Sets, Relations & Functions

In this chapter we introduce the notion of functions, the cornerstone of the entire syllabus of Mathematics in IIT JEE preparation. This section is not only important from the point of view of Algebra but also from the point of view of Calculus. It will not be an exaggeration to say that if a student wants to be well versed in Calculus then he should have a deep insight of this chapter. 

The concept of function is useful in defining the dependence of one thing (called dependent variable) on other things (called independent variables). We have to clearly understand that any relationship between two phenomenon is not necessarily a function. Further, dependence of one quantity on some other is also not always a function. You will also learn how to graphically represent various functions and how this graphical representation is useful in quickly finding out the nature of dependence of the dependent variable on the independent variable(s). The chapter starts with some basic concepts about sets, inequalities, and then defines function, types of functions, algebra of functions. This chapter also deals with various types of graphs and their transformations.

 Topics covered under IIT JEE Algebra are:-

1.       Set theory

2.       Inequalities

3.       Cartesian Product

4.       Introduction to Functions

5.       Functions- one-one, many-one, into, onto

6.       Increasing or decreasing Functions

7.       Inverse Functions

8.       Invertible Functions

9.       Even and Odd Functions

10.     Explicit and Implicit Functions

11.     Periodic Functions

12.     Bounded and unbounded Functions

13.     Constant and Identity Functions

14.     Absolute Value Function

15.     Signum Function

16.     Polynomial and Rational Function

17.     Linear Function

18.     Exponential Function

19.     Logarithmic Function

20.     Greatest Integer Function

21.     Graphical representation of a Function

22.     Algebraic operations on Functions

23.     Composite Functions

24.     Basic Transformations on Graphs

25.     Solved Examples

All these topics have been covered in detail in the coming sections. We will give an outline of some topics here. 

Sets: A set is a group of objects wherein each object is called a member of the group. A set can be represented using curly brackets so a set containing all even numbers till 10 is the set {2, 4, 6, 8, and 10}.

Relations: A relation is just a relationship between sets of information. A well-defined relation is called a function.

Function: A function is defined by its set of inputs which is called the domain and a set comprising the outputs called the codomain (or range). For example, we could define a function using the rule f(x) = x3 by saying that the domain and codomain are the real numbers, and that the ordered pairs are all pairs of real numbers (x, x3).

One-to-One Function: A function f from A to B is called one-to-one if whenever

f (a) = f (b) then a = b. No element of B is the image of more than one element in A. These functions are also termed as injective. Given any y there is only one x that can be paired with the given y. 

One-to-One Function
                            "One-to-One"

One-to-One Function
                            NOT "One-to-One"



Onto Function: A function f from A to B is said to be onto if for every b ∈ B, ∃ an a ∈ A such that f (a) = b. Such functions are also called surjective. 


"Onto"
(all elements in B are used)



Invertible Function: A function that is one-to-one as well as onto is called invertible. Let f be a function whose domain is the set X, and whose range is the set Y. Then f is invertible if there exists a function g with domain Y and range X, with the property: f(x) = y iff g(y) = x. 

If the function f is invertible, the function g is unique. This ‘g’ is called the inverse of f and written as g = f -1

Even Function: Let f(x) be a real valued function of a real variable. Then f is even if the following equation holds for all x and -x in the domain of f:

f(x) = f(-x)

Geometrically, the graph of an even function is symmetric with respect to the y-axis.

Odd Function: Again, let f(x) be a real valued function of a real variable. Then f is odd if the following equation holds for all x and -x in the domain of f:

f (-x) = - f(x) or f(x) + f(-x) =0.

 View this video for more clarity on even and odd functions

 Increasing function: A function f is said to be increasing if whenever a >b, then

f (a) = f(b). Further a function is said to be strictly increasing if

When a > b, then f(a) > f(b). The graph of an increasing function looks somewhat like this:


Increasing Function

You may also refer the video on increasing function

 Decreasing Function: A function y=f(x) is said to be decreasing if whenever

x < y, f(x) = f(y). Moreover, it is said to be strictly decreasing if when

x < y then f(x) > f(y).



decreasing Function

Bounded Functions: A function f defined on a set B is said to be bounded if, the set of its values is bounded. Mathematically, a function f is said to be bounded if there exists a real number M such that | f(x)| = M, for all x in B.

If f(x) = S for all x in X, then the function is said to be bounded above by S. Similarly, if f(x) =T for all x in X, then the function f is said to be bounded below by T.

Periodic Function: In simple words, a periodic function is a function that repeats its values in regular intervals or periods. Most significant examples include the trigonometric functions. A function which is not periodic is called aperiodic.

Example: Sine function is periodic with period 2 π, since for all values of x, the function repeats itself on intervals of length 2 π.

Signum Function: Signum function is the real valued function defined for real x as follows:

                   +1, if x > 0

 Sgn(x) =    0, if x = 0

                   -1, if x < 0

Graphical Representation of Signum Function

For all real numbers x, we have sgn(-x) = -sgn(x). 

Explicit Function: An explicit function is a function in which the dependent variable is represented explicitly in the form of an independent variable. It may also be defined as a function in which the dependent variable is expressed in form of some independent variable.  

Example: y=5x3-3 is an explicit function.

 Implicit Function: If the dependent variable has not been represented explicitly in the form of independent variable then it is called an implicit function. 

Examples:

x2 + y2 +4 = 0

y4 + 3x3 +1 = 0 

Identity Function: If K is a set, the identity function f on K is defined to be the function with domain and codomain K which satisfies

f(x) = x for all elements x in K. 

Constant Function: A linear function of the form y = k, where k is a constant is called a constant function. It may also be written as f(x) = k.

Remark: The graph of a constant function is a horizontal line. 


Constant Function

Exponential function: An exponential function with base b is defined by

f (x) = bx, where b > 0, b ≠ 1 and x is any real number. 

Greatest Integer Function: A greatest integer function is also called the step function because its graph is in step form. It is written as f(x) = [x], where f(x) is the greatest integer less than or equal to x. The greatest integer function rounds any number down to the nearest integer. 

Examples: The greatest integer less than or equal to the number [8.3] is [8].

The greatest integer less than or equal to the number [-8.3] is [-8]. 

 
Graphical representation of a greatest integer function


Linear Function: A function of the form y = ax+b is said to be linear. Here ‘b’ is a constant and ‘a’ is the coefficient of x. The graph of a linear function is a straight line.
 
Plotting of linear functions
Set Theory and Functions are important from IIT JEE perspective. Objective questions are framed on this section. AIEEE definitely has 1-2 questions every year directly on these topics. It is very important to master these concepts at early stage as this forms the basis of your preparation for IIT JEE and AIEEE Physics.

Related resources:

• Refer the Complete syllabus of JEE Mathematics

• Get an idea of the types of questions asked through our Solved Papers of Previous Years.

• Look here for the reference books of Mathematics

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