What Is Function And Relation

Relation and Function is an integral topic of algebra in the form 11 Maths syllabus. Every bit an important mathematical concept, it is also included under diverse scholastic and competitive exams. Often students are perplexed regarding the two essential concepts of Relation and Function. For all those going through like confusion, we have devised helpful notes in this blog which will help you lot empathise relations and functions.

Relation and Function Theory

In the relation and function theory, we always represent an ordered pair as (INPUT, OUTPUT). In this, the Relation shows the relationship between INPUT and OUTPUT. On the other hand, a role represents a relation which is derived through 1 OUTPUT for every unmarried INPUT.

All the functions are relations, simply non all the relations are functions.

Now that yous are familiar with the theory of relations and functions, let'south focus on the definition, types and examples to understand them better.

What is a Function?

A office tin can exist described every bit a relation which represents that there should be only one output for every input. In other words, functions can be understood as a unique relation which follows the rule i.eastward., each and every value of X should be related to only one value of Y, so it will be called as a role.

A function comprises of Domain and Range. A Domain is a collection of first values in the ordered pair i.e., it is the set of all the inputs of the 10 variable. Whereas Range refers to the collection of the 2nd values in the ordered pair i.eastward., information technology is the output of all the Y variables.

Domain	Range
-1	-3
1	3
3	9

Case of Functions:

For example, the Relation is {(-2,3), (4,5), (6,-5), (-2,3)}.

And so, the Domain would be {-ii, 4,6 } and {-5,3,5}.

Note: If any number appears more than in one case, it will only be written in one case in the domain as well as range.

Types of Functions

Now that you lot are familiar with its principal concept, mentioned beneath are some types of functions you must know about to understand relations and functions in a better fashion.

Injective Function or One to One Function: In a function f: P→Q is considered to be I to One function only if every element of P in that location is a distinct element of Q.

Many to One Function: A office in which ii or more than elements of set P are mapped to the same element of set Q.

Surjective Office or Onto Function: Information technology refers to a function where for every chemical element of set Q in that location is a pre-image in set up P.

Bijective Role or One-One and Onto Function: In the function f, each element of P is matched with a discrete element of Q and there is a preimage of every chemical element of Q in P.

What are Relations?

In one case you are familiar with functions, and then it will be easier to grasp the concept of relations. In simple terms, relations can be only understood as a subset of the Cartesian product or a bunch of points in an ordered pair.

For case: {(-2,1), (4,3), (vii,-3)}

Representation of Relations

Autonomously from the common prepare notation, at that place are many other means of representing a relation. Popular ways to do so are using tables, mapping diagram or plotting it on the XY axis.

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Types of Relations

Similar to the types of functions, there are various relations through which nosotros can larn nearly their dissimilar properties and these are:

Universal Relations
Empty Relations
Inverse Relations
Reflexive Relations
Identity Relations
Transitive Relations
Symmetric Relations

Recommended Read: Career afterward BSc Maths

Class 11 NCERT Book

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Relations and Functions: PDF

Solved Examples

Example 1: Place the range and domain the relation beneath: {(-2, iii), {four, v), (6, -5), (-ii,3)}

Solution: Since the x values are the domain, the respond is, therefore,
⟹ {-two, four, six}
The range is {-v, 3, 5}.

Instance 2: Check whether the following relation is a function: B = {(ane, 5), (one, 5), (3, -eight), (3, -8), (3, -8)}

Solution: B = {(1, 5), (one, 5), (3, -8), (3, -8), (iii, -viii)}
Though a relation is non classified as a role if there is a repetition of ten – values, this problem is a flake tricky because x values are repeated with their respective y-values.

Instance 3: Determine the domain and range of the following function: Z = {(1, 120), (2, 100), (3, 150), (four, 130)}.

Solution: Domain of z = {i, 2, 3, 4 and the range is {120, 100, 150, 130}

Example 4: Check if the following ordered pairs are functions:

West= {(1, 2), (2, 3), (3, 4), (4, 5)
Y = {(1, 6), (2, 5), (one, 9), (four, 3)}

Solution:

All the kickoff values in Due west = {(ane, 2), (2, 3), (3, 4), (4, 5)} are not repeated, therefore, this is a function.
Y = {(1, vi), (2, v), (one, 9), (4, 3)} is not a office considering the first value i has been repeated twice.

Example five: Decide whether the following ordered pairs of numbers are a function. R = (ane,1); (2,2); (three,1); (4,ii); (5,1); (6,vii)

Solution: There is no repetition of ten values in the given ready of ordered pairs of numbers.
Therefore, R = (one,ane); (2,2); (3,i); (four,two); (v,1); (6,vii) is a office.

Courtesy: Neha Agrawal Mathematically Inclined

Relations and Functions Practice Questions

1. Check whether the following relation is a function:

a. A = {(-3, -1), (2, 0), (5, one), (3, -eight), (6, -1)}

b. B = {(1, 4), (3, five), (one, -5), (three, -5), (i, five)}

c. C = {(5, 0), (0, v), (8, -8), (-8, viii), (0, 0)}

d. D = {(12, fifteen), (11, 31), (xviii, 8), (15, 12), (3, 12)}

2. The Cartesian product B x B has 9 elements among which are establish (–one, 0) and (0,1). Find the prepare B and the remaining elements of B x B.

3. Redefine the function: f(x) = |x – 1| – |x + 4|. Write its domain also.

4. Find the domain and range of the existent function f(x) = x/1+x2.

5. If A = {a, b, c, d} & B = {e, f, m}. Is R = {(a, due east) (a, f) (a, m) (b, eastward) (b, f) (b, grand) (c, e) (c, f) (d, g)} a function from A to B.Requite reasons to back up your respond.

6. Let D be the domain of existent valued role f defined by so, write D.

7. Let A = {a, b, c} and the relation R exist defined on A every bit follows:

R = {(a, a), (b, c), (a, b)}.

Then, write the minimum number of ordered pairs to be added in R to brand R reflexive and transitive.

eight. If A = {a, b, c, d} and the role f = {(a, b), (b, d), (c, a), (d, c)}, write f – one

9. exist divers past respectively. So detect g o f.

x. Is g = {(1, 1), (2, three, (3, 5), (4, 7)} a function? If g is described by 1000(x) = αx + β, and then what value should be assigned to α and β?

11. Determine the range and domains of the relation R divers by R = {(x – ane), (x + 2) : 10 ∈ (2, iii, four, 5)}

12.Let A = {three, 4, 5} and B = {half-dozen, 8, 9, 10, 12}. Permit R be the relation 'is a factor of' from A to B. Observe R.

Relation and Function Worksheet

We hope that know yous are totally clear near the concept of relations and functions. If you want to seek practiced aid regarding which career path is all-time for y'all after class twelfth, reach out to our Leverage Edu experts and we will guide you in making an informed decision towards a rewarding career. Sign upward for an e-coming together with us today!