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\}
DomR = \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.