Functions
1.0 Definitions
2.0 Relation
3.0 Types of Relation
4.0 Functions
5.0 Standard Real Functions and their Graphical Representation
5.10 Reciprocal Function
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
6.0 Operations on Real Functions
7.0 Types of Functions
8.0 Composition of a Function
9.0 Inverse of a Function
2.3 Domain and Range of a Relation
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
Let $R$ be a relation from $A$ to $B$. Then the set of all first components or co-ordinates of the ordered pairs belonging to $R$ is the domain. It is represented as $Dom(R)$.
The second component of the ordered pairs in $R$ is called the range of $R$. It is represented as $Range(R)$
Thus,
$$Dom(R) = \{ a:(a,b) \in R\} $$
and
$$Range(R) = \{ b:(a,b) \in R\} $$
The set $B$ is also at times called as a co-domain of $R$.
Example 12. Find the domain and range of the following relations,
| i. $R = \{ (2,3),(4,5),(3,2),(5,4)\} $ | ii. $R = \{ (1,6),(2,7),(3,8),(4,9)\} $ |
| iii. $R = \{ (x,y):x,y \in A,\;b\;is\;a\;multiple\;of\;a\} $ and $R$ is a relation defined on set $A = \{ 1,2,3,4,5,6\} $ | iv. A relation $R$ on the set $A = \{ 0,1,2,...,9,10\} $ defined by $2x + 3y = 12$ |
| v. $R = \{ (x,x + 5):\;x \in \{ 0,1,2,3,4,5\} \} $ |
Solution:
| Relation | Domain | Range |
| i. $R = \{ (2,3),(4,5),(3,2),(5,4)\} $ | Dom(R)={2,3,4,5} | Range(R)={2,3,4,5} |
| ii. $R = \{ (1,6),(2,7),(3,8),(4,9)\} $ | Dom(R)={1,2,3,4} | Range(R)={6,7,8,9} |
iii. $R = \{ (x,y):x,y \in A,\;b\;is\;a\;multiple\;of\;a\} $ and $R$ is a relation defined on set $A = \{ 1,2,3,4,5,6\} $ | $R = \{ (1,1),(1,2),(1,3),(1,4),(1,5),$ $(1,6),(2,2), (2,4),(2,6),(3,3),(3,6),$ $(4,4),(5,5),(6,6)\} $ Dom(R)={1,2,3,4,5,6} | $R = \{ (1,1),(1,2),(1,3),(1,4),(1,5),(1,6),$ $(2,2), (2,4),(2,6),(3,3),(3,6),$ $(4,4),(5,5),(6,6)\} $ Range(R)={1,2,3,4,5,6} |
| iv. A relation $R$ on the set $A = \{ 0,1,2,...,9,10\} $ defined by $2x + 3y = 12$ | Dom(R)={0,3,6} | Range(R)={4,2,0} |
| v. $R = \{ (x,x + 5):\;x \in \{ 0,1,2,3,4,5\} \} $ | Dom(R)={0,1,2,3,4,5} | Range(R)={5,6,7,8,9,10} |
Relation on a set:
Let $A$ be a non-void set. Then a relation from $A$ to itself i.e. a subset of $A \times A$ is called a relation on set $A$.
