In the previous lesson, you learned about Cartesian coordinate system, graph of equations and linear equations. These are prerequisites to learn about function in this lesson.

A function is a relation between two sets – set A and set B, where each element of set A is related to exactly one element of set B. Note the word “*exactly*” which is very important.

The elements is called domain of function (or inputs) and the elements is called the range of function( or outputs).

## Rules for Function

A function must follow certain rules, otherwise it is not a function.

- Every element of set A must be assigned to an element of set B.
- Each element of set A must match with an element in set B.
- An element of set B can be output of two or more elements of set A.
- An element of set A cannot be input of two or more elements of set A. (See rule 2).

### What are the different ways to represent functions ?

There are 3 ways to represent functions.

**Numerically** – you can represent a function as set of ordered pairs (x, y) where x is the domain of the function and y is the range of the function.

For example, you can represent above diagram as

$\left\{\right(1,A),(2,B),(x,y\left)\right\}$

**Algebraically** – you can represent a function using algebraic notations. A function is denoted as .

For example, you are given an expression .

The above expression can be changed to following form

$y=1\u2013x$

or

$f\left(x\right)=1\u2013x$

**Graphically** – you can represent a function using graph. A reveal lot of information about function such a increasing function, decreasing function, odd or even and other characteristics.

To represent a function using graph find the points required to plot the graph of the function. This is done using algebraic expression of the graph.

For example, let’s draw the graph of .

x |
f (x) |
points (x, (f (x)) |

-1 | 2 | (-1, 2) |

0 | 1 | (0, 1) |

1 | 0 | (1, 0) |

2 | -1 | (2, -1) |

The graph of the function using above points is given below.

## Piecewise-Defined Functions

A domain of a function is all real numbers. Sometimes a function is piecewise-defined. It means there are more than one function for different values of domain.

You need to specify two or more equations to represent a function with specific domain values.

For example,

$f\left(x\right)=\left\{\begin{array}{ll}{x}^{2}\u20131& x0\\ x\u20131& x\ge 0\end{array}\right)$

**Points to Remember**

- If x value falls under the domain, then the function f(x) is defined for x and if x does not falls under the domain of f(x) then it is undefined for x.
- When the domain of the function is not mentioned then the implied domain is set of all real numbers.
- The range of the function must be a possible value after inserting x into the function f (x). The range falls between to