Question

What is a function?

Answer 1

A realtion between x and y that every x produces only one unique y

Answer 2

can't wait to see this...
mathematicians been arguing over this one for centuries

Answer 3

So the same x cannot give different y's

Answer 4

Look, I'm only in hs, that's my basic definition, don't throw any \[\Large \epsilon~\delta\]
stuff at me, I'm outty

Answer 5

A function is a relation between two sets \(A\) and \(B\)\[f:A \to B\]such that an element of \(A\) say \(x\) points to ONLY one element of \(B\), say \(y\).

In that case, we say that \(f(x) = y.\)

Answer 6

@doulikepiecauseidont there are thankfully no epsilons and deltas here. Your definition is sufficient.

Answer 7

its a SET!!!!!!!!!!!!!!!!!!!

Answer 8

everything is a set!!!!!!!!!!!!!!!!!!! @satellite73 :)

Answer 9

What's a set?

Answer 10

some stuff in a box, and sometimes nothing in a box

Answer 11

the box and the stuff is the set, the stuff is the elements of the set

Answer 12

we call the empty set a set for simplicity

Answer 13

iz liek a collection of tings
u wriet a set liek dis:
wardrobe = {shirts, money, undies, 18+ magazines}

Answer 14

It's amazing how definitions get so stupid at this level.

Answer 15

a function is a set of ordered pairs in which the relation on these ordered pairs is given by a rule that passes 2 conditions

1) well difedness i.e x=y implies f(x) = f(y)
2) defined everywhere i.e. f(x) is defined for all x

Answer 16

well definedness* this is a word I swear

Answer 17

but u just said that f(x) = (x-5)/(2x-1) wasn't defined for all real numbers. ;-;

Answer 18

its not, and thus its domain is \(\mathbb{R}-\{\frac{1}{2}\}\)

Answer 19

that function is not defined for x = 1/2

Answer 20

yeah... I was correcting myself.

Answer 21

by the way, I was just joking in that thread. and I'm joking here as well.

Answer 22

the tricky part here is that it does not make sense to ask for the domain of a function, because we need the domain to define a function. i.e. there is no way to talk about the function without knowing the domain

Answer 23

Okay thanks @zzr0ck3r and @Zeta (:

Answer 24

but we need to make adjustments for people to understand the concept.

same thing with \(y=mx+b\) being called a linear function, it is not! It is only linear if b=0, else it is an affine function

Answer 25

@zeta we cool

Answer 26

<3

Answer 27

not joking this time. i'd never heard the term "affine function". i always thought a linear function was, if not defined, understood by the fact that its graph is a straight line. no matter where its origination is - it's still a straight line.

but yeah, your statement makes sense too because i also used to use "linear change of y with x."

can you explain it further? interesting discussion.

Answer 28

the definition of a linear function says that {0} is in the range

Answer 29

because by definition of a linear function \(L\) we have that \(L(\alpha x+\beta y)=\alpha L(x)+\beta L(y)\) So this implies \(L(\alpha x)=\alpha L(x)\)

so

\(L(0)=L(0*x)=0*L(x)=0\)

Answer 30

@Kasie see? centuries

Answer 31

lol

Answer 32

i knew satellite was gonna come in somewhere.

that definition of a linear function sounds alright to me. but why do we have to make it pass through origin? what's the underlying fact that i seem to miss here?

Answer 33

@zeta notice here that the greek letters are scalars from the field of scalars on our vector space where x, y live. But with R the elements of R are the scalar field for the vector space R. ITS ALL R.

Answer 34

...i understand...

but why did we define it that way?

Answer 35

well the reason for any definition is to classify something/s in a way that is useful. it happens to be very useful to have a difference between linear and affine.

Answer 36

in all branches of math...but I would say we mostly utilize that part of the definition in optimization.

Answer 37

Yes, I see satellite... they can't even decide if they're joking or not...

Answer 38

we are not:)

Answer 39

mainly because 0 is very useful.

Answer 40

but the word "linear" should be used for \(mx + c\).

\(y = mx\) is something that can be said to be a "direct variation" i guess?

in any case, i'm with the masses. linear is what linear is taught to be.

Answer 41

but its not....thats my point.

Answer 42

google it:)

Answer 43

linear function, and click on first thing

Answer 44

wikipedia?

Answer 45

it better...lol

Answer 46

ah, you're talking about linear maps... i see.

do you mean to target the polynomial definition as well?

Answer 47

here

Answer 48

wiki is misleading, there is no calculus version of linear functions, not that I have ever seen and in fact I learned this in an advanced calc course, and all subsequent classes have always used the one and only definition.

Answer 49

http://cfsv.synechism.org/c1/sec15.pdf

Answer 50

OK, thanks! Gonna read that paper.

Let's end it here.

Answer 51

lol

Answer 52

this is not arguing to me, just good discussion:)

Answer 53

but it is exactly what satalite was talking about