# 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, 2016 at 13:11
• @lhf Could you please explain this for a beginner? Thanks! Apr 20, 2016 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. Apr 20, 2016 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. Apr 20, 2016 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.) Apr 20, 2016 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 state your source? Thanks :) Apr 20, 2016 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. Apr 20, 2016 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, 2016 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 $0$. Jul 29, 2016 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 Apr 20, 2016 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. Apr 20, 2016 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 $x\mapsto x^2: \mathbb R \to \mathbb R^+$. Apr 20, 2016 at 13:27
• @user331377: It's up to you. For reference, the commands for $\to$ and $\mapsto$ are \to and \mapsto respectively. Apr 20, 2016 at 13:30
• Are you only on this site for reputation points? Expect downvotes if you aren’t going to post quality answers. Mar 28, 2018 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 $R$).