# Function definition notation [closed]

When I've a function $\text{f}$ and it can be depended on $x$ (where $x$ will be a real number) or other real number(s) (constant(s)), can I define the function as follows?:

$$\exists\space\text{f}\left(x\right)\mapsto\left\{\text{f}\left(x\right)\in\mathbb{R}:\forall\space x\right\}$$

Or is there a more common or good notation?

## closed as unclear what you're asking by Alex M., Morgan Rodgers, suomynonA, Rohan, BruceETDec 31 '16 at 8:13

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• @Leonhard So: $$\text{f}:\text{X}\mapsto\mathbb{R}\space\forall\space x\in\space\text{X}$$? – Poilys Dec 30 '16 at 21:18
• The other way, often clearer, is to define it using words, not symbols. – GEdgar Dec 30 '16 at 22:18
• @Leonhard This is not good notation; it should be $\to$, not $\mapsto$, plus $X$ is not currently defined (domain is mentioned to be $\mathbb{R}$), nor should we have to specify that this holds for all $x$ in the domain (that is already implied by this notational format). – Morgan Rodgers Dec 30 '16 at 22:44

The "signature" of the function (a programming term, I believe, which might not be standard amongst mathematicians; it basically means "domain and codomain" in this context) is written $f: S \to \mathbb{R}$, whatever $S$ is. Note that this does not involve the input to the function, so (for instance) it's really quite weird to write something like $f: x \to \mathbb{R} \ (x \in \mathbb{R})$; in this instance, $x$ is the input to $f$ and hence shouldn't appear in the function's signature. Notice also that $\mapsto$ tells us what happens to a particular input to the function, while $\to$ tells us what the domain and codomain of the function are.

For example, we may write the factorial function in the following ways (by no means an exhaustive list!):

• $f: \mathbb{N} \to \mathbb{R}$ given by $n \mapsto n!$
• $f: \mathbb{N} \to \mathbb{R}$ given by $x \mapsto x!$
• $f: \{ 1, 2, 3, \dots \} \to \{ x: x \in \mathbb{R} \}$ given by sending $i$ to $i \times (i-1) \times \dots \times 2 \times 1$.
• $G: \mathbb{N} \to \mathbb{R}$ given by $r \mapsto r!$
• $n \mapsto n!$ (this expression, without specifying the "signature" of the function, will suffice if the domain and codomain are clear from context)
• So, for my application I can write: $$\text{f}:x\to\mathbb{R}\space\space\space\space\space x\in\mathbb{R}$$? – Poilys Dec 30 '16 at 21:58
• You're looking for $f: \mathbb{R} \to \mathbb{R}$, unless I've misunderstood your example. Can you be more explicit about what $f$ is, so I can check if I understand you correctly? – Patrick Stevens Dec 30 '16 at 21:58
• (By the way, I explicitly said in my answer that $f: x \to \mathbb{R} \ (x \in \mathbb{R})$ would be a strange expression.) – Patrick Stevens Dec 30 '16 at 21:59
• Can you tell me exactly what $f$ is? It seems to be a function which takes an optional argument; is that correct? – Patrick Stevens Dec 30 '16 at 22:08
• You could write $f:X\times Y \to \mathbb{R}$ where $$f(x,y)=\begin{cases} g(x) : y=0 \\ c : y\neq 0, \end{cases}$$ where $y\in \{0,1\}$ is a variable that determines whether $x$ is relevant, and $g:X\to \mathbb{R}$. We don't have functions that have optional arguments in math. – mrob Dec 30 '16 at 22:36

If the function $f$ depends only from one real variable $x$ and has a real value, than you can write: $$F:X\to \mathbb{R} \qquad X \subset \mathbb{R}$$ remeber that teh function i well defined only if for all $x \in X$ we have a value $f(x) \in \mathbb{R}$.

• Why the capital letters? And, when the function $\text{f}$ depend on a real number and not on $x$ does that notation is valid in that case? – Poilys Dec 30 '16 at 21:48
• So can I also write: $$\text{f}:x\mapsto\mathbb{R}\space\space\space x\in\mathbb{R}$$? – Poilys Dec 30 '16 at 21:51
• @Poilys The Capital $X$ denotes a set of elements $x$ and note that $\rightarrow$ is different from $\mapsto$. – FuzzyPixelz Dec 30 '16 at 22:19
• @Poilys: Capital letters because $X$ is the domain of the function, that is a set , and, in this case it is a subset of $\mathbb{R}$. ( I've edit a typo in my answer). – Emilio Novati Dec 30 '16 at 22:35

I prefer : $$f:\begin{Bmatrix} X\rightarrow \Bbb R \\ x\mapsto f(x) \end{Bmatrix}$$ Where $\operatorname{dom}\text{f}=X\subset \Bbb R$

You could write $f:X\times Y \to \mathbb{R}$ where $$f(x,y)=\begin{cases} g(x) : y=0 \\ c : y\neq 0, \end{cases}$$ where $y\in \{0,1\}$ is a variable that determines whether $x$ is relevant, and $g:X\to \mathbb{R}$. We don't have functions that have optional arguments in math.

• I'm assuming the downvote on this answer is because the question is worded poorly. This is the only answer that addresses what the OP meant. You can see that by reading the other comments on other answers. – mrob Jan 1 '17 at 5:57