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Consider a function introduced by $f \colon X \rightarrow Y$. While $X$ is the domain of the function, my understanding is that $Y$ is the codomain -- and not the image! However, there are many possible codomains which one can specify. Is the convention to specify the smallest codomain which can be written in a 'simple' way?

To take a concrete example, consider the function $f(x) = x + 1$ defined on the domain $X = [0, 1]$. If I understand correctly, one can introduce this function as $f \colon [0, 1] \rightarrow \mathbb{R}$, or $f \colon [0, 1] \rightarrow \mathbb{R^+}$, or $f \colon [0, 1] \rightarrow [0, 5]$. But wouldn't it be clearest to write $f \colon [0, 1] \rightarrow [1, 2]$?

afreelunch
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    $f \colon [0, 1] \rightarrow \mathbb{R}$, or $f \colon [0, 1] \rightarrow \mathbb{R^+}$, or $f \colon [0, 1] \rightarrow [0, 5]$ are all considered as different functions. Which one of these functions you want to consider depends on what you are trying to do and there is no such thing as general convention. (If you know the exact range of the function then perhaps you would be better off taking the range as the codomain.) – geetha290krm Jun 15 '22 at 10:05
  • Related: https://math.stackexchange.com/questions/777563/codomain-of-a-function – Karl Jun 15 '22 at 10:23
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    $\mathbb R$ is arguably a simpler codomain when $[1,2]$ requires no emphasis and distracts from the flow of the text. – ryang Jun 15 '22 at 11:38

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Is the convention to specify the smallest codomain which can be written in a 'simple' way?

The smallest codomain that can be written in a simple way could be $f(X)\cap Y$ or even $f(X)$, which works irrespective of $f$, $X$ and $Y$. Where applying a function to a set has the meaning of $f(X)=\{f(x)\mid x\in X\}$.

Like with (m)any other things in math, you only bother if it matters. There are lot of cases where it's enough to know that $Y=\Bbb R^n$ or $Y=\Bbb C$, or $f$ is not known like in "be $f:S^n\to \Bbb R^n$ continuous". If it matters that $Y$ is the range of $f$, then one can say that the function "is onto" or "is surjective" or is an "epimorphism" in the case of a homomorphism.

But wouldn't it be clearest to write $f:[0,1]→[1,2]$?

Even that doesn't tell whether $[1,2]$ is the image of $f$. You could write it as $f:[0,1]\twoheadrightarrow[1,2]$, though. But IMO there's no point in steering the reader's attention to some details if they don't matter.

emacs drives me nuts
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