Function notation to relate Domain and Range

I was solving some functions problems and those exercises asked for stating the domain and range of the functions. In this process, I had my doubts about the function notation. I would like something to relate the Domain and Range. Considering the function $$f$$ I've seen notations like $$\text{Dom}(f)$$ and $$\text{Ran}(f)$$, but I would like an alternative to this.

Take the example $$f(x)=\frac{4-t^2}{2-t}=\frac{(2-t)(2+t)}{2-t}=2+t$$ Once $$t\neq2$$, the domain is $$(-\infty, 2)\cup(2, \infty)$$. The range is $$t\neq4$$, which is the point $$(2, 4)$$, where the function is undefined. Therefore, $$\text{Dom}(f)=\mathbb{R}-\{2\}$$ and $$\text{Ran}(f)=\mathbb{R}-\{4\}$$. The example given is $$f:\mathbb{R}-\{2\}\to \mathbb{R}$$, and here is my doubt.

Once $$f:A\to B$$, where the domain is A and codomain B. I know that the difference between Codomain and Range is that Codomain contains elements that might be the imagens, and Range is exactly the images produced. Therefore, $$\text{Range}\subseteq \text{Codomain}$$.

Taking the example again, I can say $$f(\mathbb{R}-\{2\})=\mathbb{R}-\{4\}$$, but are there something wrong with $$f:\mathbb{R}-\{2\}\to \mathbb{R}-\{4\}$$? I can't use this that way? I found that way very straight. I would like to know/undertand better and improve my math notation, so recommendations and corrections are welcome.

2 Answers

Let $$A$$ and $$B$$ be sets. The notation $$f:A\to B$$ says precisely that $$f$$ is a function whose domain is $$A$$ and whose codomain is B. This, so far, says nothing about the range (or image) of $$f$$. It is common to refer to the range of $$f$$ as the image of $$f$$, and denote it by $$\text{im}(f)$$. As you said, the image of $$f$$ is a subset of the codomain of $$f$$. Now, if you let $$S\subset A$$ be some subset, then we define $$f(S)=\{f(s)\in B:s\in S\}.$$ In other words, $$f(S)$$ is the set of all outputs of $$f$$ when applying $$f$$ to every element of $$S$$. Note that $$f(S)$$ is a set. So if you say $$f(S)=R$$ for some $$S\subset A$$ and some $$R\subset B$$, then this means that when you apply $$f$$ to every element of $$S$$, you obtain the set $$R$$. Note that this is not the same thing as saying that $$f$$ is a function from $$S$$ to $$R$$. I hope this helps.

It depends on context.

When you are first learning to find the range of an expression like $$(4-t^2)/(2-t)$$, then it is likely that no one is focused on the codomain. In that case, then you could write something like "We may consider $$f$$ as a function $$f:\mathbb R-\{2\}\to\mathbb R-\{4\}$$."

But if you are discussing the codomain, or care about whether the function is surjective/onto (全射的), then you shouldn't change the codomain. If you are told "the codomain of $$f(t)=(4-t^2)/(2-t)$$ is $$\mathbb R$$" or "the codomain of all functions under discussion is $$\mathbb R$$", then all you are allowed to say are things like "The range/image of $$f$$ is $$\mathbb R-\{4\}$$" and "we can define a new function $$g:\mathbb R-\{2\}\to\mathbb R-\{4\}$$ given by $$g(t)=f(t)$$.".

• Thank you @Mark S., for this question I just wondered about the notation$f:\mathbb R-\{2\}\to\mathbb R-\{4\}$ because it seemed very convenient to represent the range/image, despite it actually describing the codomain. I was wondering about some notation similar to$f:\mathbb R-\{2\}\to\mathbb R-\{4\}$, so I can show the Domain and Range/Image. Sep 7, 2020 at 17:37
• @欲しい未来 Ah. In that case, I direct you to Michael's answer with $\mathrm{im}(f)$ and the Wikipedia page for image. Sep 7, 2020 at 17:47