How to Name Mathematical Functions?

Notation 1: $f(x)=y$

This notation is the most common one out there. Using our example, the notation would be: 

$f(x)=4x+5$

We read this $f(x)$ as “function of x (or f of x) is four x plus five (the expression)" 

• The value $x$ is your input value,
• $f$ is the function which is $4x+5$,
• $f(x)$ is the output,

Usually, we equate $f(x)$, the output value in a function, with $y$, the dependent variable in an equation; Because of this, we often interchange $f(x)$ and $y$. Similarly, we can also say that $x$, the input value in a function, is the same as $x$, the independent variable in an equation.

Notation 2: $f(x)=(x,y)|y=...$

Sometimes, we'll see notations similar to this one. Using our example, the naming would be:

$f(x)=(x,y|y=4x+5)$

We read this as: "f of x is the set of ordered pairs (x,y) such that y=…".

To understand this notation, we must view the "y=..." as an equation or a formula. Not a function by itself! (there's a difference between them). Furthermore, let's look at it part by part: 

• $f(x)$ is the output value, 
• $(x,y)$ is an ordered pair, 
• The vertical bar means "such that"
• $y=\dots$ represents the equation

Here, the ordered pair indicates that whatever equation is after the vertical bar, we have to view it as a function.

Notation 3: $f:X\to Y$

Another notation we might encounter is by their sets; Looking into it, capital letters $X$ and $Y$ here are groups and not individual entries. Additionally, the arrow sign means "mapped to."

This notation means the following: (1) the function is an ordered pair between the elements of sets $X$ and $Y$, or (2) the elements of $X$ are mapped to the elements of $Y$.

Can I Only Use $f$?

We can use other letters (or even name them using words) to name functions (not just $f$). It's up to one's preference on how to call it. 

Let's apply what we have learned to the radius-area ($r$, $A$) relation, $A=\pi r^2$. The three possible notations are:

• $A(r)=\pi r^2$
• $A(r)=(r,A|A=\pi r^2)$
• $f:r\to A$

We use $r$ and $A$ instead of $x$ or $y$ for us to remember what variables we are referring to easily. 

There are a lot of other ways on how to name functions. Unless otherwise stated, we'll stick with the most common notation: $f(x)=y$.

Summary

There are many function notations - ways of naming them.

The goal of analytical notation is to simplify how we express it.

The three common notations are: (1) $f(x)=y$, (2) $f(x)=(x,y|y=...)$, and (3) $f:X\to Y$

We can use other letters (or even name them using words) to name functions