A Function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. The third and final chapter of this part highlights the important aspects of functions.
Function – Definition
A function or mapping (Defined as) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called Codomain of function ‘f’.
Function ‘f’ is a relation on X and Y such that for each, there exists a unique such that . ‘x’ is called pre-image and ‘y’ is called image of function f.
A function can be one to one or many to one but not one to many.
Injective / One-to-one function
A functionis injective or one-to-one function if for every , there exists at most one such that .
This means a function f is injective ifimplies .
- is injective.
- is injective.
- is not injective as
Surjective / Onto function
A functionis surjective (onto) if the image of f equals its range. Equivalently, for every , there exists some such that . This means that for any y in B, there exists some x in A such that .
- is surjective.
- is not surjective since we cannot find a real number whose square is negative.
Bijective / One-to-one Correspondent
A functionis bijective or one-to-one correspondent if and only if fis both injective and surjective.
Prove that a functiondefined by is a bijective function.
Explanation − We have to prove this function is both injective and surjective.
If, then and it implies that .
Hence, f is injective.
So,which belongs to R and .
Hence, f is surjective.
Since f is both surjective and injective, we can say f is bijective.
Inverse of a Function
The inverse of a one-to-one corresponding function, is the function , holding the following property −
The function f is called invertible, if its inverse function g exists.
- A Function , is invertible since it has the inverse function .
- A Function is not invertible since this is not one-to-one as .