Question

All relations can be made functions by defining a proper domain.. is this right?

Answer 1

i suppose range ... consider arcsin(x), the domain is -1 to +1, if you restrict range to particular branch you have defined a function

Answer 2

ok.. i get it..

but tell me one thing.. you said domain is -1 to +1, so when we consider a relation.. to make it a function we directly write what the domain SHOULD be..

and then restrict the range? is that correct?

Answer 3

and domain of the function is the inputs that we give.. so why did you say domain is -1 to +1?

Answer 4

oh wait.. you said arc sin.. it means sin inverse.. sorry sorry

Answer 5

yes i get it now.. thank you :)

Answer 6

yes the domain does not change. function are single valued by definition. this definition of function is very important. But if you take arcsin(1), this both equal to pi/2 as well as 5pi/2 ... so in order to make it single valued, you must specify the range to make it single valued.

Answer 7

also note that relation is pretty broad definition. closed curves such as circles also make relation (plane relation) ... you can make them functions by same procedure (removing lower half or upper half of circle)

Answer 8

i actually thought of circles, so they are not functions because for one input i can get multiple output.. right?

similarly square root function, by defining the range..

one more doubt is, when i say m defining the range, instead of that, can't i say m defining the CODOMAIN? i do understand the difference between them, range are the outputs and Codomain are all the possible outputs ..

Answer 9

I had to check a little bit for it as i wasn't sure myself
http://answers.yahoo.com/question/index?qid=20080221171006AABKvyn
According to this article Codomain is a broad set that contains range.
for example in a relation of circle of radius 5 centered at origin, mapping f(x) = sqrt(25 - x) is restricted to [0,5] is a function. The codomain can be any set containing interval [-5,5]

Answer 10

looks like this isn't strongly defined
http://www.proofwiki.org/wiki/Definition:Codomain_(Set_Theory)/Relation
http://www.proofwiki.org/wiki/Definition:Range

both are the second element in cartesian prodcut i.e AxB ... codomain being B.
Image seems to be narrower definition
http://www.proofwiki.org/wiki/Definition:Image_Set_of_Relation

strictly speaking, we define function by restricting the range/codomain/image of relation. all seems to work as long as we make function single valued.