Is the codomain and range of an onto function the same? As far as I understand, if $f:A\to B$, range is all possible values in B. So if it's an onto function, then all values in B must be mapped to something in A right? Am I correct?
[Math] Codomain and range of onto Functions
functions
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If $f:X\to Y$ is a function from $X$ to $Y$, we usually call $X$ the domain, $Y$ the codomain, and $f(X)$ the range. Some authors call $Y$ the range, in which case $f(X)$ is called the image.
What is the point of having values in the codomain that can not be output by the function, how does that aid in describing the function?
Here are a few reasons why we allow some functions to not be surjective.
As Lubin mentioned, the range of a function can be difficult to determine. For example, determining the range of a polynomial of high even degree (such as $P(x) = x^6 - 3x^2 + 6x$) amounts to finding the zeroes of a high-degree polynomial (such as $P'(x) = 6x^5 - 6x + 6$, whose roots are not expressible as radicals), a difficult task in general. We could get around this by defining the codomain of every function $f$ to be $\operatorname{im} f$ (that is, $\{y\,:\,f(x)=y\text{ for some }x\in X \}$), but that doesn't really add any information.
It's nice to separate surjective functions from other functions because surjective functions are dual to injective functions. When I say "dual" I'm referring to, for example, the following fact: a function $f:A\to B$ is injective if and only if there is a function $g:B\to A$ such that $g\circ f=1_A$ (by $1_A$ I mean the identity function on $A$); a function $f:A\to B$ is surjective if and only if there is a function $g:B\to A$ such that $f\circ g=1_B$. When you study the branch of mathematics known as category theory, you'll see that it's very natural to have dual properties like this.
Does this also mean that the domain can include numbers that the are not inputs to the function?
As others have remarked, the domain of a function can include other objects than numbers. For example, you could define a function which takes as input a person and returns his age. In any case, a function must be defined on all possible input values. The answer to your second question is no.
And is it also then true that is a function is "onto" the codomain is the same as the image? So surely any function can be "onto" if you just change the what the codomain is?
That's exactly right. You can make any function onto by changing the codomain. But as I remarked earlier, in general we don't know what the image of a function is and so it doesn't add any information to restrict the codomain.
What I'm really trying to ask I guess is the range/image of a function is defined by the function, what defines the codomain?
The codomain usually arises naturally in the definition of the function. For example, whenever you have a function which returns a number, the natural choice of codomain is $\mathbb R$. Of course, if by "number" you mean "complex number" then the codomain could be $\mathbb C$; if by "number" you mean "quaternion" then the codomain could be $\mathbb H$.
On the other hand, owing to the set-theoretic fact that "there is no set containing everything," it's not possible to pick a single universal codomain for functions.
When I wrote up this answer I realized that I used to ask the same questions as you, but I stopped once I had learned enough mathematics. I can't give you a single profound reason why we don't make all functions surjective besides a pragmatic one: surjectivity is a useful notion, and getting rid of it would be unprofitable.
Best Answer
Yes, by definition a function $f:A\to B$ is onto if the range ($f(A)$) equals the codomain ($B$).
I'm not sure you wrote what you meant to write. All values in $B$ must be mapped to by something in $A$, yes.