# What does this notation mean? $x \mapsto f(x)$

What does this notation mean? $x \mapsto f(x)$

I've seen it at the beginning of functions but don't know what it is.

• It means $x$ in the domain gets sent to $f(x)$ in the codomain. – Michael Albanese Apr 8 '15 at 3:11
• It says that the if value of the input of your function $f$ is $x$, then the corresponding output is $f(x)$. – Lubin Apr 8 '15 at 3:11
• It may be useful to note how the symbol "$\mapsto$" is actually typeset: \mapsto. There's usually a reason for the name behind how symbols are typeset, especially in this case. \mapsto--maps to what? An element in the codomain. Maybe that will further cement the idea. :) – Daniel W. Farlow Apr 8 '15 at 3:47
• @YoTengoUnLCD: if you consider it tautologous, fair enough. (But there are people who would write their functions as $x\mapsto xf$ or $x\mapsto x^f$ or similar. Not stating this can lead to genuine confusion - does $fg$ mean "do $f$ then $g$" or "do $g$ then $f$"? Writing explicitly that you will denote the image of $x$ under $f$ by $f(x)$ avoids this potential confusion. Also, for instance, with more arguments, you might write $g(x,y)$, but would prefer to write $x*y$ rather than $*(x,y)$; writing this out explicitly avoids confusing your reader.) – Billy Apr 8 '15 at 3:48
• In an expression of the form $x \mapsto f(x)$, the RHS $f(x)$ can be replaced by any expression that involves $x$. e.g. $x \mapsto x^2 + 2x + 3$. You can think of this as a way to specify an "anonymous" function which send $x$ to whatever specified by RHS. Aside from algebra, this notation also appear in mathematical logic/computer science under the name Lambda calculus. – achille hui Apr 8 '15 at 5:34

The comments are spot-on, but I thought you might like a more or less "authoritative reference." The following is from John Durbin's Modern Algebra:

It is sometimes convenient to write $x\mapsto y$ to indicate that $y$ is the image of $x$ under a mapping.

In your example, $x\mapsto f(x)$ means the exact same thing; that is, $f(x)$ is the image of $x$ under a mapping.

To further solidify this reasoning, consider a very simple example. Say you have the sets $S=\{x,y,z\}$ and $T=\{1,2,3\}$, and define the mapping $\alpha\colon S\to T$ by $\alpha(x)=1,\alpha(y)=3,\alpha(z)=1$. Your question is about what, for example, $x\mapsto \alpha(x)$ means. Well, for the mapping $\alpha\colon S\to T$ defined above, you can see that $$x\mapsto 1\equiv x\mapsto\alpha(x),\quad y\mapsto 3\equiv y\mapsto\alpha(y),\quad z\mapsto 1\equiv z\mapsto\alpha(z).$$

Added: A somewhat less standard notation is $x\stackrel{\alpha}{\to}y$ to indicate that $y$ is the image of $x$ under the mapping $\alpha$--as you can tell, this communicates the same thing in terms of what is being mapped to where, but it specifies what mapping is being considered, something that is generally unnecessary to do because context makes it clear (this notation is also referred to in the text I mentioned).

The notation $\mapsto$ denotes the actual function mapping.

Any function $f$ from domain $A$ to codomain $B$ can be denoted as $f:A\to B$.

But if we'd like to specify the function, then we should specify what our arguments $a$ in $A$ get mapped to. Perhaps they could map to $2a$, or $a^2$, or $\sin a$, or any $f(a)$. That is, $a\mapsto f(a)$. Then, $f(a)$ is called the image of $a$ under $f$.

Now, this is usually denoted as $y=f(a)$, where for each argument $a$, there is a unique $y$ in $B$. The convenience of the notation $a \mapsto f(a)$ is that one does not have to create a new variable $y$ to serve as the image of $a$ under $f$.

It also allows for shorter, yet cluttered sentences. Instead of writing "Consider a function $f:A\to B$ defined by $f(a) = \cdots$," one may write "Consider the function $f: \underset{a}{A} \underset{\mapsto}{\to} \underset{f(a)}{B}$."

When speaking about functions, $a\mapsto f(a)$ by itself describes how the mapping is carried out, while $f:A\to B$ tells us which sets (or objects) $a$ is coming from and going to.

However, I've also used it, and have seen it used for substitutions in certain cases when an equals sign doesn't feel right.

For example, consider $a_n = a_{n-1} + n^2$. Then for $n\mapsto (n-1)$, we have $a_{n-1} = a_{n-2} + (n-1)^2$.

f(x) is a function. So you know when you write an equation for a line where y=mx+b? well f(x) is in place of y. f(x) is a term separate from x. f(x) is graphed on the y axis and x is graphed on the x axis. f(x) is a function. The f stands for function so it is written f(x) but when you say it out loud you say "f of x" or "function of x". It doesn't change the equation in any way. The only difference is that y is just used for lines but f(x) is used for things that have a physical purpose in the real world or represent something. f(x) also usually has a specific domain of values that are possible for x, as where y can have any value for x. An example is if you were graphing gravity's effect on a certain airborne object. Since the equation represents something in the real world or has a purpose other than graphing a line, you use f(x) instead of y.