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    Hi mathematicians,

    I don't really understand Domains, Function, Composite Function. Can someone please explain it to me. Thank you
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    Domain is the restrictions of the values that can be placed into a function.

    It can be restricted out of choice, or "physical" inabilities. Like not being able to divide by 0s.

    The range is like the output of the function when placing the values from the domain in.

    So let's say:

    f(x) = 3ln(x)

    Domain:x > 0
    Range: - infinity < f(x) < infinity

    Composite Functions:
    You're placing another function into another, combining them... The terminology I'm using isn't fully correct I don't think but you place one within another.

    f(x) = 3x
    g(x) = e^x

    fg(x): you are going to have f(x) but wherever you have x you place g(x)

    So 3(e^x)

    gf(x) = e^(3x)

    Yeah?
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    Let's say you have a function: y = f(x).

    Domain is effectively all the possible 'x' values your function can take.

    Range is therefore all the possible 'y' values your function can take. Let's look at an example:

    y = x²

    Here, you know that any number can be squared, so your domain is going to be any real number. This can be written as any  x \in (-\infty, +\infty) . What about the range? When you square a number, it always becomes positive. So therefore your range is just going to be all positive numbers (including 0 as 0² = 0). So your range is going to be all  y \in [0, +\infty) . It doesn't always have to be written in interval form, there are many ways you can write the domain and range.

    Now, how do you remember which on is which? Notice the 'in' at the end of the word 'domain'. That's how I used to remember that domain was the 'x' values as its the values that can go 'in' the function.

    With composite function, yes, you are basically putting a function "inside" the other function. What you have to remember here is to work from right to left, instead of left to right. By this, I mean

     f(x) \circ g(x) = f(g(x)) .

    Here, we are doing "f(x) composite with g(x)", so where we have all the 'x' values in f(x), we replace them with the function g(x). Let's look at an example.

    If we have two functions: f(x) = x² + x + 3 and g(x) = x³ + 2x + 1. Then if we wanted to do composition of these two functions, it would be

     f(x) \circ g(x)
     =(x^3 + 2x + 1)^2 + (x^3 + 2x + 1) + 3 .

    See here how every time I see an 'x' in f(x), I have replaced it with the whole function of g(x). Like I said about working right to left, if we were to do the composition

     g(x) \circ f(x)
     = (x^2 + x + 3)^3 + 2(x^2 + x + 3) + 1

    where here, I have put f(x) inside g(x) and so replaced every 'x' in g(x) with f(x).

    Do you have any questions?
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    The domain is actually part of the definition of a function. A function isn't just some formula, it is comprised of three things:

    - A set D called the domain
    - A set C called the codomain
    - An association of precisely one element of C to every element of D.

    Now where people are talking about the domain being 'the biggest set of allowable values' is just a convention.

    That is, when specifying, say a function on the real numbers, they give a formula and tacitly imply that the domain is every value the formula makes sense for.

    However, the domain and codomain are important parts of the function. For example if

    f\colon \mathbb{R} \rightarrow \mathbb{R}^{\geq 0} is given by f(x)=x^2 and
    f'\colon \mathbb{R}^{\geq 0} \rightarrow \mathbb{R}^{\geq 0} is given by f'(x)=x^2

    then f and f' are different functions. This becomes important when you want to talk about inverses and compositions
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    (Original post by Mark85)
    The domain is actually part of the definition of a function. A function isn't just some formula, it is comprised of three things:

    - A set D called the domain
    - A set C called the codomain
    - An association of precisely one element of C to every element of D.

    Now where people are talking about the domain being 'the biggest set of allowable values' is just a convention.
    Isn't that effectively the same thing? Normally when you define a function, you define the domain and range and everything (i.e like your example, f mapping numbers in X to Y or whatever). But at A level, you're "working backwards" and are trying to define the function from what you have been given, and so part of that is coming up with the domain and range so you are looking at the 'x' values your function can take and the 'y' values it gives you?
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    (Original post by claret_n_blue)
    Isn't that effectively the same thing? Normally when you define a function, you define the domain and range and everything (i.e like your example, f mapping numbers in X to Y or whatever). But at A level, you're "working backwards" and are trying to define the function from what you have been given, and so part of that is coming up with the domain and range so you are looking at the 'x' values your function can take and the 'y' values it gives you?
    Well, which of the 2 functions he described is invertible as it is?

    Notice that, in some cases, you have to restrict a given function in order to define it's inverse. He was simply stating that the mapping is of great significance when it comes to inverting/composing functions.
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    (Original post by claret_n_blue)
    Isn't that effectively the same thing? Normally when you define a function, you define the domain and range and everything (i.e like your example, f mapping numbers in X to Y or whatever). But at A level, you're "working backwards" and are trying to define the function from what you have been given, and so part of that is coming up with the domain and range so you are looking at the 'x' values your function can take and the 'y' values it gives you?
    Well, it is a backwards approach. Even if you work under the assumption that the domain should be the largest set of real numbers to which some rule applies - the convention with the codomain is more confused... should it be the image of the function or should it be the whole of \mathbb{R}? I think the way this is treated in the curriculum is confusing since they later go on to talking about surjectivity and inverting functions etc. but having not clarified the issues of domain and codomain, people probably assume that the codomain of any function is the same things as its image and thus the whole concept of surjectivity seems redundant.

    These issues don't seem like much of a big deal at first but you only have to get up to defining trig functions and their inverses to get into big problems.
 
 
 
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