We expand more of our knowledge of mathematical functions by discussing their notation - our way of naming it. Typically, we make use of mathematical symbols and expressions.

The goal of analytical notation is to simplify how we express it. It may be challenging to understand functions through words. In addition, it may be cumbersome if we use ordered pairs or mapping.

To illustrate, let's consider an algebraic equation with independent variable \(x\) and dependent variable \(y\): \(y=4x+5\). Let's express this in terms of a function using three notations. Despite different forms, they all mean the same.

## 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"
- \(y=…\) 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: \(f: X\rightarrow{Y}\)

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:

- \(A(r)=\pi{r^2}\)
- \(A(r)={(r,A|A=\pi{r^2})}\)
- \(f:r\rightarrow{A}\)

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) \(f:X\rightarrow{Y}\)

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