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Relations and Functions

Relations and Functions are important concepts in mathematics, especially for students preparing for JEE Mains. In this blog post, we will explore what relations and functions are, the different types of relations and functions, and how to solve problems related to them.


A relation is a set of ordered pairs. It is simply a way of showing how two sets of elements are related to each other. For example, the relation "is greater than" between two numbers can be represented as a set of ordered pairs, such as (2, 3), (4, 5), etc.


A function is a special type of relation where each element of the first set, known as the domain, is related to exactly one element in the second set, known as the range. In other words, a function assigns a unique output to each input. For example, the function f(x) = 2x + 1 assigns the value of 2x + 1 to each input x.

A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function.


NECESSARY CONDITIONS FOR FUNCTION


  • Set A and B must be non-empty sets


TYPES OF FUNCTIONS


There are various types of functions in mathematics which are explained below in detail. The different function types covered here are:

  • One – one function (Injective function)

If each element in the domain of a function has a distinct image in the co-domain, the function is said to be a one-one function



  • Many – one function

If there are at least two elements in the domain whose images are the same, the function is known as many to one.



  • Onto–function (Surjective Function)

A function is called an onto function if each element in the co-domain has at least one pre–image in the domain.



  • Into – function

If there exists at least one element in the co-domain which is not an image of any element in the domain then the function will be Into function.



  • Algebraic Functions

A function that consists of a finite number of terms involving powers and roots of independent variable x and fundamental operations such as addition, subtraction, multiplication, and division is known as an algebraic equation.

Cubic Function

A cubic polynomial function is a polynomial of degree three and can be expressed as;

F(x) = ax3 + bx2 + cx + d and a is not equal to zero.


  • Greatest Integer Function

The greatest integer function always gives integral output. The Greatest integral value that has been taken by the input will be the output.

For example: [4.5] = 4

[6.99] = 6



  • Fractional Part Function

It always give fractional value as output.

Example

{4.5} = 0.5

{6.99} = 0.99




  • Even and Odd Function

If f(x) = f(-x) then the function will be even function & f(x) = -f(-x) then the function will be odd function

Example

f(x) = x2sinx

f(-x) = -x2sinx


  • Periodic Function

A function is said to be a periodic function if there exist a positive real number T such that f(u – t) = f(x) for all x ε Domain.

Example

f(x) = sinx

f(x + 2π) = sin (x + 2π) = sinx fundamental

then period of sinx is 2π

  • Composite Function

Let A, B, C be three non-empty sets

Let f: A B & g : G C be two functions then gof : A C. This function is called composition of f and g

given g of (x) = g(f(x))

Example

f(x) = x2 & g(x) = 2x

f(g(x)) = f(2x) = (2x)2 = 4x

  • Constant Function

P= set of real numbers

The function f : P → P defined by b = f (a) = a for each a ϵ P is called the identity function.

  • Domain of f = P

  • Range of f = P

  • Graph type: A straight line passing through the origin.

  • Identity Function


To solve problems related to relations and functions, it is important to understand the different types of relations and functions and how they work. For example, if you are given a relation and asked to determine if it is a function, you can use the vertical line test.


If a vertical line can intersect the relation in more than one point, it is not a function. In addition, it is also important to be familiar with the mathematical operations and techniques used to manipulate relations and functions, such as composition of functions, inverse functions, and domain and range.

In conclusion, understanding relations and functions is important for students preparing for JEE Mains as it is a fundamental concept that is heavily tested in the exam. By understanding the different types of relations and functions and practicing solving related problems, students can improve their chances of success in the exam.


MCQs on Relations and Functions:


1. Which of the following is a relation?

a) y = 2x + 1

b) x^2 + y^2 = 4

c) x + y = 3

d) All of the above


2. A function is a _________ of ordered pairs.

a) set

b) pair

c) both a and b

d) None of the above


3. A function is one-to-one if:

a) No two different inputs have the same output.

b) Two different inputs have the same output.

c) No two inputs have the same output.

d) None of the above


4. A function is onto if:

a) Every element in the range has at least one corresponding element in the domain.

b) Every element in the domain has at least one corresponding element in the range.

c) Every element in the domain and range has at least one corresponding element.

d) None of the above


5. A function is bijective if it is:

a) one-to-one and onto.

b) one-to-one and not onto.

c) many-to-one and onto.

d) None of the above


6. What is the vertical line test used for?

a) Determining if a relation is a function.

b) Determining if a function is a one-to-one.

c) Determining if a function is onto.

d) None of the above


7. Which of the following is an inverse function?

a) f(x) = 2x + 1

b) f(x) = x^2

c) f(x) = 2x

d) f(x) = 1/x


8. What is the domain of a function?

a) The set of all outputs of a function.

b) The set of all inputs of a function.

c) The set of all ordered pairs of a function.

d) None of the above


9. What is the range of a function?

a) The set of all outputs of a function.

b) The set of all inputs of a function.

c) The set of all ordered pairs of a function.

d) None of the above


10. What is the composition of functions?

a) The combination of two or more functions.

b) The inverse of a function.

c) The domain and range of a function.

d) None of the above



Answers:

  1. d) All of the above

  2. a) set

  3. a) No two different inputs have the same output.

  4. a) Every element in the range has at least one corresponding element in the domain.

  5. a) one-to-one and onto

  6. a) Determining if a relation is a function

  7. d) f(x) = 1/x

  8. b) The set of all inputs of a function

  9. a) The set of all outputs of a function

  10. a) The combination of two or more functions.


Problems On Relations and functions:

  1. Prove that the relation R = {(a, b) | a - 2b = 0} is a function.

  2. Determine if the relation R = {(a, b) | a^2 + b^2 = 4} is a function.

  3. Show that the function f(x) = 2x + 1 is one-to-one.

  4. Prove that the function f(x) = x^2 is not onto.

  5. Show that the function f(x) = x^2 + 1 is not one-to-one.

  6. Determine if the function f(x) = x^3 is bijective.

  7. Find the inverse function of f(x) = 2x + 3.

  8. Determine the domain and range of the function f(x) = 1/x.

  9. Find the composition of the functions f(x) = x^2 and g(x) = x + 2.

  10. Find the inverse function of the function f(x) = 3x + 5 and check if it is a function.

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