0Functions :
A function can have the same range mapped as that of in relation, such that a set of inputs is related to exactly one output. A function f from a set A to a set B is a rule which associates each element of set A to a unique element of set B.
• Set A is domain and set B is codomain of the function
• Range is the set of all possible resulting values given by the function. For example: 2 x is a function where values of x will be the domain and value given by 2 x is the range.
Types of Function:
1. One-One Function:
• A function f from set A to set B is called one-one function if no two distinct elements of A have the same image in B.
• Mathematically, a function f from set A to set B if f x f y ( ) = ( ) implies that x y = for all x y A , .
• One-one function is also called an injective function.
• For example: If a function f from a set of real numbers to a set of real numbers, then f x x ( ) = 2 is one-one function.
Onto Function:
• A function f from set A to set B is called onto function if each element of set B has a preimage in set A or range of function f is equal to the codomain i.e., set B.
• Onto function is also called surjective function.
• For example: If a function f from a set of natural numbers to a set of natural numbers, then f x x ( ) = −1 is onto the function. 3. Bijective Function:
• A function f from set A to set B is called a bijective function if it is both one-one function and onto function.
• For example: If a function f from a set of real numbers to a set of real numbers, then f x x ( ) = 2 is one-one function and onto function.
Composition of function and invertible function
• Composition of function: Let f A B : → and g B C : → then the composite of g and f , written as g f is a function from A to C such that ( g f a g f a = )( ) ( ( )) for all a A .
• Properties of composition of function: Let f A B : → , g B C : → and h C A : → then a. Composition is associative i.e., h gf hg f ( ) = ( ) b. If f and g are one-one then g f is also one-one c. If f and g are onto then g f is also onto.
Invertible function:
If f is bijective then there is a function 1 f B A : − → such that ( )( ) 1 f f a a − = for all a A and ( )( ) 1 f f b b − = for all b B 1 f − is the inverse of the function f and is always unique.
