Question

what is the difference between the domain of a function and natural domain of a function?

Answer 1

i could be wrong, but i think the "natural domain" is taken by assumption to be the largest set of numbers on which the function can act, whereas the domain is something that should be specified
the truth is that when a function is defined, the domain should be part of the definition.
in other words, you are supposed to say what the function does and also on what set the function acts

Answer 2

natural domains seem to be sets of numbers; integers, rationals, reals, etc

domain itself are all the values from the natural domain that actually work in the function

Answer 3

aha, so natural domain is related to domain, as codomain is related to range?

Answer 4

those are terms i aint heard in awhile; but yes

Answer 5

But is there a difference between "range" and "image", then?

Answer 6

correct me if i am wrong, but suppose i have a function, say \[C(x)=7x+\frac{48}{x}\] then without specifying the domain at the outset, the "natural domain" would be all real numbers except zero.

on the other hand, if i said this function represented the cost of building a fence if one side has length \(x\) then the domain would be \(x>0\) since the side cannot be negative

Answer 7

From wikipedia: "Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f."

Answer 8

From: Doctor Rick

Hi, Ayah.

Every function has a domain, the set of (input) values over which it
is defined. If I don't state what the domain is, by convention we
take the domain to be all (real) numbers for which the expression
defining the function can be evaluated. We call this the "natural
domain" of the function.

Let's look at an example. Take the function f(x) = 1/(1 - x). This
function can be evaluated for all x except x = 1, because replacing x
with 1 means dividing by zero -- an undefined operation. Therefore,
1/(1 - x) is not defined for x = 1:

{x | x != 1} (the pair of symbols != means "is not equal to")

If I wanted to, I could state a domain explicitly, for instance

g(x) = 1/(1 - x), for x < 1

In this case, the domain is

{x | x < 1}

Why? Because I said so! It isn't "natural"; I had to declare
something to make it so. By stating the domain explicitly, I am
saying that this function g(x) has no value for any inputs greater
than or equal to 1, even though I would have no trouble evaluating
1/(1 - x) for, say, x = 2. I have restricted the domain.

In summary, the natural domain of a function (or rather, of the
expression used to define the function) is one particular domain --
the one we assume when no domain is stated explicitly.

When we do state the domain explicitly, it must be a subset of the
natural domain. In my example, I can't say h(x) = 1/(1 - x) for all
real numbers, because it has no value for x = 1.

So yes, you are right @satellite73