Function notation is the expression used to represent a function. A function can be represented in various ways. The function notations give some information about a function. Every function has an inverse function for it, which has a notation like $f^{-1}(x)$. After the completion of this article you will be able to:

1) Define function notation.

2)
Know the advantages of function notation.

3)
Write the same function in different notations.

4)
Understand the inverse function and its notation.

A function notation gives the information about the functions.

For example, an equation is written as $y$ = $2x\ +\ 3$, while the function notation $f(x)$ = $2x\ +\ 3$ shows that it is a function of $x$ depending on the values of $x$. The value of $f(2)$ will be the value of the expression $(2x\ +\ 3)$ at $x$ = $2$.

Given below are different function notations:

1) $f(x)$ = $2x\ +\ 3$

2) $f:\ x\ \rightarrow\ 2x\ +\ 3$

3) $f$ = $\{(x,\ y)|y$ = $2x\ +\ 3\}$

4) $f\overset{x}{\rightarrow}(2x+3)$

Each notation is given some information about the function.
The functional notation gives us information about the function.

For example, $f(x,\ y,\ z)$ = $3x\ +\ 7y\ -\ 2z$ tells that the function has three variables $x,\ y,$ and $z$ and the graph will be a three-dimensional graph.

If a function is given as $f:\ x\ \rightarrow\ x^{2}\ -\ 1$, then the value of $f(2)$ is $2^{2}\ -\ 1$ = $4\ -\ 1$ = $3$. The function notation makes it easier to find the value of function at a certain point.
Example: Find the value of f = {(x, y)|y = $x^2 + 3x - 3$} at x = 2.

Solution:

$y$ = $x^{2}\ +\ 3x\ -\ 3$
       
      = $2^{2}\ +\ 3.2\ -\ 3$ = $4\ +\ 6\ -\ 3$ = $7$

The value of function at $x$ = $3$ is $7$.
When a function is injective, there exists one and only one inverse for it and the function is called an invertible function. Thus, if $f(x)$ = $y$ is invertible, then there exists a unique function g such that $g(y)$ = $x$.

The function $y$ = $f(x)$ has an inverse which can be represented as $f^{-1}(x)$. The inverse of the function $y$ = $f(x)$ will be the function $x$ = $f(y)$. This function is represented as $f^{-1}(x)$. Let a function be f(x) = $x^2$. Then the inverse function will be $f^{-1}(x) = \sqrt{x}$.
Let us few examples and understand:

Example 1: Find the value of $f(-2)$ for the given functions.

a) $f(x)$ = $\frac{3x - 2}{x+1}$

b) $f:\ x\ \rightarrow\ 2x\ +\ 3e^{x}$

Solution:

To find the value of $f(-2)$ for these functions, we should put $x$ = $-2$ in each function and evaluate.

a) $f(x)$ = $\frac{3x - 2}{x+1}$

Putting $x$ = $-2$,

$f(-2)$ = $\frac{3\times (-2) - 2}{-2+1}$ = $\frac{-8}{-1}$ = $8$.

b) $f:\ x\ \rightarrow\  2x\ +\ 3e^{x}$

This can be written as $f(x)$ = $2x\ +\ 3e^{x}$

Putting $x$ = $-2$,

$f(-2)$ = $2\ \times\ -\ 2\ +\ 3e^{-2}$ = $-3.59399415029$
Example 2: Match the functions in column A to their equivalent functions in column B.

A B
$f(x)$ = $\frac{3x-4y}{3y+2x}$ $f(x)$ = $e^{3x} - 2x$
$f$ = ${(x, y)|y = 3x^3 - 2x^2 -1}$ $f\overset{x}{\rightarrow}(2x+3logx)$
$f:\ x\ \rightarrow\ 2x\ +\ 3logx$ $f\overset{x,y}{\rightarrow}$$(\frac{3x-4y}{3y+2x})$
$f\overset{x}{\rightarrow}(e^{3x}-2x)$ $y$ = $3x^3 - 2x^2 -1$

Solution:

The functions equivalent to each other are given here.

$A1\ \rightarrow\ B3$

$A2\ \rightarrow\ B4$

$A3\ \rightarrow\ B2$

$A4\ \rightarrow\ B1$