Relations and Functions

Representation of Functions

Suppose a printing machine prints 100 lines at the start and gradually increases its printing speed by 15 lines per sec. The printing speed does not count for any fraction of seconds. The representation of this statement can vary but the result will be the same. In this section, we will learn about the various representations of functions which will show the relationship between two elements of sets.

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Representation of Functions

The function is the connection or the link between two sets and can be represented in different ways. Consider the above example of the printing machine. The function that shows the relationship between the numbers of seconds (x) and the numbers of lines printed (y). We are quite familiar with functions and now we will learn how to represent them.

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Algebraic Representation

Here, we will represent the function by a simple algebraic equation which is: f(x) = y = 100 + 15(x). For different values of x, the values of y (= f(x)) change accordingly. What if, one wants to know about the numbers of lines printed by the machine in 15 seconds? Simple, Numbers of words printed = y = 100 + 15 (15) = 325.

Table Representation

In this method, we represent the relationship in the form of a table. For each value of x (input), there is one and only one value of y (output). The table representation of the problem:

x (second) y (number of lines)
1 100
2 130
4 160
6 190
8 220
10 250
12 280
14 310
15 325

Graphical Representation

Here, we will draw a graph showing the connection between the two elements of two sets say x and y such that x ∈ X and y ∈ Y. Plotting the satisfying points of x and y in the respective axes. Drawing a straight line passing through these points will represent the function in a graphical way. Graphical representation of the above problem:

representation of function

Solved Examples for You

Question 1: Consider an auto-driver who charges Rs. 15 for the first 7 km and subsequently charges an additional fare of Rs. 5 for each km. Find the cost one has to pay for 12 km by representing in tabular form.

Answer : Let x be the difference in distance (km) the auto ran and y be the fare of the auto for different distances (in Rs.). The equation of the given problem is: f(x) = y = 15 + 5 (x). The difference in distance travelled by auto is 12 – 7 = 5 km and we are required to find the fare for it. In tabular form, the above problem is

Actual distance x (difference in distance, km) y (fare in Rs.)
7 0 15
8 1 20
9 2 25
10 3 30
11 4 35
12 5 40

One has to pay Rs. 40 for travelling a distance of 12 km.

Question 2: What does representation mean?

Answer: Representation aids us to organize, record, and communicate mathematical ideas and it helps us to solve problems. In addition, we can represent mathematical ideas externally and internally. With these mathematical representation objects, we can represent objects and actions to make it easier to understand them.

Question 3: What is representation theory?

Answer:Itis a division of mathematics that helps us to study the intellectual algebraic constructions by expressing their elements as linear conversions of vector spaces, and studies modules over these abstract algebraic structures.

Question 4: What is a visual representation in math?

Answer: Usually, the data or information in mathematics is represented visually as this method organizes, extend, or replace other methods of presentation. In addition, visual representation in mathematics involves creating and forming models that can represent mathematical information, so that anyone can understand it easily.

Question 5: Why is it helpful to represent the same mathematical data in multiple ways?

Answer:  Multiple representations is a way to symbolize, describe and to refer to the same mathematical entity. Moreover, they are used to develop, top understand, and to communicate different mathematical features of the same object or operation, as well as connections between different properties.

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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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