Relations and Functions


Have you ever thought that a particular person has particular jobs or functions to do? Consider the functions or roles of postmen. They deliver letters, postcards, telegrams and invites etc. What do firemen do? They are responsible for responding to fire accidents. In mathematics also, we can define functions. They are responsible for assigning every single object of one set to that of another.

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A function is a relation that maps each element x of a set A with one and only one element y of another set B. In other words, it is a relation between a set of inputs and a set of outputs in which each input is related with a unique output. A function is a rule that relates an input to exactly one output.


It is a special type of relation. A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B and no two distinct elements of B have the same mapped first element.  A and B are the non-empty sets. The whole set A is the domain and the whole set B is codomain.

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A function f: X →Y is represented as f(x) = y, where, (x, y) ∈ f and x ∈ X and y ∈ Y.

For any function f, the notation f(x) is read as “f of x” and represents the value of y when x is replaced by the number or expression inside the parenthesis. The element y is the image of x under f and x is the pre-image of y under f.


Every element of the set has an image which is unique and distinct. If we notice around, we can find many examples of functions.

If we lift our hand upward, it is a function. Waving our hand freely, it is a function.  A walk in a circular track, yes it is a type of function. Now you can think of other examples too! A graph can represent a function. The graph is the set of all pairs of the Cartesian product.

Does this mean that every curve in the world defines a function? No, not every curve drawn is a function. How to find it? Vertical line test. If any curve intercepts a vertical line at more than one point, it is a curve only not a function.

Solved Example for You

Question 1: Which of the following is a function?


3.Answer : Figure 3 is an example of function since every element of A is mapped to a unique element of B and no two distinct elements of B have the same pre-image in A.

Question 2: Give an example of a function?

Answer: An example of a function is the relationship x → x. The reason for this is that every element in x has a relation with y. Moreover, no element in x has two or more than two relationships.

Question 3: Explain what is a function and what is not?

Answer: A function refers to a relation such that every input has only one output. For example, y is a function of x and x is not a function of y (y = 9 consist of multiple outputs). Moreover, y is not a function of x (x = 1 consist of multiple outputs), x does not happen to be a function of y (y = 2 has multiple outputs).

Question 4: What is the classification of functions?

Answer: The classification of function takes place by the type of mathematical equation which shows their relationship. Some functions can be algebraic. Trigonometric functions like f(x) = sin x are those that involve angles. Some functions have logarithmic and exponential and logarithmic relationships and their classification are as such.

Question 5: What are basic polynomial functions?

Answer: (x) =c, f(x) =x, f(x) =x2, and f(x) =x3 are basic polynomial functions.

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2 responses to “Relations”

  1. Al quba says:

    The example for onto function doesn’t qualify as a function in the first place. Does it??
    It is a relation but not a function because a single element in the domain has been mapped to two elements in the co domain. Isn’t it??
    Please tell me if I’m correct or not.
    It is really confusing.

    • Eli Simeon says:

      An onto function exists if and only the co-domain is equal to the range that is every element in set A (the domain) is mapped to every element in set B (the range/codomain) i.e without leaving out any element. Irrespective of whether it is a one to one mapping or not. Therefore it is a function. Put simply, take set A as a set of sons and set B as a set of fathers, a function requires that every son has one father (which is normal) yet every father can have more than one son(which is also normal) so if one element in set A maps to more than one element in set B it is not a function (and we will need DNA test to know who really is the father) but if more than one element in set A maps to one element in set B it is still a function (the elements are just brothers). What and onto function requires is that every father has a son. Sorry if I made it a bit complicated I feel that if I continue I might make it worse just it study a bit more and from different sources, videos or books and you will understand it better.

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