(x) = x^2), is there a reason why we choose to not define functions as one-to-many , for example f(x) = +-sqrt(x). Equations that fail the vertical line test are not functions?

My own thinking on this is that we don't want to get into the habit of saying an input can result in two different outputs, but i don't exactly understand why? Is it some vague idea of being well-defined? lack of ambiguity? Or we can't undo a function that is one-to-many ?

Thoughts would be helpful!

First, you may quite properly make anything be a definition in your own writings, provided you explicitly give that definition and then use it consistently.

Second, you may use commonly accepted definitions without necessarily defining them explicitly, but then should use them consistently.

Mathematicians commonly use a definition of "function" that insists that a function have an unambiguous resultant once every independent argument is specified. It is a common definition because an absence of ambiguity is frequently desirable.

EDIT: We can't "undo" all functions that are many to one so that is not the explanation

 

[Dr.Peterson]

I understand functions can only be one-to-one or many-to-one ( like f(x) = x^2), is there a reason why we choose to not define functions as one-to-many , for example f(x) = +-sqrt(x). Equations that fail the vertical line test are not functions?

My own thinking on this is that we don't want to get into the habit of saying an input can result in two different outputs, but i don't exactly understand why? Is it some vague idea of being well-defined? lack of ambiguity? Or we can't undo a function that is one-to-many ?

Thoughts would be helpful!

"Why" questions can be answered in many different ways!

You might just say that we have long been in the habit of expecting any expression we write to have only one value; generally that is what you expect when you do any calculation. So being able to expect that is a reasonable request.

Ultimately, it just makes things easier to talk about. It's a matter of convenience (like any definition).

However, there are situations where we do allow "functions" to be "one to many"; have you taken a course in complex variables yet, or seen the distinction between arcsin and Arcsin? In such a case, we are extending the meaning of "function" because, for one reason or another, that is now more convenient for us.

 

[apple2357]

"Why" questions can be answered in many different ways!

You might just say that we have long been in the habit of expecting any expression we write to have only one value; generally that is what you expect when you do any calculation. So being able to expect that is a reasonable request.

Ultimately, it just makes things easier to talk about. It's a matter of convenience (like any definition).

However, there are situations where we do allow "functions" to be "one to many"; have you taken a course in complex variables yet, or seen the distinction between arcsin and Arcsin? In such a case, we are extending the meaning of "function" because, for one reason or another, that is now more convenient for us.

Thanks for your thoughts! No i haven't studied complex variables and didn't realise that there was a difference between arcsin and capital a 'Arcsin'? Is there?

The way i think about it is , suppose the input is the name of a person and the output is the height, while it's reasonable to think about two different people having the same height ( many-to-one) it doesn't make sense for the same person to have two different heights ( one-to-many). Does that help a younger person understand some reasoning behind the definition?

 

[Dr.Peterson]

Thanks for your thoughts! No i haven't studied complex variables and didn't realise that there was a difference between arcsin and capital a 'Arcsin'? Is there?

The way i think about it is , suppose the input is the name of a person and the output is the height, while it's reasonable to think about two different people having the same height ( many-to-one) it doesn't make sense for the same person to have two different heights ( one-to-many). Does that help a younger person understand some reasoning behind the definition?

Did you read the links I provided?

There are many relations in everyday life that are one-to-many; but very many are many-to-one, and since those (like your example, which is one I use) are particularly easy to work with, we give those a name ("function"). I think that's really all there is to it.

An example I give is recording the temperature: at any given time your thermometer will report only one temperature, so temperature is a function of time. If you got two temperatures at once, your thermometer would be broken.

I used to use phone books as an example of a function (from names to numbers) and its inverse (caller ID). But today many people have more than one phone number, so this is not necessarily a function! It's not senseless to have a non-function; but it's a lot easier when you do.

 

[apple2357]

Did you read the links I provided?

There are many relations in everyday life that are one-to-many; but very many are many-to-one, and since those (like your example, which is one I use) are particularly easy to work with, we give those a name ("function"). I think that's really all there is to it.

An example I give is recording the temperature: at any given time your thermometer will report only one temperature, so temperature is a function of time. If you got two temperatures at once, your thermometer would be broken.

I used to use phone books as an example of a function (from names to numbers) and its inverse (caller ID). But today many people have more than one phone number, so this is not necessarily a function! It's not senseless to have a non-function; but it's a lot easier when you do.

Thanks. I missed the link . so will read up about it!

 

[apple2357]

I have been reading this book

MATHEMATICS and the IMAGINATION Edward Kasner and James Newman

And in it they say this:

For instance, the word "function" probably expresses the most important idea in the whole history of mathematics. Yet, most people hearing it would think of a "function" as meaning an evening social affair, while others, less socially minded, would think of their livers. The word "function" has at least a dozen meanings, but few people suspect the mathematical one. The mathematical meaning (which we shall elaborate upon later) is expressed most simply by a table. Such a table gives the relation between two variable quantities when the value of one variable quantity is de·termined by the value of the other.


Do we agree that 'function' is such an important idea in mathematics?

 

[Dr.Peterson]

I have been reading this book

MATHEMATICS and the IMAGINATION Edward Kasner and James Newman

And in it they say this:

For instance, the word "function" probably expresses the most important idea in the whole history of mathematics. Yet, most people hearing it would think of a "function" as meaning an evening social affair, while others, less socially minded, would think of their livers. The word "function" has at least a dozen meanings, but few people suspect the mathematical one. The mathematical meaning (which we shall elaborate upon later) is expressed most simply by a table. Such a table gives the relation between two variable quantities when the value of one variable quantity is determined by the value of the other.

Do we agree that 'function' is such an important idea in mathematics?

I'm not sure whether I'd call it THE most important idea (there are a lot of contenders), but it was essential, for example, in making calculus what it is.

You may be interested in this page I wrote long ago, and especially the page(s) it links to.

 

[apple2357]

I'm not sure whether I'd call it THE most important idea (there are a lot of contenders), but it was essential, for example, in making calculus what it is.

You may be interested in this page I wrote long ago, and especially the page(s) it links to.

Really interesting read. Thank you. There is a link that goes back nearly 20 years! How time flies!

Can you clarify what you meant by this:

It's also worth knowing that when we work with complex numbers, the
square root is NOT a function; there is no way to consistently define
the square root as having only one value.

 

[apple2357]

Really interesting read. Thank you. There is a link that goes back nearly 20 years! How time flies!

Can you clarify what you meant by this:

It's also worth knowing that when we work with complex numbers, the
square root is NOT a function; there is no way to consistently define
the square root as having only one value.

Edit...a little later

I have just seen this with one of the links you provided earlier. Is this what you meant? ( though it is not easy to follow!) from wiki

For example, let {\displaystyle f(z)={\sqrt {z}}\,}

be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of {\displaystyle z=1}

in the complex plane, and then further along curves starting at {\displaystyle z=1}

, so that the values along a given curve vary continuously from {\displaystyle {\sqrt {1}}=1}

. Extending to negative real numbers, one gets two opposite values of the square root such as {\displaystyle {\sqrt {-1}}=\pm i}

, depending on whether the domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for [FONT=&quot]n[/FONT]th roots, logarithms and inverse trigonometric functions.

 

[Dr.Peterson]

I have just seen this with one of the links you provided earlier. Is this what you meant? ( though it is not easy to follow!) from wiki

For example, let

be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of

in the complex plane, and then further along curves starting at

, so that the values along a given curve vary continuously from

. Extending to negative real numbers, one gets two opposite values of the square root such as

, depending on whether the domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for nth roots, logarithms and inverse trigonometric functions.

Yes, this is discussing the same point.

Put simply, there is no way to define a "principal" square root, as we do with real numbers, so that everything works consistently; so we can't turn the square root into a well-behaved function over the complex numbers. You don't need to understand all the complex analysis; the simplest difficulty to see in choosing a principal value is that if you simply define \(\displaystyle \sqrt{-1} = i\), then the usual product rule fails: \(\displaystyle \sqrt{-1}\sqrt{-1} = i^2 = -1\), but \(\displaystyle \sqrt{-1\cdot -1} = \sqrt{1} = +1\). No choice you can make will allow square roots to behave the way they do for positive real numbers. So you are forced to deal with the radical as a multivalued "function".

 

[apple2357]

Yes, this is discussing the same point.

Put simply, there is no way to define a "principal" square root, as we do with real numbers, so that everything works consistently; so we can't turn the square root into a well-behaved function over the complex numbers. You don't need to understand all the complex analysis; the simplest difficulty to see in choosing a principal value is that if you simply define \(\displaystyle \sqrt{-1} = i\), then the usual product rule fails: \(\displaystyle \sqrt{-1}\sqrt{-1} = i^2 = -1\), but \(\displaystyle \sqrt{-1\cdot -1} = \sqrt{1} = +1\). No choice you can make will allow square roots to behave the way they do for positive real numbers. So you are forced to deal with the radical as a multivalued "function".

Good explanation. Thank you

 

[pka]

Put simply, there is no way to define a "principal" square root, as we do with real numbers, so that everything works consistently; so we can't turn the square root into a well-behaved function over the complex numbers. You don't need to understand all the complex analysis; the simplest difficulty to see in choosing a principal value is that if you simply define \(\displaystyle \sqrt{-1} = i\), then the usual product rule fails: \(\displaystyle \sqrt{-1}\sqrt{-1} = i^2 = -1\), but \(\displaystyle \sqrt{-1\cdot -1} = \sqrt{1} = +1\). No choice you can make will allow square roots to behave the way they do for positive real numbers. So you are forced to deal with the radical as a multivalued "function".

Actually I take exception with that. Not to be argumentative but to point out another way of doing complex functions.
When non-standard analysis was a hot research topic the idea of enlargements of a systems were popular. Some of us found it was possible to define the number \(\displaystyle \bf{i}\) having the property that it is the solution to the equation of \(\displaystyle x^2+1=0\). But to do this it requires restrictions such that we allow the use of the radical symbol \(\displaystyle \sqrt a\) only if \(\displaystyle a\in\mathbb{R}~\&~a\ge 0\).

Now to my complaint. By way of example: suppose that \(\displaystyle z=\sqrt3-\bf{i}~.\)
Now what are the five fifth root of \(\displaystyle z~?\) Well written in so-called 'polar-form' \(\displaystyle z=2\exp\left(\dfrac{-\pi\bf{i}}{6}\right)\)
Let \(\displaystyle \rho=\sqrt[5]{2}\exp\left(\dfrac{-\bf{i}\pi}{30}\right)~\&~\zeta=\exp\left(\dfrac{2 \bf{i} \pi}{5}\right)\) The five roots are \(\displaystyle \rho\cdot\zeta^k.~k=0,1,2,3,4\)
We would say that \(\displaystyle \rho\) is the principal root.

 

[Dr.Peterson]

Put simply, there is no way to define a "principal" square root, as we do with real numbers, so that everything works consistently; so we can't turn the square root into a well-behaved function over the complex numbers.

Actually I take exception with that. Not to be argumentative but to point out another way of doing complex functions.

Now to my complaint. By way of example: suppose that \(\displaystyle z=\sqrt3-\bf{i}~.\)
Now what are the five fifth root of \(\displaystyle z~?\) Well written in so-called 'polar-form' \(\displaystyle z=2\exp\left(\dfrac{-\pi\bf{i}}{6}\right)\)
Let \(\displaystyle \rho=\sqrt[5]{2}\exp\left(\dfrac{-\bf{i}\pi}{30}\right)~\&~\zeta=\exp\left(\dfrac{2 \bf{i} \pi}{5}\right)\) The five roots are \(\displaystyle \rho\cdot\zeta^k.~k=0,1,2,3,4\)
We would say that \(\displaystyle \rho\) is the principal root.

My statement is, of course, made within a certain context, and depends on this phrase (and on how you interpret it). Those are intentional weasel-words to avoid having to write a dissertation to clarify exactly what I mean.

Of course you can define something to call a principal root. It just isn't going to be what we are used to at the elementary level, which is the only point I was trying to make.