Relation & Function

Relation & Function

 What is a relation? 

A relation between two sets is any ordered pairs containing one object from each set. The elements in the first set where the arrows come from are called domain. While the elements of the second set where the arrows end is called range. The reason or rule that you follow to make this mapping is the relation of the two sets. 

A relation may be classified as one-to-one, many-to-one, or one-to-many.

ONE-TO-ONE

One-to-one relation if every element in the domain is mapped to a unique element in the range.

MANY-TO-ONE

Many-to-one if two or more elements in the domain are mapped to only one element in the range.

ONE-TO-MANY

One-to-many if each element in the domain is mapped to any two or more elements in the range.

Representations of Relations

Aside from mapping, a relation may be represented in four other ways: table, ordered pairs, graph, and equation. 

Relation Expressed in a Table  

Relationship between two variables can also be presented by a table. The table below illustrates the relationship of the two quantities.  

Mr. Reyes is a dealer of poultry eggs. The table shows the cost for each number of trays. The no. of trays of eggs (x) and cost in pesos (y) presented horizontally and vertically 

Relations Expressed in Ordered Pairs  

Based on the sample problem above, the following ordered pairs were derived from the relationship between the number of trays and the total cost (no. of trays, total cost), then the ordered pairs are (1, 210), (2, 420), (3, 630), (4, 840), (5, 1050) …, and in general as (x, y).  

Relations Expressed as a Rule or Equation

A relation may also be presented through an equation in two variables or certain rule expressed in equation. Now let us model the relationship of total cost and the number of trays of eggs through the equation:   

Total Cost = (cost per tray of eggs) (number of trays) 

If:    

210 = 210 (1)

420 = 210 (2)  

630 = 210 (3)   

840 = 210 (4)  

1,050= 210 (5)  

Then, y  = 210 (x), is the rule expressed through equation in two variables. 

Relation Described Graphically  

The graphical representation on the picture illustrates the relationship between the total cost and the number of trays of eggs.

The x- axis represents the number of trays of eggs while the y- axis is the total cost in pesos. 

Relations that are Functions

  

A function is a special type of relation. It is a relation where no element in the domain is mapped or paired to more than one element in the range. Going back to the types of relations, which of them can be called function? 

By looking at the definition of a function, one-to-one relation and many-to-one relation are functions. One-to-many and many-to-many relations are just mere relation. 

The set of ordered pairs represents a function if no two distinct ordered pairs have the same abscissas (first elements). 

Example 1. Determine which of the following sets of ordered pairs is a function.  

     A= {(3, 5), (2, 7), (4, 6), (-3, -5)}  

     B= {(0, 1), (-1, 2), (0, 3), (-2, 4), (-1, 5)}  

     C= {(-1, 2), (1, -2), (-2,16), (2, -16), (-3, -54), (3,54),…)} 

Solution:  

Set A is a function since there are no distinct ordered pairs having the same first element.  

Set B is not a function since ordered pairs (0, 1) and (0, 3) have the same first element.  

Set C is an infinite set but the pattern shows that there are no abscissas with the same value; hence Set C is a function. 

Vertical Line Test

This is a test which uses vertical line to check whether the relation expressed in graph is a function or not. If every vertical line intersects the graph no more than once, then the graph represents a function.  

Solution:  

A. The graph in A is a function since the vertical line touches the graph at exactly one point even if it is moved to the right or to the left.  

B. The graph in B is not a function because if we examine the vertical line, it touches the graph more than once.  

C. Examining the vertical line in graph C, it touches the graph once. Hence, graph in C is a function.

Determine whether the given equation is a function or not

*If one value of the domain yields exactly one unique value in the range, then the equation represents a function. 

*If a rule or an equation represents a function, then it can be written in the form of y = f(x). 

Identifying whether the following real-life relationships are a function or a mere relation. 

The rule of function is also evident in some real-life relationships. For instance, the relationship of a child to his biological mother is a function since there is no child that has two or more biological mothers, the relation is one-to-one, thus, a function.  

Function Notation 

The function notation y = f(x), which is read as “y equals 𝑓 of x” or y is a function of x” is used to denote a functional relationship between x and y variables.  

 

If there is a rule relating y to x such as y = 5x + 2, and if the relation is a function, then you can also write this in function notation 𝑓(x) = 5x + 2. 𝑓(x) represents the value of the function at 𝑦. The name of the function is 𝑓.Other letters may be used to name functions.  

 

The domain of a function 𝑓 is the set of values of 𝑦 for which 𝑓 is defined. The range of a function 𝑓 is the set of all values of 𝑓(𝑦), where x is an element of the domain of 𝑓.  

PRACTICE ACTIVITY

Multiple Choice

Which of the following sets of ordered pairs represents a function?

• { (0,2), (1, 3), (2, 4), (3, 5), (4, 6) }

• { (0,1), (0, 2), (0, 3), (0, 4), (0,5 ) }

• { (0,2), (1, 3), (1, 4), (3, 5), (3, 6) }

• { (0,0), (4, 2), (4,-2), (9,3 ),(9,-3) }

Multiple Choice

Given the y = 7x – 5, which representation best describes the relation?

• Rule

• Table

• Graph

• Mapping/Diagram

Multiple Choice

Determine if graphs is a function or a mere relation.

• function

• mere relation

Multiple Choice

Determine if the graphs is a function or a mere relation.

• function

• mere relation

Multiple Choice

Determine if the graphs is a function or a mere relation.

• function

• mere relation

Multiple Choice

Determine if the graphs is a function or a mere relation.

• function

• mere relation

Multiple Choice

Given the function 𝑓(x) = 2x − 5, find the value of 𝑓(4).

• 7

• 0

• -3

• 3

Multiple Choice

$$Equation:y=3x–2,Findf(-2)$$  

• -8

• -4

• 4

• 8

Multiple Choice

$$Equation:\ y^2+x=2,Findf(-1)$$  

• $$\pm2$$

• $$\pm\sqrt{3}$$

• $$\pm1$$

• $$\pm3$$

Multiple Choice

 $Equation:y=2x^2+3x-1,Findf(0)$  

• 2

• 1

• 0

• -1