Concept of Functions

Let A and B be any two non–empty sets. Then a function ‘$$f$$’ is a rule or law which associates each element of ‘A’ to a unique element of set ‘B’.

Notation:
(i) A function is usually denoted by small letters, i.e. $$f,g,h$$ etc. and Greek letters, i.e. $$\alpha ,\beta ,\gamma ,\phi ,\psi $$etc.
(ii) If ‘$$f$$’ is a function from ‘A’ to ‘B’ then we write $$f:{\text{A}} \to {\text{B}}$$.

Ordered Pair:
Let ‘a’ and ‘b’ be any two elements then an element$$\left( {{\text{a}},{\text{b}}} \right)$$ is called an ordered pair.

Cartesian product:
e.g.      $${\text{A}} = \left\{ {1,2,3} \right\}$$, \[{\text{B}} = \left\{ {{\text{a}},{\text{b}},{\text{c}}} \right\}\]
$${\text{A}} \times {\text{B}} = \left\{ {\left( {1{\text{,a}}} \right),\left( {1,{\text{b}}} \right),\left( {1,{\text{c}}} \right),\left( {2,{\text{a}}} \right),\left( {2,{\text{b}}} \right),\left( {2,{\text{c}}} \right),\left( {3,{\text{a}}} \right),\left( {3,{\text{b}}} \right),\left( {3,{\text{c}}} \right)} \right\}$$

Let ‘A’ and ‘B’ be any two non–empty sets, then the set of all those elements of the form$$\left( {{\text{a}},{\text{b}}} \right)$$, where $${\text{a}} \in {\text{A}}$$, $${\text{b}} \in {\text{B}}$$ is called a Cartesian product.
It is denoted by $${\text{A}} \times {\text{B}} = \left\{ {\left( {{\text{a}},{\text{b}}} \right)\left| {{\text{a}} \in {\text{A}}} \right.,{\text{b}} \in {\text{B}}} \right\}$$

Binary Relation:
Let ‘A’ and ‘B’ be any two non–empty sets, then every sub–set of $${\text{A}} \times {\text{B}}$$ is called a binary relation from A to B.
It is denoted by ‘$$R$$’i.e.    $$R \subseteq {\text{A}} \times {\text{B}}$$
e.g.      $${\text{A}} = \left\{ {1,2,3} \right\}$$,                  \[{\text{B}} = \left\{ {{\text{a}},{\text{b}},{\text{c}}} \right\}\]
$${\text{A}} \times {\text{B}} = \left\{ {\left( {1{\text{,a}}} \right),\left( {1,{\text{b}}} \right),\left( {1,{\text{c}}} \right),\left( {2,{\text{a}}} \right),\left( {2,{\text{b}}} \right),\left( {2,{\text{c}}} \right),\left( {3,{\text{a}}} \right),\left( {3,{\text{b}}} \right),\left( {3,{\text{c}}} \right)} \right\}$$
$$R = \left\{ {\left( {1{\text{,a}}} \right),\left( {2,{\text{b}}} \right),\left( {3,{\text{c}}} \right)} \right\} \subseteq {\text{A}} \times {\text{B}}$$
‘$$R$$’ is a binary relation from A to B.

Function as a Binary Relation:
Let ‘A’ and ‘B’ be any two non–empty sets, then a binary relation ‘$$R$$’ from ‘A’ to ‘B’ is called a function if it satisfied the following two conditions.
(i) Domain of $$R = {\text{A}}$$,   i.e. $${{\text{D}}_{\text{A}}} = {\text{A}}$$
(ii) For each element ‘x’ of ‘A’ there exists a unique element $${\text{y}} \in {\text{B}}$$ such that
$$\left( {{\text{x}},{\text{y}}} \right) \in R$$

Domain of a Relation:
The set of the first elements of all ordered pairs in a relation is called the domain of a relation.
e.g.$$R = \left\{ {\left( {1,2} \right),\left( {3,4} \right),\left( {5,6} \right)} \right\}$$
Dom$$R = \left\{ {1,3,5} \right\}$$

Range of a Relation:
The set of the second elements of the ordered pair in a relation is called the range of a relation.
e.g.$$ R\left\{ {\left( {1,{\text{a}}} \right),\left( {2,{\text{b}}} \right),\left( {3,{\text{c}}} \right)} \right\}$$
Range$$ R = \left\{ {{\text{a}},{\text{b}},{\text{c}}} \right\}$$

Domain of the Function:
e.g.      $${\text{y}} = \frac{1}{{{\text{x}} – 2}}$$
For$${\text{x}} = 0\;\;\;\therefore {\text{y}} = \frac{1}{{0 – 2}} = – \frac{1}{2} \in \mathbb{R}$$
For$${\text{x}} = 1\;\;\;\therefore {\text{y}} = \frac{1}{{1 – 2}} = – 1 \in \mathbb{R}$$
$$\boxed{{\text{For x}} = 2,\;\;\;\therefore {\text{y}} = \frac{1}{{2 – 2}} = \frac{1}{0} = \infty }$$
For $${\text{x}} = 3,\;\;\;\therefore {\text{y}} = \frac{1}{{3 – 2}} = \frac{1}{1} = 1 \in \mathbb{R}$$
Domain$$ = \left\{ {0,1,3, \cdots } \right\}$$

For the set of all values of ‘x’ for which ‘y’ is finite, the definite real number is called the domain of the function.