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) Function is usually denoted by small letters i.e. f,g,hetc and the Greek letter i.e. \alpha ,\beta  ,\gamma ,\phi ,\psi etc.
(ii) If ‘f’ is a function from ‘A’ to ‘B’ then we writef:{\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 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 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 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 iff 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 exist 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 pair in a relation is called, 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, 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\}
The set of all those values of ‘x’ for which ‘y’ is finite, definite real number is called domain of the function.

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