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Given the function $f:X\longrightarrow Y$, $X$ is called the domain while $Y$ is called the codomain. But what do you call $f(x)=x^2$ in this context, where $x\in X$? That is to say - what is the name for the $f(x)$ notation?

And while I'm here, what is the proper way to write a function like this? Would it be $f:\mathbb{R}\to\mathbb{R},\;f(x)=x^2$?


Edit:

I figured I'd add this to add a bit of context into why I'm asking. I'm writing a set of notes in LaTeX, and I'd like to use the correct terminology for the definition of a function.

A function from set $A$ to set $B$, denoted by $$f:A\to B;x\mapsto f(x)$$ is a mapping of elements from set $A$, (the $\textit{domain}$) to elements in set $B$ (the $\textit{codomain}$) using the $\color{blue}{\sf function}$ $f(x)$. The domain of a function is the set of all valid elements for a function to map from. The codomain of a function is the set of all possible values that an element from the domain can be mapped to. The $\textit{range}$ (sometimes called the $\textit{image}$) of a function is a subset of the codomain, and is the set of all elements that actually get mapped to by the function $f$.

Here I'm pretty sure the highlighted word "function" is not right.

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  • $\begingroup$ I suggest \to or \rightarrow instead of \longrightarrow. The last one is there for if you need the arrow to be longer because you're writing something over it. As in $\overset{\text{text}}{\longrightarrow}$ instead of $\overset{\text{text}}{\to}$. $\endgroup$
    – Jim
    Feb 22, 2013 at 23:16
  • $\begingroup$ why do you think that function is not right = $\endgroup$ Feb 22, 2013 at 23:36
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    $\begingroup$ Some would call "$f(x)=x^2$" the rule of the function $f$. $\endgroup$ Feb 22, 2013 at 23:44
  • $\begingroup$ @DominicMichaelis Well, function is used in two places to mean two slightly different things; I don't like using function to describe part of a function. $\endgroup$
    – Mirrana
    Feb 22, 2013 at 23:47

3 Answers 3

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I can remember to read this text, and being puzzled with the exact same question. From what I've learned from my teacher, you're right, writing down something as "the function $f(x)$..." is sloppy notation. However, many books/people will use it this way.

If you're are very precise, $f(x)$ is not a function or an map. I don't know of a standard way to refer to $f(x)$, but here is some usage I found on the internet:

  • The output of a function $f$ corresponding to an input $x$ is denoted by $f(x)$.
  • Some would call "$f(x)=x^2$" the rule of the function $f$.
  • For each argument $x$, the corresponding unique $y$ in the codomain is called the function value at $x$ or the image of $x$ under $f$. It is written as $f(x)$.
  • If there is some relation specifying $f(x)$ in terms of $x$, then $f(x)$ is known as a dependent variable (and $x$ is an independent variable).

A correct way to notate your function $f$ is: $$f:\Bbb{R}\to\Bbb{R}:x\mapsto f(x)=x^2$$

Note that $f(x)\in\Bbb{R}$ and $f\not\in\Bbb{R}$. But the function $f$ is an element of the set of continuous functions, and $f(x)$ isn't.

In some areas of math it is very important to notate a function/map specifying it's domain, codomain and function rule. However in for example calculus/physics, you'll see that many times only the function rule $f(x)$ is specified, as the reader is supposed to understand domain/codmain from the context.

You can also check those questions:

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  • $\begingroup$ Nice, Kasper +1 $\endgroup$
    – amWhy
    Feb 23, 2013 at 1:50
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normally you say $f:X\rightarrow Y$; $x\mapsto f(x)$, as a functions takes an element from $X$ and give you one from $Y$. $$y=f(x)$$ Is an equation, and not a definition of a function in a strict sense.

The proper way would be $$f:\mathbb{R}\rightarrow \mathbb{R}; \ x\mapsto x^2$$

The Image of a function is definied as $$\operatorname{im}f:=\{f(x)|x\in X\}$$ It is often written as $$f(X):=\{f(x)|x\in X\}$$ Notice that $X$ is a set, not an element. so $f(\{x\})=\{f(x)\}\subseteq Y$ while $f(x)\in Y$

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  • $\begingroup$ I was wondering - is $f(x)$ called the image? I've seen that word used but only know it to be synonymous with Range. $\endgroup$
    – Mirrana
    Feb 22, 2013 at 23:15
  • $\begingroup$ not $f(x)$ but $f(X)$ note the different as $X$ is a set but $x$ is an element $\endgroup$ Feb 22, 2013 at 23:19
  • $\begingroup$ @agent154 You can say "$f(x)$ is the image of the element $x$ under the function $f$". Perhaps this is what you were thinking of (here of course "image" means something different than the image of the function). $\endgroup$ Feb 22, 2013 at 23:32
  • $\begingroup$ @davidMitra i would be careful with that, as the image is a set but $f(x)$ is an element of $Y$ $\endgroup$ Feb 22, 2013 at 23:34
  • $\begingroup$ The notation $f(X)$ is also written (in set theory) as $f[X]$ and $f''X$ sometimes, often because the elements of the set are sets themselves, and sometimes $f(x)\neq f''x$. $\endgroup$
    – Asaf Karagila
    Feb 22, 2013 at 23:35
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From what I understand, $f$ is the function, so saying "a function $f(x)$" would be wrong. Instead, you could say "a function $f$", or if you didn't want to assign it a name, "a function $x\mapsto x^2$", or if you want to specify the domain and codomain, "a function $f:X\to Y$".

If you just want to define the function's input-output relationship, $f(x)=x^2$ suffices, maybe with a $\forall x\in X$ at the beginning if you want to do it properly. That doesn't necessarily define the domain though, that set could just be a subset of the domain. $f(x)$ is really no different from $f(1)$, except that $x$ is a (qualified) variable. It's basically a rule defining the input and output for an improper subset of its domain.

The addition of "$:X\to Y$" is useful if you want to specify both the domain and codomain when defining the function. As far as I can tell, you can add it after a function like $f$ and the whole expression still refers to the function. You can't really use it with the $f(x)$ style, so I'd do something like

$f:\mathbb R\to\mathbb R=x\mapsto x^2$

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