Bazza's 3/4 Mathematics questions

[WhoTookMyUsername]

Does R+ include 0? (R-)?

[ellecee]

Does R+ include 0? (R-)?

Nope.
0 is neither positive nor negative.

[WhoTookMyUsername]

What is the codomain?  (f: R -> R)

Difference between range? I google searched it but i cant really tell the difference still


2) is ran or a suitable shortening of range?
3) when indicating a asymptote at x=0 or y=0 do we need to draw a dotted line over the axis?

[chocolatedaddy]

What is the codomain?  (f: R -> R)

Difference between range? I google searched it but i cant really tell the difference still


2) is ran or a suitable shortening of range?
3) when indicating a asymptote at x=0 or y=0 do we need to draw a dotted line over the axis?

Codomain is  part of the definition of the function.Codomain is the set of values that could possibly come out, Range is the values that do come out E.g. You could define a function f(x)=x^2 with the codomain of R and domain of R. But after further examination you would realize that all the numbers output are positive so the range would be R+. Hence, Range is a subset of codomain. That is the difference. The codomain f:R->R just means that f has a domain of R and a codomain of R also.

As for the asymptotes, I am fairly sure you do need to mark the asymptotes when they lie on the x or y axes.

And for asymptotes I am pretty sure you do not need to mark the asymptote if it lies on one of the axes.

[kamil9876]

^great to see a vce student who actually knows what codomain is (from memory sometimes VCE books even stuff up on the basics of functions). Just one point to clear up a slip up:

You could define a function f(x)=x^2 with the codomain of Z(Z being integers). But after further examination you would realize that all the numbers output are even so the range would be R+.

Mind clearing this up? I'm quite sure you must've meant something else because that doesn't make sense(how can the range be R+ and Z be the codomain if R+ is not a subset of Z). What did you define as the domain here?

[brightsky]

1) chocolatedaddy did a pretty good job of explaining what the codomain is. here's my two cents:

we all know that we can think of a function as a machine. you input an x-value in and the machine churns out a y-value for you. let's say we did this multiple times. we get a SET of x-values that we tried, and a SET of y-values which came out of the machine. a bit of a divergence but the definition of a function is just when each element in the SET of x-values correspond to a different element in the SET of y-values.

now sometimes when a function is too complicated or incomplete, we don't know for sure what the output would be for certain x-values. that is we know we have a SET of x-values we could try, but the SET of y-values remain ambiguous. however, what we do KNOW is that the SET of y-values lie in a bigger set of numbers, that is, the set of y-values is a subset of this bigger set. this bigger set is referred to by mathematicians as the codomain.

2) yeah that's fine

3) it's best if you do draw a dotted line over the axis, but more important is that you put Asymptote y = 0 or Asymptote x=0 beside the line.

[WhoTookMyUsername]

thanks

but if we have a function, let's say doesn't the function imply that we're using all values of (x), including non integers, positive, negative and everything, doesn't this mean the codomain is always the same as the range (all the numbers that could come out DO come out in the graph?)? Or is the codomain just R not R+ in this case?

[brightsky]

as far as i know, you can set the codomain to be anything, so long as the range is a subset of it. as far as maths methods is concerned, if you put codomain = R, you'll never go wrong (pretty sure!) but perhaps kamil can enlighten us all and give a more rigorous definition of the codomain.

[TrueTears]

as far as i know, you can set the codomain to be anything, so long as the range is a subset of it. as far as maths methods is concerned, if you put codomain = R, you'll never go wrong (pretty sure!) but perhaps kamil can enlighten us all and give a more rigorous definition of the codomain.

correct, unless you're dealing in the complex field, then R is sufficient.

http://en.wikipedia.org/wiki/Codomain

[WhoTookMyUsername]

wb for spesh if you are in complex field?

[dc302]

Then your function probably looks like:

f : C --> C

[kamil9876]

thanks

but if we have a function, let's say doesn't the function imply that we're using all values of (x), including non integers, positive, negative and everything, doesn't this mean the codomain is always the same as the range (all the numbers that could come out DO come out in the graph?)? Or is the codomain just R not R+ in this case?

Well the domain and codomain are a part of the actual definition of a function(I'd say it's good to think of a function like brightsky explained, rather than just as an expression of x). So I think it would be unfair for someone to ask "if what's the domain and codomain?". How would you know that we're not say dealing with complex numbers etc. or perhaps the domain and codomain could be the set of all 2 by 2 matrices and f(A) is just the square of the matrix A?

Of course in VCE you probably get questions like "what's the domain of?.." but from experience that's just implicitly asking for the largest subset of the real numbers which can be a domain of that function. You can make the codomain anything, so long as it makes sense ie: f(a) is in the codomain for every a in the domain.

I think the explanations in this thread have been very good, just hopefully chocolatedaddy clears up his example which doesn't make sense atm but I think he was onto something good

[dc302]

thanks

but if we have a function, let's say doesn't the function imply that we're using all values of (x), including non integers, positive, negative and everything, doesn't this mean the codomain is always the same as the range (all the numbers that could come out DO come out in the graph?)? Or is the codomain just R not R+ in this case?

As kamil said, the codomain is part of the definition. That means, this whole expression f : R -> R is one big definition along with f(x) = .... This basically means you choose the domain and you choose the codomain, although with some common sense, and of course you also choose your what your f(x) is.

It might also help to understand what the codomain helps to convey. The codomain gives an idea of the 'type' of numbers being used, and also usually (but not in spesh), the dimension. It IS ok to write the codomain as R+ in your case, but it is typically written as R. It's like saying, I have a machine that takes in spheres and turns them into cubes. The codomain is 'cubes', and this is kind of like a general description helping you to know what comes out, even though it doesn't say anything about the exact type of cube (the range).

[pi]

3) it's best if you do draw a dotted line over the axis, but more important is that you put Asymptote y = 0 or Asymptote x=0 beside the line.

I'd say to draw it dotted just above or in a different colour (not red! - blue is best), just so examiners can see it and not see it as part of the axis. But as brightsky said, definitely label it properly, I always did: Asym. x=0 or something like that

wb for spesh if you are in complex field?

You'll never need to do that btw

[chocolatedaddy]

Edited my explanation, hopefully that makes sense now.