Confused about the definition of a function

[Fritz]

To help me understand the requirements for a function to be valid, can you guys tell me if:

x^2 = y determines a function,

if y^2 = x determines a function,

if y = x^2 determines a function,

and if x = y^2 determines a function.

 

[matt grime]

Part of the definition of a function is the domain (and codomain, the image set)

Are we to presume you mean from R to R?

nb those are equalities, you need to say what x and y are.

 

[Fritz]

I don't know what you mean. The x in x^2, for example, has the value x.

In my book it says that y^2 = x does not define a function. Why?

 

[matt grime]

Can I suggest you get a better book, then?

A function from X to Y is a subset of XxY such that for all x in x there is a unique y in Y such that (x,y) is in this subset. X is the domain Y the codomain.

If we take RxR, and the subset defined by {(x,y) | y=x^2} then this defines a function.

{(x,y) | x=y^2} isn't a function since x=1 has two possible y's, 1 and -1.

You book assumes some intrinsic meaning for x and y, it appears, that we don't all share.

 

[jcsd]

I thought it was specifically the graph of the function that was the given subset of XxY (though many authors do idenitfy the function with it's graph, so the difference I guess is just a matter of semnatics).

 

[jcsd]

To answer the orginal question slightly differently then a function is a map that assigns each meber of the set X (the domain) a member of the set Y (the range).

 

[Fritz]

How do you know what the domain and range are?

Is y always the output and x the input?

You can have y = x^2 but not x = y^2. Is this because y is always the range and x the domain?

 

[Fritz]

Can I suggest you get a better book, then?

A function from X to Y is a subset of XxY such that for all x in x there is a unique y in Y such that (x,y) is in this subset. X is the domain Y the codomain.

If we take RxR, and the subset defined by {(x,y) | y=x^2} then this defines a function.

{(x,y) | x=y^2} isn't a function since x=1 has two possible y's, 1 and -1.

You book assumes some intrinsic meaning for x and y, it appears, that we don't all share.

I don't understand any of this!

 

[jcsd]

How do you know what the domain and range are?

Is y always the output and x the input?

You can have y = x^2 but not x = y^2. Is this because y is always the range and x the domain?

If you don't know/can't find out what the domain or the range is then you don't know what the function is! The range and thedomian is part of the function.

By convention 'x' usually denotes 'a member' of the domain and 'y' denotes 'a member' of the range (or of the image set if you prefer).

 

[matt grime]

That is my exact point, Fritz, what assumptions is the book making? There isn't a universally accepted use of notation for this. The question should say something like which of these defines a function of "foo" where it states what it considers to be the input. IS it always x to be the input, or is it always what is on the right hand side of the equals sign that is the input?

 

[dav2008]

Ok, here it is in simple terms.

If y is a function of x that means that for any x for which the function is defined there is only one y.

Consider y=x² If you graph it on an x-y plane you will see that for any x that you pick there is only one corresponding value of y, so you can conclude that for y=x² y is a function of x.

However, look at the same equation of y=x² and test if x is a function of y. Again, if you look at the graph you will see that for every y for which the function is defined there is not one corresponding value of x, but two. Thus, you conclude that for y=x², x is not a function of y.

 

[JasonRox]

A function must pass the Vertical Line Test. That is what matt grime said, I believe.

y^2=x can be a function depending on how you want to look at it. For now, let's say it is not. I understand where your textbook is going, but I don't think they should do that.

 

[matt grime]

Can we at least pretend to accept that functions are defined on things other than the real numbers, please?

 

[dav2008]

Can we at least pretend to accept that functions are defined on things other than the real numbers, please?

We're also pretending that he's not doing graduate work for a mathematics degree.

 

[jcsd]

We're also pretending that he's not doing graduate work for a mathematics degree.

Functions which do not have R as the domain are common throughout math, for example sequences represent functions with N as the domain and functions with complex domains are very important to both to mathematicans and physicists. The only way to know if a 'function' is infact a function is to look at the primitive concept.

 

[matt grime]

Just because at a lower level the 'correct' definitions (if such a thing can be said to exist) aren't used doesn't mean that when someone has a question that requires them we should not use them.

Just saying is x^2=y a function doesn't mean anything: is what a function of what on what domain (and hence range/codomain)? These are all necessary to answer the question at any level. Functions properly arise in the first year of an *under*graduate course in the UK.

And this is important in say taking square roots, where we don't even need to go outside the real numbers. y=sqrt(x) does not define a function from R to R (x is in the domain), but it is a function from R+ (the positive reals) to R.

 

[matt grime]

Ok, let's add a low level definition of function:

A function, f, is an unambiguous way of assigning to each element in D (the domain) an element in C (the image set)

Note, the domain, aka preimage, and image set are part of the definition still. You can't get round that fact as hopefully the example of sqrt( ) demonstrates without going beyond anyone's knowledge.

 

[JasonRox]

Would that explain the common convention that when we are speaking of...

y=sqrt(x), then we always take the positive so we can create a function out of it.

The only time you take both is when y=+-sqrt(x), which implies y^2=x and this not a function. Under this assumption y=-sqrt(x) implies that we only take the negative value and makes this a function.

I don't know if the UK goes by this convention, but I think its important to know or they make it important to know.

Personally, if you don't understand what a function is, then how will you get through the inverse functions? They generally have the same meaning, but if you can't understand the first you obviously don't understand the second.

Note: I'm just blabbing on. I just thought you'd like to know the basic convention here in North America.

 

[matt grime]

Erm, jason, the "convention" is that of mathematics, and in this case is universal.

y^2=x will not define y as a function of x in general though you are making the same mistake as everyone else by not specifying the domain (and codomain).

So, what do you mean by is or isn't a function if you do not state from where to where!

y=sqrt(x) even with the single valued choice of the root does not define a function from R to R, again x as the input since there is no real number that is the square root of a negative real number.

y=sqrt(x) from R+ to R is function since we always take it to mean the principal branch.

 

[zim777]

I Could Really Do With Some Help On This

E=mc2 is the velocity of 5 x r3 to the exponent 5 i understand the symbols but can't get the grasp of putting it as one please help E=mc2 is the velocity of 5 x r3 to the exponent 5

 

[HallsofIvy]

E=mc2 is the velocity of 5 x r3 to the exponent 5 i understand the symbols but can't get the grasp of putting it as one please help E=mc2 is the velocity of 5 x r3 to the exponent 5

This makes absolutely no sense! "E= mc2" is an equation, it can't be a velocity of anything.

In any case, only objects have "velocity". "4 x r3 to the exponent 5" (by which I assume you mean (4r³)⁵) can't have a velocity.

 

[DeadWolfe]

Poor Fritz.

He's probably just trying to pass a high school course for crying out loud.

 

[JasonRox]

You didn't understand what I said.

I do know what domains, and it is very important to know what it is. Same with the range.

All I was saying is that the convention is to avoid confusion between what is a function and what is not.