What are functions in mathematics? A function is a relation in which each element in the domain is mapped onto one and only one member in the co-domain. Thus, for every element of the domain, there is exactly one image in the co-domain. Hold on, before you despair, it isn’t as complicated as you think at all. let’s break it down. Consider the following relations:

The relations above are both functions. This is because each member of the domain maps onto only one member of the co-domain or range. In this scenario, the domains are the set of values on the left and the co-domains are the set of values on the right.

In fig. 1, the elements “a”, “b” and “c” all have only one image, which is “d”. Similarly in fig. 2 both “p” and “q” have the element “s” as their image in the co-domain while “r” has the element “t” as its image. So still each element in the domain has only one element in the co-domain.

Now, consider the following relations:

The above relations are not functions. Looking at the relation in fig. 3, we notice that the element “b” of the domain has no image in the co-domain. So for a relation to qualify as a function, all members of the domain must have corresponding images in the co-domain.

Also, in the fig. 4 relation, an element of the domain, that is “q”, has more than one image. These images of “q” are “s” and “t”. Furthermore, if an element in the domain of a relation has more than one image in the co-domain, then it does not qualify as a function.

NOTATIONS FOR FUNCTIONS IN MATHEMATICS

Now, let X and Y be two sets. In such a case, a very important type of relationship between the two sets X and Y is formed when each member of the set X is related to one and only one member of the set Y. Also, as described above, this type of relation is known as a function. It follows that this is a function from X to Y.

It is written as “

Graphically it is shown as below:

Furthermore, it can be written as the equation “

In this case, the set X is called the domain of “

From the ongoing discussion, it is evident that only two types of relations qualify as functions. These are one-to-one relations and many-to-one relations. Now let’s look at the types of functions.

TYPES OF FUNCTIONS IN MATHEMATICS

Now that we have a clearer picture of what functions really are, let’s look at some types of functions.

ONE-TO-ONE FUNCTION

A one-to-one function is a function in which each element in the domain has only one image in the co-domain and each element in the co-domain is associated with only one element in the domain.

Thus, each element in the domain is mapped onto different elements in the co-domain.

Now, the function displayed in Fig. 5 above is a one-to-one function since each element in the domain has only one image in the co-domain and each element in the co-domain is associated with only one element in the domain.

But the function displayed in Fig. 6 below is not a one-to-one function since two distinct elements “p” and “q” in the domain of the function of “g” have the same image “t”.

ONTO FUNCTIONS

A function is said to be an onto function if and only if the range of “

MANY-TO-ONE FUNCTION

Just like the many-to-one relation, a many-to-one function has several elements in the domain which have the same image in the co-domain. Therefore, fig. 6 above is an example of a many-to-one function. This is because both “p” and “q” have the same image “t” in the co-domain.

EVEN FUNCTIONS

Also a function “

ODD FUNCTIONS

Now a function “

Conclusion: Functions of mathematics

I hope you enjoyed reading about the basics of functions in mathematics. Functions make math interesting. Many other mathematical concepts are based on functions.

If you have any questions or comments, leave it in the comments section below. Don’t forget to keep loving math. If you enjoyed this article do justice to math and share it with your friends.

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