### Theory:

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 \(A\).

which is not the image of any element of \(A\).