The functions are of four types, namely:
1. One - one function
2. Many - one function
3. Onto function
4. Into function
Let us look at each of the functions in detail.
One - one function
Let \(f : A \rightarrow B\) be a one-one function. Then, each distinct element in \(A\) will have a distinct image in \(B\).
Let us look at the representation given below for a better understanding.
In other words, if for all \(a_1\), \(a_2\) \(\in A\), \(f(a_1) = f(a_2)\) implies \(a_1 = a_2\), \(f\) is an one - one function.
A one-one function is otherwise called an injection.
Many - one function
Let \(f : A \rightarrow B\) be a many - one function. Then, two or more elements in \(A\) will have a distinct image in \(B\).
Onto function
A function \(f : A \rightarrow B\) is said to be onto function if the range of \(f\) is equal to the co-domain of \(f\).
That is, every co-domain will have a preimage.
An onto function is also called surjection.
Into function
A function \(f : A \rightarrow B\) is called an into a function if there exists at least one element in \(B\)
which is not the image of any element of \(A\).