Types of Functions
(For IIT-JEE Preparation)
IITians at your home
ISO 9001:2008 Certified
India's No. 1 Online Academy
Contact: +91-9555288846
Types of Functions

The notion of a function along with some special functions like identity function, constant function, polynomial function, rational function, modulus function, signum function etc. along with their graphs are known to you as you have studied them in previous chapter.

Addition, subtraction, multiplication and division of two functions have also been studied. As the concept of function is of paramount importance in mathematics and among other disciplines as well, we would like to extend our study about function from where we finished earlier. In this topic, we would like to study different types of functions.

Consider the functions f1, f2, f3 and f4 given by the following diagrams.
types of functions In Fig 1.2, we observe that the images of distinct elements of X1 under the function f1 are distinct, but the image of two distinct elements 1 and 2 of X1 under f2 is same, namely b. Further, there are some elements like e and f in X2 which are not images of any element of X1 under f1, while all elements of X3 are images of some elements of X1 under f3. The above observations lead to the following definitions:

Definition A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x1, x2 ∈ X, f (x1) = f (x2) implies x1 = x2. Otherwise, f is called many-one.
The function f1 and f4 in Fig 1.2 (i) and (iv) are one-one and the function f2 and f3 in Fig 1.2 (ii) and (iii) are many-one.

Definition A function f : X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f (x) = y.
The function f3 and f4 in Fig 1.2 (iii), (iv) are onto and the function f1 in Fig 1.2 (i) is not onto as elements e, f in X2 are not the image of any element in X1 under f1.

Remark f : X → Y is onto if and only if Range of f = Y.

Definition A function f : X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto.
The function f4 in Fig 1.2 (iv) is one-one and onto.

Example Show that the function f : NN, given by f (x) = 2x, is one-one but not onto.
Solution The function f is one-one, for f (x1) = f (x2) ⇒ 2x1 = 2x2 ⇒ x1 = x2. Further, f is not onto, as for 1 ∈ N, there does not exist any x in N such that f (x) = 2x = 1.

Example Prove that the function f : RR, given by f (x) = 2x, is one-one and onto.
Solution f is one-one, as f (x1) = f (x2) ⇒ 2x1 = 2x2 ⇒ x1 = x2. Also, given any real number y in R, there exists y/2 in R such that f (y/2) = 2 . (y/2) = y. Hence, f is onto.
types of functions
Example Show that the function f : NN, given by f (1) = f (2) = 1 and f (x) = x – 1, for every x > 2, is onto but not one-one.
Solution f is not one-one, as f (1) = f (2) = 1. But f is onto, as given any y ∈ N, y ≠ 1, we can choose x as y + 1 such that f (y + 1) = y + 1 – 1 = y. Also for 1 ∈ N, we have f (1) = 1.

Example Show that the function f : RR, defined as f (x) = x2, is neither one-one nor onto.
Solution Since f (– 1) = 1 = f (1), f is not oneone. Also, the element – 2 in the co-domain R is not image of any element x in the domain R (Why?). Therefore f is not onto.
types of functions
Example Show that an onto function f : {1, 2, 3} → {1, 2, 3} is always one-one.
Solution Suppose f is not one-one. Then there exists two elements, say 1 and 2 in the domain whose image in the co-domain is same. Also, the image of 3 under f can be only one element. Therefore, the range set can have at the most two elements of the co-domain {1, 2, 3}, showing that f is not onto, a contradiction. Hence, f must be one-one.

Example Show that a one-one function f : {1, 2, 3} → {1, 2, 3} must be onto.
Solution Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different elements of the co-domain {1, 2, 3} under f. Hence, f has to be onto.

Do you like this Topic?
Share it on

IIT JEE Maths Study Material

Skip Navigation Links.
Collapse MathsMaths
Collapse AlgebraAlgebra
Expand SetsSets
Expand Sequences and seriesSequences and series
Expand Complex NumbersComplex Numbers
Expand Quadratic EquationsQuadratic Equations
Expand Linear inequalitiesLinear inequalities
Expand Permutations and CombinationsPermutations and Combinations
Expand ProbabilityProbability
Expand Binomial theoremBinomial theorem
Expand Matrices and determinantsMatrices and determinants
Collapse TrignometryTrignometry
Collapse CalculusCalculus
Expand Relations and FunctionsRelations and Functions
Expand Limits and DerivativesLimits and Derivatives
Expand Continuity and DifferentiabilityContinuity and Differentiability
Expand Applications of derivativesApplications of derivatives
Expand Integral calculusIntegral calculus
Expand Application of integralsApplication of integrals
Expand Differential equationsDifferential equations
Collapse Coordinate GeometryCoordinate Geometry
Vector and 3D Geometry

Free IIT JEE Study Material

  • Online Classroom Program

    • IITians @ Your Home, Attend Classes directly from Home
    • Two way interaction between Teacher and Students
    • Zero Travel Time or Cost involved, No Hidden Charges
    • Small Batch Sizes, Maximum of 12-15 Students
    • Golden Opportunity to be trained by IITians and NITians
    • All you need is a Computer / Laptop & Internet Connection
    Try Free Demo Class IIT JEE Online Classes
  • 1 on 1 Online Class

    • IITians @ your Home for 1 on 1 Class
    • Two way Communication between Teacher and Student
    • Special Doubt Removal & Problem solving Sessions
    • Customize the Course as per your desire - Duration, Timing, Faculty etc
    • All faculties are IITians and NITians
    • All you need is a Computer / Laptop & Internet Connection
    Try Free Demo Class IIT JEE 1 on 1 Online Classes
  • Correspondence Course Details

    • 600 Hours of Recorded Lectures by IITians and NITians
    • Best Study Material comprising all Concepts and Tricks
    • 10000+ Solved Question Bank + 250 Hours of Video Solutions
    • Complete NCERT solutions + 150 Hours of NCERT Video Solutions
    • Last 30 years JEE chapterwise video solutions
    • 100% Original Material prepared by JEE Experts
    Register for Free Now IIT JEE Correspondence Courses
  • AITS for JEE Main and JEE Advanced

    Register for Free Now JEE Test Series
Welcome to Kshitij Education India

Kshitij Education India as an organisation is immensely grateful to society at large and has made efforts to contribute to the Indian Education system. The free study material section of our website can be deemed as a treasure of resources sufficient for an aspirant to clear JEE Mains. Kshitij provides past year solutions of IIT, AIEEE, JEE Mains and Advanced in its free study material section. Our faculty members put in best efforts to explain each and every question in the best possible manner. Online IIT JEE Free Study Material section also features a collection of over 10,000 pages of study resources covering the entire IIT JEE curriculum. Our discussion forum stimulates a student intellectually with over 5000 threads of IIT discussion topics.

Please enter the Email Id and Contact number of the person to whom you would like to refer www.kshitij-iitjee.com

Welcome to Kshitij Education India
Our Guarantee:
  1. We're so sure you'll have the time of your life with us, we back our courses with a 100% Satisfaction Guarantee.
  2. If for any reason you aren't 100% satisfied with your classes in first 7 days, just let us know and we'll refund your fees. No questions asked.*
  3. And based on your feedback, we will take the necessary steps to ensure we never repeat any mistakes as such.

* It is mandatory to attend first 7 days of live classes and than if you found any valid reason to step back you can request us for refund.

Live Chat