Question

What is a function

Answer 1

thi isnt as simple as it looks.is it

Answer 2

it has to be
one to one and onto
anything else?

Answer 3

http://finedrafts.com/files/Larson%20PreCal%208th/Larson%20Precal%20CH1.pdf

pp 39 and on

Answer 4

i think it's one to one and many to one

Answer 5

many to one?

Answer 6

multiple x values corresponding to one y value...a horizontal line

Answer 7

1 to 1 only;

Answer 8

the perfect illustration is on pp 39

Answer 9

so horizontal line is a function, but a vertical line is not,

Answer 10

a function can be injective (one to one), surjective (onto), or bijective ( which is injective and surjective both)

Answer 11

there are test to test whether a graph is a function

Answer 12

one-to-one correspondence
exactly one dependent value for each independent value
basically what panlac said

Answer 13

so you're saying a horizontal line is not a function?

Answer 14

since horizontal line is a many-to-one correspondence

Answer 15

so what do you call it when one / many of these conditions are breached

Answer 16

I haven't seen the terms domain and range...

Answer 17

a horizontal line can be a function if it is parameterized or in a different coordinate system besides cartesian

Answer 18

breaching these definition makes it not a function; I know of no special term for that

Answer 19

is it a relation? i cant remember what that means exactly

Answer 20

it's a one-to-one correspondence
say I have an independent variable t, and a dependent variable y

examples:

t | y
---
1|2
3|-5
0|2
2|7

function^
(each t is associated with exactly one y)

as opposed to

t | y
---
1|2
3|-5
0|2
1|7
^not a function because the dependent variable t is associated with more than one variable y
that is, when t=1 y= both 2 and 7

that is not a one-to-one correspondence because the independent variable t corresponds to more than one value of the dependent variable y for at least one value of t - in this case t=1 violates the definition of a funtion

Answer 21

more subtle examples:
t | y
---
1|0
2|0
3|0
4|0

this is a function (f(t)=0) because each independent variable value is associated with a unique value of the dependent variable


t | y
---
0|1
0|2
0|3
0|4

NOT a function because the independent variable t at t=0 corresponds to more than one value of the dependent variable y
(since t=0 corresponds to y=1,2,3,4)

Answer 22

soa function needs a independent variable?

Answer 23

and a dependent variable ,

Answer 24

yes, a function has at least one dependent variable, and possibly multiple independent variables

Answer 25

and it can have lots of each , but it must have at lest one on both kinds

Answer 26

i think i get it now

Answer 27

can it have more than one independent variable?
I think so, yes (though not positive)

Answer 28

should be the case as far as I can see
the important part is the on-to-one correspondence between independent and dependent variables

Answer 29

consider \(\mathbb R^2 \mapsto \mathbb R^3\)
position on a map corresponding to place on a globe

Answer 30

*the globe

Answer 31

right, that should be a function
f(lattitude, longitude)=(whatever unique position on the globe)
so yeah, I think that is a function

Answer 32

1 domain is limited to point at 1 range, but different domains can point to the same range.

Answer 33

latitude and longitude are linearly independent ,

in general, are variables necessarily are orthogonal ?

Answer 34

very good Q @UnkleRhaukus
I don't think I'm qualified to answer that

Answer 35

I am not sure if I get the question. but if I were to think of a globe, I immediately picture the x and y axes, whereby the y is dependent to the x. so when you say orthogonal, I am not quire sure if you're looking at a symmetry in respect to x, y, or origin.

Answer 36

I'm wondering in general though if the independent variables of a function must be orthogonal in order to fulfill the definition. Clearly latitude and longitude bear a similarity to the rectangular coordinate system, but can we prove it for all coordinate systems?

Answer 37

I wonder if the question even has any meaning for cases like parametric functions...