# What does the function f: x ↦ y mean?

I am doing IGCSE Maths, and am having a few problems with function notation. I understand the form f(x).

What does the form f: x ↦ y mean? Could you also give one or two examples?

And, if possible, state your source. Thank you.

• – lhf Apr 20 '16 at 13:11
• @lhf Could you please explain this for a beginner? Thanks! – Annabelle Sykes Apr 20 '16 at 13:12
• Examples: $f:x\mapsto x^2$ is the squaring function and $g:x\mapsto x+1$ is the function which adds one. – arctic tern Apr 20 '16 at 13:14
• I really prefer $f:x\in\mathbb{R}\mapsto x^2\in[0,\infty)$. It's a more complete description. Maybe for this function it's not needed, but there is LOT of cases when it's best to tell what is the domain and codomain. – Integral Apr 20 '16 at 13:26
• Related: math.stackexchange.com/questions/1740154/… and math.stackexchange.com/questions/473247/… (The latter was posted by Deusovi in a comment under their answer.) – Martin Sleziak Apr 20 '16 at 13:48

It means that $f$ is a function that takes the value $x$ to the value $y$. For instance, $$f: x\mapsto x^2$$ is an alternate way of writing $f(x) = x^2$.

• Could you please give an example? Thanks :) – Annabelle Sykes Apr 20 '16 at 13:11
• Could you please state your source? Thanks :) – Annabelle Sykes Apr 20 '16 at 13:21
• @SWFApp: I've known this for a while so I don't have the source that I used; here is another SE question with several answers that agree with me. – Deusovi Apr 20 '16 at 13:22
• So, is there any scenario where you would need to use the "↦" notation instead of "=" notation? If your answer is a full explanation, it seems to me that ↦ is entirely useless. – twiz Jul 29 '16 at 11:15
• @twiz: Just using = leaves open the possibility that x is a constant. Sure, we assume that it's a variable, but it's not necessarily the case. For instance, what if $f(x)$ was actually the function where $x \mapsto x³-4$? Then "$f(x) = x^2$" would be an equation to solve, not a definition of a function. That sort of thing pops up all the time - for instance, when we want to find the roots of the function, we use the equation $f(x) = 0$; that does not mean that we're redefining $f$ to be the function that always gives $0$. – Deusovi Jul 29 '16 at 11:19

$f:x \mapsto y$ means that $f$ is a function which takes in a value $x$ and gives out $y$.

But,
$f: \mathbb{N} \to \mathbb{N}$ means that $f$ is a function which takes a natural number as domain and results in a natural number as the result.

• So if f = x+2, f: 1 ↦ y = 3 – Annabelle Sykes Apr 20 '16 at 13:14
• Because you're wrong: the $\to$ and $\mapsto$ arrows mean different things. Also, $\mathbb{W}$ is not the set of positive numbers: that's $\mathbb{R}^+$. Whole numbers are not nonnegative numbers, either; they are natural numbers including 0. Oh, and $\to$ talks about the sets of the domain and range, while $\mapsto$ talks about the elements: you conflated them. – Deusovi Apr 20 '16 at 13:18
• Whole numbers are the nonnegative integers. And you conflated two different arrows: $\to$ and $\mapsto$. They have different definitions. $f(x) = x^2$ can be described as $x\mapsto x^2: \mathbb R \to \mathbb R^+$. – Deusovi Apr 20 '16 at 13:27
• @user331377: It's up to you. For reference, the commands for $\to$ and $\mapsto$ are \to and \mapsto respectively. – Deusovi Apr 20 '16 at 13:30
• Are you only on this site for reputation points? Expect downvotes if you aren’t going to post quality answers. – Prince M Mar 28 '18 at 6:31

As it is evident from math.stackexchange notation — the symbol $\mapsto$ reads as "maps to".
This is backed up by Wikipedia article on functions:

... the notation $\mapsto$ ("maps to", an arrow with a bar at its tail) ...

There is another arrow-symbol, which also used for mapping $\rightarrow$, which might be a bit confusing. The difference between two (as it is mentioned in the linked answer, as well as in the answer by MathEnthusiast):

• $\mapsto$ maps an element of one set to an element of another set;
• $\rightarrow$ maps a set to a set.

Example (borrowed from here):

$$f:R \rightarrow R$$ $$x \mapsto x^2$$

It means that: under $f$, any element $x \subset R$ gets mapped to the element $x∗x=x^2$ (which is also an element of $R$).