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Copyright © 2004, New Age International (P) Ltd., PublishersPublished by New Age International (P) Ltd., Publishers

All rights reserved.No part of this ebook may be reproduced in any form, by photostat, microfilm,xerography, or any other means, or incorporated into any information retrievalsystem, electronic or mechanical, without the written permission of the publisher.All inquiries should be emailed to rights@newagepublishers.com

PUBLISHING FOR ONE WORLD

NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS4835/24, Ansari Road, Daryaganj, New Delhi - 110002Visit us at www.newagepublishers.com

ISBN (13) : 978-81-224-2979-4

Preface v

Preface

As the title of the book ‘How to learn calculus of one variable’ Suggests we have tried to present the entirebook in a manner that can help the students to learn the methods of calculus all by themselves we have felt thatthere are books written on this subject which deal with the theoretical aspects quite exhaustively but do nottake up sufficient examples necessary for the proper understanding of the subject matter thoroughly. Thebooks in which sufficient examples are solved often lack in rigorous mathematical reasonings and skip accuratearguments some times to make the presentation look apparently easier.

We have, therefore, felt the need for writing a book which is free from these deficiencies and can be used asa supplement to any standard book such as ‘Analytic geometry and calculus’ by G.B. Thomas and Finny whichquite thoroughly deals with the proofs of the results used by us.

A student will easily understand the underlying principles of calculus while going through the worked-outexamples which are fairly large in number and sufficiently rigorous in their treatment. We have not hesitated towork-out a number of examples of the similar type though these may seem to be an unnecessary repetition. Thishas been done simply to make the students, trying to learn the subject on their own, feel at home with theconcepts they encounter for the first time. We have, therefore, started with very simple examples and graduallyhave taken up harder types. We have in no case deviated from the completeness of proper reasonings.

For the convenience of the beginners we have stressed upon working rules in order to make the learning allthe more interesting and easy. A student thus acquainted with the basics of the subject through a wide rangeof solved examples can easily go for further studies in advanced calculus and real analysis.

We would like to advise the student not to make any compromise with the accurate reasonings. They shouldtry to solve most of examples on their own and take help of the solutions provided in the book only when it isnecessary.

This book mainly caters to the needs of the intermediate students whereas it can also used with advantagesby students who want to appear in various competitive examinations. It has been our endeavour to incorporateall the finer points without which such students continually feel themselves on unsafe ground.

We thank all our colleagues and friends who have always inspired and encouraged us to write this bookeverlastingly fruitful to the students. We are specially thankful to Dr Simran Singh, Head of the Department ofLal Bahadur Shastri Memorial College, Karandih, Jamshedpur, Jharkhand, who has given valuable suggestionswhile preparing the manuscript of this book.

Suggestions for improvement of this book will be gratefully accepted.

DR JOY DEV GHOSH

MD ANWARUL HAQUE

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Contents vii

Contents

Preface v

1. Function 1

2. Limit and Limit Points 118

3. Continuity of a Function 151

4. Practical Methods of Finding the Limits 159

5. Practical Methods of Continuity Test 271

6. Derivative of a Function 305

7. Differentiability at a Point 321

8. Rules of Differentiation 354

9. Chain Rule for the Derivative 382

10. Differentiation of Inverse Trigonometric Functions 424

11. Differential Coefficient of Mod Functions 478

12. Implicit Differentiation 499

13. Logarithmic Differentiation 543

14. Successive Differentiation 567

15. L’Hospital’s Rule 597

16. Evaluation of Derivatives for Particular Arguments 615

17. Derivative as Rate Measurer 636

18. Approximations 666

viii Contents

19. Tangent and Normal to a Curve 692

20. Rolle’s Theorem and Lagrange’s Mean Value Theorem 781

21. Monotonocity of a Function 840

22. Maxima and Minima 870

Bibliography 949

Index 950

Function 1

1

Function

To define a function, some fundamental concepts arerequired.

Fundamental Concepts

Question: What is a quantity?Answer: In fact, anything which can be measured orwhich can be divided into parts is called a quantity.But in the language of mathematics, its definition isput in the following manner.

Definition: Anything to which operations ofmathematics (mathematical process) such as addition,subtraction, multiplication, division or measurementetc. are applicable is called a quantity.

Numbers of arithmetic, algebraic or analyticexpressions, distance, area, volume, angle, time,weight, space, velocity and force etc. are all examplesof quantities.

Any quantity may be either a variable or a constant.

Note: Mathematics deals with quantities which havevalues expressed in numbers. Number may be real orimaginary. But in real analysis, only real numbers as

values such as –1, 0, 15, 2 , p etc. are considered.

Question: What is a variable?Answer:Definitions 1: (General): If in a mathematical discus-sion, a quantity can assume more than one value,then the quantity is called a variable quantity or sim-ply a variable and is denoted by a symbol.Example: 1. The weight of men are different for dif-ferent individuals and therefore height is a variable.

2. The position of a point moving in a circle is avariable.Definition: 2. (Set theoritic): In the language of settheory, a variable is symbol used to represent anunspecified (not fixed, i.e. arbitrary) member (elementor point) of a set, i.e., by a variable, we mean an elementwhich can be any one element of a set or which canbe in turn different elements of a set or which can bea particular unknown element of a set or successivelydifferent unknown elements of a set. We may think ofa variable as being a “place-holder” or a “blank” forthe name of an element of a set.

Further, any element of the set is called a value ofthe variable and the set itself is called variable’sdomain or range.

If x be a symbol representing an unspecifiedelement of a set D, then x is said to vary over the setD (i.e., x can stand for any element of the set D, i.e., xcan take any value of the set D) and is called a variableon (over) the set D whereas the set D over which thevariable x varies is called domain or range of x.

Example: Let D be the set of positive integers and xÎ D = {1, 2, 3, 4, …}, then x may be 1, 2, 3, 4, … etc.

Note: A variable may be either (1) an independentvariable (2) dependent variable. These two terms havebeen explained while defining a function.

Question: What is a constant?Answer:Definition 1. (General): If in a mathematicaldiscussion, a quantity cannot assume more than one

2 How to Learn Calculus of One Variable

vale, then the quantity is called a constant or aconstant quantity and is denoted by a symbol.Examples: 1. The weights of men are different fordifferent individuals and therefore weight is a variable.But the numbers of hands is the same for men ofdifferent weights and is therefore a constant.2. The position of a point moving in a circle is avariable but the distance of the point from the centreof the circle is a constant.3. The expression x + a denotes the sum of twoquantities. The first of which is variable while thesecond is a constant because it has the same valuewhatever values are given to the first one.

Definition: 2. (Set theoritic): In the language of settheory, a constant is a symbol used to represent amember of the set which consists of only one member,i.e. if there is a variable ‘c’ which varies over a setconsisting of only one element, then the variable ‘c’is called a constant, i.e., if ‘c’ is a symbol used torepresent precisely one element of a set namely D,then ‘c’ is called a constant.

Example: Let the set D has only the number 3; thenc = 3 is a constant.

Note: Also, by a constant, we mean a fixed elementof a set whose proper name is given. We often refer tothe proper name of an element in a set as a constant.Moreover by a relative constant, we mean a fixedelement of a set whose proper name is not given. Weoften refer to the “alias” of an element in a set as arelative constant.

Remark: The reader is warned to be very carefulabout the use of the terms namely variable andconstant. These two terms apply to symbols only notto numbers or quantities in the set theory. Thus it ismeaningless to speak of a variable number (or avariable quantity) in the language of set theory forthe simple reason that no number is known to humanbeings which is a variable in any sense of the term.Hence the ‘usual’ text book definition of a variable asa quantity which varies or changes is completelymisleading in set theory.

Kinds of Constants

There are mainly two kinds of constants namely:

1. Absolute constants (or, numerical constants).2. Arbitrary constants (or, symbolic constants).

Each one is defined in the following way:1. Absolute constants: Absolute constants have thesame value forever, e.g.:(i) All arithmetical numbers are absolute constants.Since 1 = 1 always but 1 ¹ 2 which means that thevalue of 1 is fixed. Similarly –1 = –1 but –1 ¹ 1 (Anyquantity is equal to itself. this is the basic axiom ofmathematics upon which foundation of equationstakes rest. This is why 1 = 1, 2 = 2, 3 = 3, … x = x anda = a and so on).(ii) p and logarithm of positive numbers (as log2, log3, log 4, … etc) are also included in absolute constants.2. Arbitrary constants: Any arbitrary constant isone which may be given any fixed value in a problemand retains that assigned value (fixed value)throughout the discussion of the same problem butmay differ in different problems.

An arbitrary constant is also termed as a parameter.

Note: Also, the term “parameter” is used in speakingof any letter, variable or constant, other than thecoordinate variables in an equation of a curve definedby y = f (x) in its domain.

Examples: (i) In the equation of the circle x2 + y2 =a2, x and y, the coordinates of a point moving along acircle, are variables while ‘a’ the radius of a circle mayhave any constant value and is therefore an arbitraryconstant or parameter.(ii) The general form of the equation of a straight lineput in the form y = mx + c contains two parametersnamely m and c representing the gradient and y-inter-cept of any specific line.

Symbolic Representation of Quantities,Variables and Constants

In general, the quantities are denoted by the letters a,b, c, x, y, z, … of the English alphabet. The letters from“a to s” of the English alphabet are taken to representconstants while the letters from “t to z” of the Englishalphabet are taken to represent variables.

Question: What is increment?Answer: An increment is any change (increase orgrowth) in (or, of) a variable (dependent or

Function 3

independent). It is the difference which is found bysubtracting the first value (or, critical value) of thevariable from the second value (changed value,increased value or final value) of the variable.

That is, increment= final value – initial value = F.V – I. V.

Notes: (i) Increased value/changed value/final value/second value means a value obtained by makingaddition, positive or negative, to a given value (initialvalue) of a variable.(ii) The increments may be positive or negative, inboth cases, the word “increment” is used so that anegative increment is an algebraic decrease.

Examples on Increment in a Variable

1. Let x1 increase to x2 by the amount ∆ x. Then wecan set out the algebraic equation x1 + ∆ x = x2 which⇒ ∆ x = x2 – x1.2. Let y1 decrease to y2 by the amount ∆ y. Then wecan set out the algebraic equation y1 + ∆ y = y2 which⇒ ∆ y = y2 – y1.

Examples on Increment in a Function

1. Let y = f (x) = 5x + 3 = given value … (i)Now, if we give an increment ∆ x to x, then we also

require to give an increment ∆ y to y simultaneously.Hence, y + ∆ y = f (x + ∆ x) = 5 (x + ∆ x) + 3 = 5x +

5∆ x + 3 … (ii)∴ (ii) – (i) ⇒ y + ∆ y – y = (5x + 5∆ x + 3) – (5x + 3)

= 5x + 5∆ x + 3 – 5x – 3 = 5∆ xi.e., ∆ y = 5∆ x

2. Let y = f (x) = x2 + 2 = given value,then y + ∆ y = f (x + ∆ x) = (x + ∆ x)2 + 2 = x2 +

∆ x2 + 2x ∆ x + 2⇒ ∆ y = x2 + ∆ x2 + 2x ∆ x + 2 – x2 – 2 = 2x ∆ x +

∆ x2Hence, increment in y = f (x + ∆ x) – f (x) where

f (x) = (x2 + 2) is ∆ y = x2 + ∆ x2 + 2x ∆ x + 2 – x2 – 2= 2x ∆ x + ∆ x2

3. Let yx

= 1 = given value.

Then, y + ∆ y = 1

x x+ ∆

Hence, increment in y = f (x + ∆x) – f (x) where

f (x) = 1

x

⇒ ∆ y = 1

x x+ ∆ – 1

x =

x x x

x x x

− ++

∆∆

a fa f =

–∆∆x

x x x+a f4. Let y = log x = given value.

Then, y + ∆ y = log (x + ∆ x)

and ∆ y = log (x + ∆ x) – log x = log x x

x

+FHG

IKJ

∆ =

log 1 +FHG

IKJ

∆ xx

5. Let y = sin θ, given valueThen, y + ∆ y = sin (θ + ∆ θ)

and ∆ y = sin (θ + ∆ θ) – sin θ = 2cos 2

2

θ θ+FHG

IKJ

∆·

sin ∆ θ2

FHG

IKJ .

Question: What is the symbol used to represent (or,denote) an increment?Answer: The symbols we use to represent smallincrement or, simply increment are Greak Letters ∆and δ (both read as delta) which signify “an increment/change/growth” in the quantity written just after it asit increases or, decreases from the initial value toanother value, i.e., the notation ∆ x is used to denotea fixed non zero, number that is added to a givennumber x0 to produce another number x = x0 + ∆ x. ify = f (x) then ∆ y = f (x0 + ∆ x) – f (x0).Notes: If x, y, u. v are variables, then increments inthem are denoted by ∆ x, ∆ y, ∆ u, ∆ v respectivelysignifying how much x, y, u, v increase or decrease,i.e., an increment in a variable (dependent orindependent) tells how much that variable increasesor decreases.

Let us consider y = x2

When x = 2, y = 4x = 3, y = 9

4 How to Learn Calculus of One Variable

∴ ∆ x = 3 – 2 = 1 and ∆ y = 9 – 4 = 5⇒ as x increases from 2 to 3, y increases from 4

to 9.⇒ as x increases by 1, y increases by 5.

Question: What do you mean by the term “function”?Answer: In the language of set theory, a function isdefined in the following style.

A function from a set D to a set R is a rule or, law(or, rules, or, laws) according to which each elementof D is associated (or, related, or, paired) with a unique(i.e., a single, or, one and only one, or, not more thanone) element of R. The set D is called the domain ofthe function while the set R is called the range of thefunction. Moreover, elements of the domain (or, theset D) are called the independent variables and theelements of the range (or, range set or, simply the setR) are called the dependent variables. If x is theelement of D, then a unique element in R which therule (or, rules) symbolised as f assigns to x is termed“the value of f at x” or “the image of x under the rulef” which is generally read as “the f-function of x” or, “fof x”. Further one should note that the range R is theset of all values of the function f whereas the domainD is the set of all elements (or, points) whose eachelement is associated with a unique elements of therange set R.

Functions are represented pictorially as in theaccompanying diagram.

One must think of x as an arbitrary element of thedomain D or, an independent variable because a valuef of x can be selected arbitrarily from the domain D aswell as y as the corresponding value of f at x, adependent variable because the value of y dependsupon the value of x selected. It is customary to writey = f (x) which is read as “y is a function of x” or, “y isf of x” although to be very correct one should saythat y is the value assigned by the function fcorresponding to the value of x.

Highlight on the Term “The Rule or theLaw”.

1. The term “rule” means the procedure (orprocedures) or, method (or, methods) or, operation(or, operations) that should be performed over theindependent variable (denoted by x) to obtain thevalue the dependent variable (denoted by y).

Examples:1. Let us consider quantities like

(i) y = log x (iv) y = sin x(ii) y – x3 (v) y = sin–1x

(iii) y x= (vi) y = ex, … etc.In these log, cube, square root, sin, sin–1, e, … etc

are functions since the rule or, the law, or, the function

f = log, ( )3, , sin, sin–1 or, e, … etc has been

performed separately over (or, on) the independentvariable x which produces the value for the dependentvariable represented by y with the assistance of the

rule or the functions log, ( )3, , sin, sin–1 or, e, …

etc. (Note: An arbitrary element (or point) x in a setsignifies any specified member (or, element or point)of that set).2. The precise relationship between two sets ofcorresponding values of dependent and independentvariables is usually called a law or rule. Often the ruleis a formula or an equation involving the variablesbut it can be other things such as a table, a list ofordered pairs or a set of instructions in the form of astatement in words. The rule of a function gives thevalue of the function at each point (or, element) of thedomain.

Examples:

(i) The formula f xx

a f =+

1

12 tells that one should

square the independent variable x, add unity and thendivide unity by the obtained result to get the value ofthe function f at the point x, i.e., to square theindependent variable x, to add unity and lastly todivide unity by the whole obtained result (i.e., squareof the independent variable x plus unity).

D

x y f x = ( )

R

Function 5

(ii) f (x) = x2 + 2, where the rule f signifies to squarethe number x and to add 2 to it.(iii) f (x) = 3x – 2, where the rule f signifies to multiplyx by 3 and to subtract 2 from 3x.

(iv) C r= 2 π an equation involving the variables C(the circumference of the circle) and r (the radius of

the circle) which means that C r= 2 π = a functionof r.

(v) y s= 64 an equation involving y and s which

means that y s= 64 a functions of s.

3. A function or a rule may be regarded as a kind of

machine (or, a mathematical symbol like , log, sin,

cos, tan, cot, sec, cosec, sin–1, cos–1, tan–1, cot–1,sec–1, cosec–1, … etc indicating what mathematicaloperation is to be performed over (or, on) the elementsof the domain) which takes the elements of the domainD, processes them and produces the elements of therange R.

Example of a function of functions:

Integration of a continuous function defined on someclosed interval [a. b] is an example of a function offunctions, namely the rule (or, the correspondence)that associates with each object f (x) in the given set

of objects, the real number f x dxa

b

� � .Notes: (i) We shall study functions which are givenby simple formulas. One should think of a formula asa rule for calculating f (x) when x is known (or, given),i.e., of the rule of a function f is a formula giving y interms of x say y = f (x), to find the value of f at anumber a, we substitute that number a for x whereverx occurs in the given formula and then simplify it.

(ii) For x D f x R∈ ∈, � � should be unique meansthat f can not have two or more values at a givenpoint (or, number) x.(iii) f (x) always signifies the effect or the result ofapplying the rule f to x.

(iv) Image, functional value and value of the functionare synonymes.

Notations:

We write 1. " "f D R: → or " "D Rf

→ for “f is afunction with domain D and range R” or equivalently,“f is a function from D to R”.

2. f x y: → or, x yf

→ or, x f x→ � � for “afunction f from x to y” or “f maps (or, transforms) xinto y or f (x)”.

3. f D R: → defined by y = f (x) or, f D R: → byy = f (x) for “(a) the domain = D, (b) the range = R, (c)the rule : y = f (x).4. D (f) = The domain of the function f where Dsignifies “domain of”.5. R (f) = The range of the function f where R signifies“range of”.

Remarks:(i) When we do not specify the image of elements ofthe domain, we use the notation (1).(ii) When we want to indicate only the images ofelements of the domain, we use the notation (2).(iii) When we want to indicate the range and the ruleof a function together with a functional value f (x), weuse the notation (3).(iv) In the language of set theory, the domain of afunction is defined in the following style:

D (f): x x D: ∈ 1 � where, D1 = the set ofindependent variables (or, arguments) = the set of allthose members upon which the rule ‘f ’ is performedto find the images (or, values or, functional values).(v) In the language of set theory, the range of afunction is defined in the following way:

R f f x x D f x R� � � � � �� = ∈ ∈: , = the set of allimages.(vi) The function f n is defined by f n (x) = f (x) · f (x) …n. times

= [f (x)]n, where n being a positive integer.(vii) For a real valued function of a real variable bothx and y are real numbers consisting of.(a) Zero

6 How to Learn Calculus of One Variable

(b) Positive or negative integers, e.g.: 4, 11, 9, 17,–3, –17, … etc.

(c) Rational numbers, e.g.: 9

5

17

2, ,

− … etc.

(d) Irrational numbers e.g.: 7 14, ,− … etc.(viii) Generally the rule/process/method/law is notgiven in the form of verbal statements (like, find thesquare root, find the log, exponential, … etc.) but inthe form of a mathematical statement put in the formof expression containing x (i.e. in the form of a formula)which may be translated into words (or, verbalstatements).(ix) If it is known that the range R is a subset of someset C, then the following notation is used:

f D C: → signifying that(a) f is a function(b) The domain of f is D(c) The range of f is contained in C.

Nomenclature: The notation " "f D C: → is read fis a function on the set D into the set C.”

N.B: To define some types of functions like “intofunction and on to function”, it is a must to define a

function " "f D C: → where C = codomain andhence we are required to grasp the notion of co-domain. Therefore, we can define a co-domain of afunction in the following way:

Definition of co-domain: A co-domain of a functionis a set which contains the range or range set (i.e., setof all values of f) which means R C⊆ , where R = theset of all images of f and C = a set containing imagesof f.

Remember:1. If R C⊂ (where R = the range set, C = co-domain)i.e., if the range set is a proper subset of the co-domain,then the function is said to be an “into function”.

2. If R = C, i.e., if the range set equals the co-domain,then the function is said to be an “onto function”.3. If one is given the domain D and the rule (orformula,) then it is possible (theoretically at least) tostate explicitly a function as any ordered pair and oneshould note that under such conditions, the rangeneed not be given. Further, it is notable that for each

specified element ' ' ,a D∈ the functional value f (a)is obtained under the function ‘f’.

4. If a D∈ , then the image in C is represented by f(a) which is called the functional value (correspondingto a)and it is included in the range set R.

Question: Distinguish between the terms “a functionand a function of x”.Answer: A function of x is a term used for “an imageof x under the rule f” or “the value of the function f at(or, for) x” or “the functional value of x” symbolisedas y = f (x) which signifies that an operation (or,operations) denoted by f has (or, have) been performedon x to produce an other element f (x) whereas theterm “function” is used for “the rule (or, rules)” or“operation (or, operations)” or “law (or, laws)” to beperformed upon x, x being an arbitrary element of aset known as the domain of the function.Remarks: 1. By an abuse of language, it has beencustomary to call f (x) as function instead of f when aparticular (or, specifies) value of x is not given onlyfor convenience. Hence, wherever we say a “functionf (x) what we actually mean to say is the function fwhose value at x is f (x). thus we say, functions x4, 3x2

+ 1, etc.

2. The function ‘f’ also represents operator like n ,

( )n, | |, log, e, sin, cos, tan, cot, sec, cosec, sin–1, cos–1, tan–1, cot–1, sec–1 or cosec–1 etc.3. Function, operator, mapping and transformationare synonymes.4. If domain and range of a function are not known, itis customary to denote the function f by writing y = f(x) which is read as y is a function of x.

Question: Explain the terms “dependent andindependent variables”.Answer:1. Independent variable: In general, an independentvariable is that variable whose value does not depend

fDC

Rx1 y1x5 y5y6x2 y2x6x3 y3x4 y4

Function 7

upon any other variable or variables, i.e., a variable ina mathematical expression whose value determinesthe value of the whole given expression is called anindependent variable: in y = f (x), x is the independentvariable.

In set theoretic language, an independent variableis the symbol which is used to denote an unspecifiedmember of the domain of a function.2. Dependent variable: In general a dependentvariable is that variable whose value depends uponany other variable or variables, i.e., a variable (or, amathematical equation or statement) whose value isdetermined by the value taken by the independentvariable is called a dependent variable: in y = f (x), y isthe dependent variable.

In set theoretic language, a dependent variable isthe symbol which is used to denote an unspecifiedmember of the range of a function.e.g.: In A f r r= =� � π 2, r is an independent variableand A is a dependent variable.

Question: Explain the term “function or function ofx” in terms of dependency and independency.Answer: When the values of a variable y aredetermined by the values given to another variable x,y is called a function of (depending on) x or we saythat y depends on (or, upon) x. Thus, any expressionin x depends for its value on the value of x. This iswhy an expression in x is called a function of x put inthe form: y = f (x).

Question: What are the symbols for representing theterms “a function and a function of a variable”?Answer: Symbols such as f, F, φ etc are used todenote a function whereas a function of a variable isdenoted by the symbols f (x), φ x f t� � � �, , F t� � ,φ t� � and can be put in the forms: y = f (x); y x= φ � � ;y = f (t); y = F (t); y t= φ � � , that y is a function of(depending on) the variable within the circular bracket( ), i.e., y depends upon the variable within circularbracket.

i.e., y = f (x) signifies that y depends upon x, i.e., yis a function of x.

S = f (t) signifies that s depends upon t, i.e., s is afunction of t.

C r= φ� � signifies that c depends upon r, i.e., c isa function of r.

Notes:1. Any other letter besides f F, ,φ etc may be usedjust for indicating the dependence of one physicalquantity on an other quantity.2. The value of f /functional value of f correspondingto x = a / the value of the dependent variable y for aparticular value of the independent variable issymbolised as (f (x))x = a = f (a) or [f (x)]x = a = f (a) whileevaluating the value of the function f (x) at the pointx = a.3. One should always note the difference between“a function and a function of”.4. Classification of values of a function at a point x= a.There are two kinds of the value of a function at apoint x = a namely(i) The actual value of a function y = f (x) at x = a.(ii) The approaching or limiting value of a function y= f (x) at x = a, which are defined as:(i) The actual value of a function y = f (x) at x = a:when the value of a function y = f (x) at x = a isobtained directly by putting in the given value of theindependent variable x = a wherever x occurs in agiven mathematical equation representing a function,we say that the function f or f (x) has the actual valuef (a) at x = a.(ii) The approaching value of a function y = f (x) at x= a: The limit of a function f (x) as x approaches somedefinite quantity is termed as the approaching (or,limiting) value of the function y = f (x) at x = a. Thisvalue may be calculated when the actual value of thefunction f (x) becomes indeterminate at a particularvalue ‘a’ of x.5. When the actual value of a function y = f (x) is

anyone of the following forms: 0

0, 0 0

0, ,× ∞

∞∞

∞ − ∞, , ∞ ∞0 1, , imaginary, any real number

0for a particular value ‘a’ of x, it is said that the functionf (x) is not defined or is indeterminate or is meaninglessat x = a.6. To find the value of a function y = f (x) at x = ameans to find the actual value of the function y = f (x)at x = a.

8 How to Learn Calculus of One Variable

Pictorial Representation of a Function, itsDomain and Range.

1. Domain: A domain is generally represented byany closed curve regular (i.e., circle, ellipse, rectangle,square etc) or irregular (i.e. not regular) whosemembers are represented by numbers or alphabets ordots.2. Range: A range is generally represented byanother closed curve regular or irregular or the someclosed curve regular or irregular as the domain.3. Rule: A rule is generally represented by an arrowor arc (i.e., arc of the circle) drawn from each memberof the domain such that it reaches a single member ormore than one member of the codomain, the codomainbeing a superset of the range (or, range set).

Remarks:1. We should never draw two or more than two arrowsfrom a single member of the domain such that it reachesmore than one member of the codomain to show thatthe venn-diagram represents a function. Logic behindit is given as follows.

If the domain are chairs, then one student can notsit on more than one chair at the same time (i.e., onestudent can not sit on two or more than two chairs atthe same time)

Fig. 1.1 Represents a function

Fig. 1.2 Represents a function

Fig. 1.3 Does not represent a function

Fig. 1.4 Represents a function

2. In the pictorial representation of a function theword “rule” means.(i) Every point/member/element in the domain D is

joined by an arrow →� � or arc ∩� � to some point inrange R which means each element x D∈

corresponds to some element y R C∈ ⊆ .(ii) Two or more points in the domain D may be joinedto the same point in R C⊆ (See Fig. 1.4 where thepoints x2 and x3 in D are joined to the same point y2 in

R C⊆ .(iii) A point in the domain D can not be joined to twoor more than two points in C, C being a co-domain.(See Fig. 1.3)(iv) There may be some points in C which are notjoined to any element in D (See Fig. 1.4 where thepoints y4, y5 and y6 in C are not joined to any point inD.

Precaution: It is not possible to represent anyfunction as an equation involving variables always.At such circumstances, we define a function as a setof ordered pairs with no two first elements alike e.g., f= {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10), (6, 12), (7, 14)}whose D = domain = {1, 2, 3, 4, 5, 6, 7}, R = range = {2,4, 6, 8, 10, 12, 14} and the rule is: each second elementis twice its corresponding first element.

But f = {(0, 1), (0, 2), (0, 3), (0, 4)} does not define afunction since its first element is repeated.

D = domain

x y

R = range

f = rule =a function

f = rule = a function

D = domainC = codomain

R = rangex1 y1

y5

x2 y2x3 y3

y4

D = domainC = codomain

x1 y1x2

y2

y3x3 y4 y6x4 y5

D = domainC = codomain

R = rangex1 y1 y4x2 y2 y5x3x4 y3 y6

Function 9

Note: When the elements of the domain and the rangeare represented by points or English alphabet withsubscripts as x1, x2, … etc and y1, y2, … etcrespectively, we generally represent a function as aset of ordered pairs with no two first elements alike,i.e., f: {x, f (x): no two first elements are same} or, {x, f(x): no two first elements are same} or, {(x, y): x D∈

and y f x R= ∈� � } provided it is not possible torepresent the function as an equation y = f (x).

Question: What is meant whenever one says afunction y = f (x) exist at x = a or y = f (x) is defined at(or, f or) x = a?Answer: A function y = f (x) is said to exist at x = a or,y = f (x) is said to be defined at (or, f or) x = a providedthe value of the function f (x) at x = a (i.e. f (a)) is finitewhich means that the value of the function f (x) at x =a should not be anyone of the following forms

0

00 0

0, , ,× ∞

∞∞

, ∞ ∞– , ∞0 , 1∞ , imaginary

value, a real number

0.

Remarks:(i) A symbol in mathematics is said to have beendefined when a meaning has been given to it.(ii) A symbol in mathematics is said to be undefinedor non-existance when no meaning is attributed tothe symbol.

e.g.: The symbols 3/2, –8/15, sin–1(1/2), log (1/2)are defined or they are said to exist whereas the

symbols − ÷−9 5 5 01

, , ,cos log (–3), 52 are

undefined or they are said not to exist.(iii) Whenever we say that something exists, we meanthat it has a definite finite value.

e.g.:(i) f (a) exists means f (a) has a finite value.

(ii) limx a

f x→

� � exists means limx a

f x→

� � has a finitevalue.(iii) f ' (a) exists means f ' (a) has a finite value.

(iv) f x dxa

b

� � exists means that f x dxa

b

� � has afinite value.

Classification of Functions

We divide the function into two classes namely:(i) Algebraic(ii) Transcendental which are defined as:

(i) Algebraic function: A function which satisfiesthe equation put in the form:

Ao [f (x)] m + A1 [f (x)]

m – 1 + A2 [f (x)] m – 2 + … + Am

= 0, where A0, A1, … Am are polynomials is called analgebraic function.

Notes:1. A function f: R → R defined by f (x) = a0 xn + a1 xn– 1 + … + am – 1 x + am where a0, a1, a2, … am areconstants and n is a positive integer, is called apolynomial in x or a polynomial function or simply apolynomial. One should note that a polynomial is aparticular case of algebraic function as we see ontaking m = 1 and A0 = a constant in algebraic function.2. The quotient of two polynomials termed as arational function of x put in the form:

a x a x a x a

b b x b x

n nm m

mn

0 11

1

0 1

+ + + +

+ + +

−–

...

...

is also an algebraic function. It is defined in everyinterval only in which denominator does not vanish.If f1 (x) and f2 (x) are two polynomials, then general

rational functions may be denoted by R xf x

f x� � � �� �=

1

2

where R signifies “a rational function of”. In case f2(x) reduces itself to unity or any other constant (i.e., aterm not containing x or its power), R (x) reducesitself to a polynomial.3. Generally, there will be a certain number of valuesof x for which the rational function is not defined andthese are values of x for which the polynomial indenominator vanishes.

e.g.: R xx x

x x� � = − +

− +

2 5 1

5 6

2

2 is not defined when x

= 2 or x = 3.4. Rational integral functions: If a polynomial in x isin a rational form only and the indices of the powersof x are positive integers, then it is termed as a rationalintegral function.

10 How to Learn Calculus of One Variable

5. A combination of polynomials under one or moreradicals termed as an irrational functions is also an

algebraic function. Hence, y x f x= = � �; y =

x f x5 3 = � �; y x

x=

+2 4 serve as examples for

irrational algebraic functions.6. A polynomial or any algebraic function raised toany power termed as a power function is also an

algebraic function. Hence, y x n R f xn

= ∈ =, ;� � � �

y x f x= + =2 3

1� � � � serve as examples for powerfunctions which are algebraic.

Remarks:

1. All algebraic, transcendental, explicit or implicitfunction or their combination raised to a fractionalpower reduces to an irrational function. Hence,

y x f x= =5 3 � �; y x x f x= + =sin� � � �

12 serve

as examples for irrational functions.

2. All algebraic, transcendental, explicit or implicitfunction or their combination raised to any power isalways regarded as a power function. Hence, y = sin2

x = f (x); y = log2 | x | = f (x) serve as examples for powerfunctions.

Transcendental function: A function which is notalgebraic is called a transcendental function. Hence,all trigonometric, inverse trigonometric, exponentialand logarithmic (symoblised as “TILE”) functions aretranscendental functions. hence, sin x, cos x, tan x,cot x, sec x, cosec x, sin–1 x, cos–1 x, tan–1 x, cot–1 x,sec–1 x, cosec–1 x, log |f (x) |, log | x |, log x2, log (a + x2),ax (for any a > 0), ex, [f (x)]g (x) etc serve as examplesfor transcendental functions.

Notes: (In the extended real number system)(A)

(i) ex = ∞ when x = ∞

(ii) ex = 1 when x = 0(iii) ex = 0 when x = − ∞ .

(B) One should remember that exponential functionsobeys the laws of indices, i.e.,(i) xe · ey = ex + y

(ii) xe / ey = ex – y

(iii) (ex)m = emx

(iv) ee

x

x

− = 1

(C)(i) log0 = − ∞(ii) log 1 = 0(iii) log ∞ = ∞

Further Classification of Functions

The algebraic and the transcendental function arefurther divided into two types namely (i) explicitfunction (ii) implicit function, which are defined as:

(i) Explicit function: An explicit function is afunction put in the form y = f (x) which signifies that arelation between the dependent variable y and theindependent variable x put in the form of an equationcan be solved for y and we say that y is an explicitfunction of x or simply we say that y is a function of x.hence, y = sin x + x = f (x); y = x2 – 7x + 12 = f (x) serveas examples for explicit function of x’s.Remark: If in y = f (x), f signifies the operators (i.e.,functions) sin, cos, tan, cot, sec, cosec, sin–1, cos–1,tan–1, cot–1, sec–1, cosec–1, log or e, then y = f (x) iscalled an explicit transcendental function otherwise itis called an explicit algebraic function.

(ii) Implicit function: An implicit function is afunction put in the form: f (x, y) = c, c being a constant,which signifies that a relation between the variables yand x exists such that y and x are in seperable in anequation and we say that y is an implicit function of x.Hence, x3 + y2 = 4xy serves as an example for theimplicit function of x.Remark: If in f (x, y) = c, f signifies the operators (i.e.,functions) sin, cos, tan, cot, sec, cosec, sin–1, cos–1,tan–1, cot–1, sec–1, cosec–1, log, e and the orderedpain (x, y) signifies the combination of the variables xand y, then f (x, y) = c is called an implicit algebraicfunction of x, i.e., y is said to be an implicit algebraicfunction of x, if a relation of the form:

Function 11

ym + R1 ym – 1 + … + Rm = 0 exists, where R1, R2, …

Rm are rational function of x and m is a positive integer.

Note: Discussion on “the explicit and the implicitfunctions” has been given in detail in the chapter“differentiation of implicit function”.

On Some Important Functions

Some types of functions have been discussed inprevious sections such as algebraic, transcendental,explicit and implicit functions. In this section definitionof some function used most frequently are given.1. The constant function: A function f: R → Rdefined by f (x) = c is called the “constant function”.

Let y = f (x) = c∴ y = c which is the equation of a straight line

parallel to the x-axis, i.e., a constant functionrepresents straight lines parallel to the x-axis.

Also, domain of the constant function = D (f) ={real numbers} = R and range of the constant function= R (f) = {c} = a singleton set for examples, y = 2; y = 3are constant functions.

Remarks:(i) A polynomial a0 x

n + a1 xn – 1 + … a m – 1 x + am

(whose domain and range are sets of real numbers)reduces to a constant function when degree ofpolynomial is zero.(ii) In particular, if c = 0, then f (x) is called the “ zerofunction” and its graph is the x-axis itself.

2. The identity function: A function f: R → Rdefined by f (x) = x is called the “identity function”whose domain and range coincide with each other,i.e., D (f) = R (f) in case of identity function.

Let y = f (x) = x∴ y = x which is the equation of a straight line

passing through the origin and making an angle of45° with the x-axis, i.e., an identity function representsstraight lines passing through origin and making anangle of 45° with the x-axis.

3. The reciprocal of identity function: A function

f: R – {0} → R defined by f xx

� � = 1 is called the

reciprocal function of the identity function f (x) = x orsimply reciprocal function.

Let y f xx

= =� � 1

∴ xy = 1 which is the equation of a rectangularhyperbola, i.e., the reciprocal of an identity functionrepresents a rectangular hyperbola.

Also, D (f) = {real number except zero} = R – {0}and R (f) = {real numbers}

4. The linear function: A function put in the form: f(x) = mx + c is called a “linear function” due to the factthat its graph is a straight line.

Also, D (f) = {real numbers except m = 0} and R (f)= {real number except m = 0}

Question: What do you mean by the “absolute valuefunction”?Answer: A function f: R → R defined by f (x) = | x |

=x x

x x

,

,

≥− <���

0

0 is called absolute value (or, modulus or,

norm) function.Notes: (A) A function put in the form | f (x) | is calledthe “modulus of a function” or simply “modulus of afunction” which signifies that:

(i) | f (x) | = f (x), provided f x� � ≥ 0 , i.e., if f (x) ispositive or zero, then | f (x) | = f (x).(ii) | f (x) | = –f (x), provided f (x) < 0, i.e., if f (x) isnegative, then | f (x) | = –f (x) which means that if f (x)is negative, f (x) should be multiplied by –1 to make f(x) positive.(B) | f (x) | = sgn f (x) × f (x) where sgn

f xf x

f xf x� � � �� � � �= ≠, 0

= 0, f (x) = 0i.e., sgn f (x) = 1 when f (x) > 0

= –1 when f (x) < 0 = 0 when f (x) = 0

where ‘sgn’ signifies “sign of ” written briefly for theword “signum” from the Latin. Also, domain of abso-lute value function = D (f) = {real numbers} and rangeof absolute value function = R (f) = {non negative realnumbers} = R+ ∪ {0}.

(C) 1. (i) | x – a | = (x – a) when x a− ≥� � 0| x – a | = –(x – a) when x a−

12 How to Learn Calculus of One Variable

(ii) | 3| = 3 since 3 is positive.| –3 | = –(–3) since –3 is negative. For this reason,

we have to multiply –3 by –1.2. If the sign of a function f (x) is unknown (i.e., wedo not know whether f (x) is positive or negative),then we generally use the following definition of theabsolute value of a function.

f x f x f x� � � � � �= =2 2

3. Absolute means to have a magnitude but no sign.4. Absolute value, norm and modulus of a functionare synonymes.5. Notation: The absolute value of a function isdenoted by writing two vertical bars (i.e. straight lines)within which the function is placed. Thus the notationto signify “the absolute value of” is “| |”.6. | f 2 (x) | = f 2 (x) = | f (x) |2 = (–f (x))2

7. In a compact form, the absolute value of a function

may be defined as f x f x� � � �= 2

= f (x), when f x� � ≥ 0= –f (x), when f (x) < 0

8. f x f x f x f x1 2 1 2� � � � � � � �= ⇔ = ±e.g.: x x x x− = + ⇔ − = ± +2 3 2 3� � � �

which is solved as under this line. x − =2� �x + ⇒ − =3 2 3� � which is false which means this

equation has no solution and x x− = − + ⇒2 3� � � �x x x x x− = − − ⇒ + = − ⇒ = − ⇒2 3 2 3 2 1

x = − 12

.

9. f x k k f x k� � � �≤ ⇔ − ≤ ≤ which signifiesthe intersection of f x k� � ≥ − and f x k� � ≤ ,∀ >k 0 .

10. f x k f x k� � � �≥ ⇔ ≥ or f x k� � ≤ − whichsignifies the union of f x k� � ≥ and f x k� � ≤ − ,∀ >k 0 .11. | f (x) |n = (f (x)n, where n is a real number.

12. | |f x� � ≥ 0 always means that the absolute valueof a functions is always non-negative (i.e., zero orpositive real numbers)13. | f (x) = | –f (x) |

14. | |f x f x� � � �≥15. | f 1 (x) · f 2 (x) |

= | f 1 (x) | · | f 2 (x) |

16.f x

f x

f x

f xf x1

2

1

22 0

� �� �

� �� � � �= ≠,

17. f x f x f x f x1 2 1 2� � � � � � � �+ ≤ +

18. f x f x f x f x1 2 1 2� � � � � � � �− ≥ −19. | 0 | = 0, i.e. absolute value of zero is zero.20. Modulus of modulus of a function (i.e. mod of | f(x) | ) = | f (x) |

Remarks: When

(a) | x | = x, when x x x≥ ⇔ =0 , ∀ ∈ ∞x 0 ,� �and | x | = –x , when x x x< ⇔ = −0 ,

∀ ∈ −∞x , 0� � .(b) | x | = | –x | = x, for all real values of x

(c) x x=2

(d) x a a x a≤ ⇔ − ≤ ≤ and x a≥ ⇔ x a≥

and x a≤ − .

Geometric Interpretation of Absolute Valueof a Real Number x, Denoted by | x |

The absolute value of a real number x, denoted by | x| is undirected distance between the origin O and thepoint corresponding to a (i.e. x = a) i.e, | x | signifiesthe distance between the origin and the given point x= a on the real line.

Explanation: Let OP = xIf x > o, P lies on the right side of origin ‘O’, then

the distance OP = | OP | = | x | = x

xx ′ P x ( )

–a a

P x ( )0

Function 13

If x = O, P coincides with origin, the distance OP =| x | = | o | = o

If x > O, P lies on the left side of origin ‘o’, then thedistance OP = | OP | = | –OP | = | –x | = x

Hence, | x | =x, provided x > o means that the absolute valueof a positive number is the positive numberitself.o, provided x = o means the absolute value ofzero is taken to be equal to zero.–x, provided x < o means that the absolute valueof a negative number is the positive value ofthat number.

Notes:1. x is negative in | x | = –x signifies –x is positive in |x | = –x e.g.: | –7| = –(–7) = 7.2. The graphs of two numbers namely a and –a onthe number line are equidistant from the origin. Wecall the distance of either from zero, the absolute valueof a and denote it by | a |.

3. x a x a= ⇔ = ±

4. x a x a x a x2 2 2 2

= ⇔ = ⇔ = ⇔ =

± ⇔ = ± ⇔ =a x a x a2 2

.

5. x x=2

signifies that if x is any given number,

then the symbol x2 represents the positive square

root of x2 and be denoted by | x | whose graph issymmetrical about the y-axis having the shape ofEnglish alphabet 'V '. which opens (i) upwards if y =| x | (ii) downwards if y = – | x | (iii) on the right side ifx = | y | (iv) on the left side if x = – | y |.

An Important Remark

1. The radical sign " "n indicates the positive root

of the quantity (a number or a function) written under

it (radical sign) e.g.: 25 5= + .2. If we wish to indicate the negative square root of aquantity under the radical sign, we write the negative

sign (–) before the radical sign. e.g.: − = −4 2 .

3. To indicate both positive square root and negativesquare root of a quantity under the radical sign, wewrite the symbol ± (read as “plus or minus”) beforethe radical sign.

e.g.: ± = ±1 1

± = ±4 2

± = ±16 4

Remember:1. In problems involving square root, the positivesquare root is the one used generally, unless there is

a remark to the contrary. Hence, 100 10= ;

169 132

= =; x x .

2. x y x y2 2 2 2

1 1+ = ⇔ = − ⇔ x2

=

12

− ⇔y x y x y= − ⇔ = ± −1 12 2

e.g.: cos sin cos2 2

1θ θ θ= − ⇔ =

1 12 2

− ⇔ = ± −sin cos sinθ θ θ

one should note that the sign of cosθ isdetermined by the value of the angle ' 'θ and the

value of the angle ' 'θ is determined by the quadrant

in which it lies. Similarly for other trigonometrical

functions of θ , such as, tan2 θ = sec2 θ – 1 ⇔ tan

θ = ± − ⇔ = −sec tan sec2 21 1θ θ θ

cot cosec cot2 2

1θ θ θ= − ⇔ =

± − ⇔ = −cosec cot cosec2 2

1 1θ θ θ

sec tan sec2 2

1θ θ θ= + ⇔ =

± ⇔ =1 + tan sec 1 + tan2 2θ θ θ , w h e r e

the sign of angle ' θ ' is determined by the quadrant inwhich it lies.3. The word “modulus” is also written as “mod” and“modulus function” is written as “mod function” inbrief.

14 How to Learn Calculus of One Variable

On Greatest Integer Function

Firstly, we recall the definition of greatest integerfunction.

Definition: A greatest integer function is the functiondefined on the domain of all real numbers such thatwith any x in the domain, the function associatesalgebraically the greatest (largest or highest) integerwhich is less than or equal to x (i.e., not greater thanx) designated by writing square brackets around x as[x].

The greatest integer function has the property ofbeing less than or equal to x, while the next integer is

greater than x which means x x x≤ < + 1 .

Examples:

(i) x x= ⇒ = ������

=3

2

3

21 is the greatest integer in

3

2.

(ii) x = 5 ⇒ [x] = [5] = 5 is the greatest integer in 5.

(iii) x x= ⇒ = =50 50 7 is the greatest

integer in 50 .

(iv) x = 2.5 ⇒ [x] = [–2.5] = –3 is the greatest integerin –2.5.(v) x = 4.7 ⇒ [x] = [–4.7] = –5 is the greatest integerin –4.7.(vi) x = –3 ⇒ [x] = [–3] = –3 is the greatest integerin –3.

To Remember:1. The greatest integer function is also termed as“the bracket, integral part or integer floor function”.2. The other notation for greatest integer function is� � or [[ ]] in some books inspite of [ ].3. The symbol [ ] denotes the process of finding thegreatest integer contained in a real number but notgreater than the real number put in [ ].

Thus, in general y = [f (x)] means that there is agreatest integer in the value f (x) but not greater thanthe value f (x) which it assumes for any x R∈ .

This is why in particular y = [x] means that for aparticular value of x, y has a greatest integer which isnot greater than the value given to x.4. The function y = [x], where [x] denotes integralpart of the real number x, which satisfies the equalityx = [x] + q, where 0 1≤

Function 15

(x) x = [x] + {x} where { } denotes the fractional part

of x x R, ∀ ∈(xi) x x x x R− < ≤ ∀ ∈1 ,(xii) x x x≤ < + 1 for all real values of x.

Question: Define “logarithmic” function.Answer: A function f R: 0 , ∞ →� � defined by f (x) =loga x is called logarithmic function, wherea a≠ >1 0, . Its domain and range are 0 , ∞� � and Rrespectively.

Question: Define “Exponential function”.Answer: A function f: R → R defined by f(x) = ax,where a ≠ 1 , a > 0. Its domain and range are R and0 , ∞� � respectively.

Question: Define the “piece wise function”.Answer: A function y = f (x) is called the “piece wisefunction” if the interval (open or closed) in which thegiven function is defined can be divided into a finitenumber of adjacent intervals (open or closed) overeach of which the given function is defined in differentforms. e.g.:

1. f x x x� � = + ≤

16 How to Learn Calculus of One Variable

numbers, the function must be called a real function(or real valued function) but not a real function of areal variable because a function of a real variablesignifies that it is a function y = f (x) whose domainand range are subsets of the set of real numbers.

Question: What do you mean by a “single valuedfunction”?Answer: When only one value of function y = f (x) isachieved for a single value of the independent variablex = a, we say that the given function y = f (x) is asingle valued function, i.e., when one value of theindependent variable x gives only one value of thefunction y = f (x), then the function y = f (x) is said tobe single valued, e.g.:1. y = 3x + 22. y = x2

3. y = sin –1 x, − ≤ ≤π π2 2

y

serves as examples for single valued functions be-cause for each value of x, we get a single value for y.

Question: What do you mean by a “multiple valuedfunction”?Answer: when two or more than two values of thefunction y = f (x) are obtained for a single value of theindependent variable x = a, we say that the givenfunction y = f (x) is a multiple (or, many) valuedfunction, i.e. if a function y = f (x) has more than onevalue for each value of the independent variable x,then the function y = f (x) is said to be a multiple (or,many) valued function, e.g.:

1. x y x x y2 2 2

9 9+ = ⇒ = ± − ⇒ has tworeal values, ∀ 0 and | y |

= y2

= –y for y > 0).

Question: What do you mean by standard functions?Answer: A form in which a function is usually writtenis termed as a standard function.

e.g.: y = xn, sin x, cos x, tan x, cot x, sec x, cosec x,sin–1 x, cos–1 x, tan–1 x, cot–1 x, sec–1 x, cosec–1 x, logax, log ex, ax, ex, etc. are standard functions.

Question: What do you mean by the “inversefunction”?Answer: A function, usually written as f –1 whosedomain and range are respectively the range anddomain of a given function f and under which theimage f –1 (y) of an element y is the element of which ywas the image under the given function f, that is,

f y x f x y− = ⇔ =1 � � � � .

Remarks:1. A function has its inverse ⇔ it is one-one (or,one to one) when the function is defined from itsdomain to its range only.2. Unless a function y = f (x) is one-one, its inversecan not exist from its domain to its range.3. If a function y = f (x) is such that for each value ofx, there is a unique values of y and conversely foreach value of y, there is a unique value of x, we saythat the given function y = f (x) is one-one or we saythat there exists a one to one (or, one-one) relationbetween x and y.4. In the notation f –1, (–1) is a superscript written atright hand side just above f. This is why we shouldnot consider it as an exponent of the base f which

means it can not be written as ff

−=

1 1.

5. A function has its inverse ⇔ it is both one-oneand onto when the function is defined from its domainto its co-domain.

xy f x = ( )

D

fR

yy f x = ( )–1

D

f –1

R

Function 17

Pictorial Representation of InverseFunction

To have an arrow diagram, one must follow thefollowing steps.

1. Let f D R: → be a function such that it is one-one (i.e. distinct point in D have distinct images in Runder f).2. Inter change the sets such that original range of fis the domain of f–1 and original domain of f is therange of f–1.3. Change f to f–1.

Therefore, f D R: → defined by y = f (x) s.t it isone-one ⇔ →

−f R D

1: defined by f–1 (y) = x is an

inverse function.

On Intervals

1. Values and range of an independent variable x: Ifx is a variable in (on/over) a set C, then members(elements or points) of the set C are called the valuesof the independent variable x and the set C is calledthe range of the independent variable x, whereas xitself signifies any unspecified (i.e., an arbitrary)member of the set C.2. Interval: The subsets of a real line are calledintervals. There are two types (or, kinds) of an intervalnamely (i) Finite and (ii) Infinite.(i) Finite interval: The set containing all real numbers(or, points) between two real numbers (or, points)including or excluding one or both of these two realnumbers known as the left and right and points issaid to be a finite interval. A finite interval is classifiedinto two kinds namely (a) closed interval and (b) openinterval mainly.(a) Closed interval: The set of all real numbers xsubject to the condition a x b≤ ≤ is called closedinterval and is denoted by [a, b] where a and b arereal numbers such that a < b.

In set theoretic language, [a, b] = {x: a x b≤ ≤ , xis real}, denotes a closed interval.

Notes:1. The notation [a, b] signifies the set of all realnumbers between a and b including the end points aand b, i.e., the set of all real from a to b.2. The pharase “at the point x = a” signifies that xassumes (or, takes) the value a.3. A neighbourhood of the point x = a is a closedinterval put in the form [a – h, a + h] where h is apositive number, i.e.,

[ a – h, a + h] = { x a h x a h: − ≤ ≤ + , h is asmall positive number}4. All real numbers can be represented by points ona directed straight line (i.e., on the x-axis of cartesiancoordinates) which is called the number axis. Hence,every number (i.e. real number) represents a definitepoint on the segment of the x-axis and converselyevery point on the segment (i.e., a part) of the x-axisrepresents only one real number. Therefore, thenumbers and points are synonymes if they representthe members of the interval concerned.(Notes 1. It isa postulate that all the real numbers can be representsby the points of a straight line. 2. Neigbourhoodroughly means all points near about any specifiedpoint.)(b) Open interval: The set of all real numbers x subjectto the condition a < x < b is called an open intervaland is denoted by (a, b), where a and b are two realnumbers such that a < b.

In the set theoretic language, (a, b) = {x: a < x < b,x is real}

Notes:1. The notation (a, b) signifies the set of all realnumbers between a and b excluding the end points aand b.2. The number ‘a’ is called the left end point of theinterval (open or closed) if it is within the circular orsquare brackets on the left hand side and the numberb is called the right end point of the interval if it iswithin the circular or square brackets on the right hand side.

a b

a b

18 How to Learn Calculus of One Variable

3. Open and closed intervals are represented by thecircular and square brackets (i.e., ( ) and [ ] )respectively within which end points are writtenseparated by a comma.(c) Half-open, half closed interval (or, semi-open, semiclosed interval): The set of all real numbers x such

that a x b< ≤ is called half open, half closed interval(or, semi-open, semi closed interval), where a and bare two real numbers such that a < b.

(a, b) = {x: a x b< ≤ , x is real}Note: The notation (a, b] signifies the set of all realnumbers between a and b excluding the left end pointa and including the right end point b.

(d) Half closed, half open (or, semi closed, semi openinterval): The set of all real numbers x such that

a x b≤ < is called half-closed, half open interval(or, semi closed, semi open interval), where a and b betwo real numbers such that a < b.

In set theoretic language, [a, b) = {x: a x b≤ < , xis real}

Note: The notation [a, b) signifies the set of all realnumbers between a and b including the left end pointa and excluding the right end point b.

2. Infinite interval

(a) The interval −∞ ∞,� � : The set of all real numbersx is an infinite interval and is denoted by −∞ ∞,� � orR.

In set theoretic language,

R = −∞ ∞,� � = {x: −∞ < < ∞x , x is real}

(b) The interval a , ∞� � : The set of all real numbers xsuch that x > a is an infinite interval and is denoted

by a , ∞� � .

In set theoretic language,

a , ∞� � = {x: x > a, x is real}or, a , ∞� � = {x: a x< < ∞ , x is real}

(c) The interval a , ∞� : The set of all real numbersx such that x a≥ is an infinite interval and is denoted

by a , ∞� .

In set theoretic language,

a , ∞� = {x: x a≥ , x is real}or, a , ∞� = {x: a x≥ > ∞ , x is real}

(d) The interval −∞ , a� � : The set of all real numbersx such that x < a is an infinite interval and is denoted

by −∞ , a� � .

−∞ , a� � = {x: x < a, x is real}or, −∞ , a� � = {x: −∞ <

Function 19

2. a x b≤ ≤ signifies the intersection of the two

sets of values given by x a≥ and x b≤ .

3. x a≥ or x b≤ signifies the union of the two

sets of values given by x a≥ and x b≤ .4. The sign of equality with the sign of inequality(i.e., ≥ ≤or ) signifies the inclusion of the specifiednumber in the indicated interval finite or infinite. Thesquare bracket (i.e., [,)also (put before and/after anyspecified number) signifies the inclusion of thatspecified number in the indicated interval finite orinfinite.5. The sign of inequality without the sign of equality(i.e. > or

20 How to Learn Calculus of One Variable

f (1) = 12 – 1 + 1 = 1

and f1

2

1

2

1

21

2���� =���� −

���� +

= − + =1

4

1

21

3

4

2. If f xx

� � = 1 , find f h fh

1 1+ −� � � �.

Solution: ∴ =f xx

� � 1 ...(1)

∴ + =+

f hh

11

1� � ...(2)

f 11

11� � = = ... (3)

∴ + − =+

− = −+

f h fh

h

h1 1

1

11

1� � � � ...(3)

∴ =+ −4 1 1� � � � � �

h

f h f

h

=− +

=+

h h

h h

/ 1 1

1

� �

Type 2: (To evaluate a piecewise function f (x) at apoint belonging to different intervals in which differentexpression for f (x) is defined). In general, a piece wisefunction is put in the form

f (x) = f1 (x), when x > a= f2 (x), when x = a

f3 (x), when x < a, ∀ ∈x Rand one is required to find the values (i) f (a1) (ii) f

(a) and (iii) f (a0), where a, a0 and a1 are specified (or,given) values of x and belong to the interval x > awhich denote the domains of different function f1 (x),f2 (x) and f3 (x) etc for f (x).

Note: The domains over which different expressionf1 (x), f2 (x) and f3 (x) etc for f (x) are defined are intervals

finite or infinite as x > a, x < a, x a≥ , x a≤ , a < x <

b, a x b≤ < , a x b< ≤ and a x b≤ ≤ etc andrepresent the different parts of the domain of f (x).

Working rule: It consists of following steps:

Step 1: To consider the function f (x) = f1 (x) to findthe value f (a1), provided x = a1 > a and to and to putx = a1 in f (x) = f1 (x) which will provide one the valuef (a1) after simplification.Step 2: To consider the function f (x) = f2 (x) to findthe value f (a), provided x = a is the restriction againstf2 (x) and put x = a in f2 (x). If f (x) = f2 (x) when the

restrictions imposed against it are x a≥ , x a≤ ,

a x b≤ < , a x b< ≤ , a x b≤ ≤ or any otherinterval with the sign or equality indicating theinclusion of the value ‘a’ of x, we may consider f2 (x)to find the value f (a). But if f (x) = f2 (x) = constant,when x = a is given in the question, then f2 (x) = givenconstant will be the required value of f (x) i.e. f (x) =given constant when x = a signifies not to find thevalue other than f (a) which is equal to the givenconstant.Step 3: To consider the function f (x) = f3 (x) to findthe value f (a2) provided x = a2 < a and x = a2 in f (x) =f3 (x) which will provide one the value f (a2) aftersimplification.

Remember:

1. f (x) = f1 (x), when (or, for, or, if) a x a≤ < 2 signifiesthat one has to consider the function f (x) = f1 (x) tofind the functional value f1 (x) for all values of x (givenor specified in the question) which lie in between a1and a2 including x = a1.

2. f (x) = f2 (x), when (or, for, or, if) a x a2 3< ≤ ,signifies that one has to consider the function f (x) =f2 (x),to find the functional value f2 (x) for all values ofx (given or specified in the question) which lie inbetween a2 and a3 including x = a3.3. f (x) = f3 (x), when (or, for, or, if) a4 < x < a5 signifiesthat one has to consider the function f (x) = f3 (x) tofind the functional value f3 (x) for all values of x (givenor specified in the question) which lie in between a4and a5 excluding a4 and a5.

Solved Examples1. If f R R: → is defined by

f (x) = x2 – 3x, when x > 2= 5, when x = 2

= 2x + 1, when x < 2, ∀ ∈x R

Function 21

find the values of (i) f (4) (ii) f (2) (iii) f (0) (iv) f (–3)(v) f (100) (vi) f (–500).

Solution: 1. �4 2> ∴, by definition, f (4) = (x2

– 3x) for x = 4 = 42 – 3 (4) = 16 – 12 = 4(ii) � 2 = 2, ∴ by definition, f (2) = 5(iii) 0 < 2, ∴ by definition, f (0) = 2 (0) + 1 = 1(iv) –3 < 2, ∴ by definition, f (–3) = 2 (–3) + 1 = –5(v) 100 > 2, ∴ by definition, f (100) = (100)2 – 3

(100) = 10000 – 300 = 9700(vi) –500 < 2, ∴ by definition f (–500) = 2 (–500) +

1 = –1000 + 1 = –999

2. If f (x) = 1 + x, when − ≤ 0, insuch cases, we may put h = 0.0001 for easiness toguess in which domain (or, interval) the point

represented by x a h= ± lies.e.g.: If a function is defined as under

f (x) = 1 + x, when − ≤

22 How to Learn Calculus of One Variable

Definition: If f D R: → defined by y = f (x) be areal valued function of a real variable, then the domainof the function f represented by D (f) or dom (f) isdefined as the set consisting of all real numbersrepresenting the totality of the values of theindependent variable x such that for each real valueof x, the function or the equation or the expression inx has a finite value but no imaginary or indeterminatevalue.

Or, in set theoretic language, it is defined as:

If f D R: → be real valued function of the realvariable x, then its domain is D or D (f) or dom (f)

= { x R f x∈ : � � has finite values }= { x R f x∈ : � � has no imaginary or indeterminate

value.}

To remember:1. Domain of sum or difference of two functions f (x)

and g (x) = dom f x g x� � � �± = dom (f (x)) ∩ dom(g (x)).2. Domain of product of two functions f (x) and g (x)

g (x) = dom f x g x� � � �⋅ = dom (f (x)) ∩ dom (g (x)).3. Domain of quotient of two functions f (x) and g (x)

=���

���

= ∩ ∩dom dom domf x

g xf x g x

� �� � � � � �� �

x g x: � �� ≠ 0

= ∩ −dom domf x g x� �� � � �� � x g x: � �� ≠ 0 i.e.,the domain of a rational function or the quotientfunction is the set of all real numbers with the exceptionof those real numbers for which the function indenominator becomes zero.

Notes: 1. The domain of a function defined by aformula y = f (x) consists of all the values of x but novalue of y (i.e., f (x)).2. (i) The statement “f (x) is defined for all x” signifies

that f (x) is defined in the interval −∞ ∞,� � .(ii) The statements “f (x) is defined in an intervalfinite or infinite” signifies that f (x) exists and is realfor all real values of x belonging to the interval. Hence,

the statement “f (x) is defined in the closed interval[a, b]” means that f (x) exists and is real for all realvalues of x from a to b, a and b being real numberssuch that a < b. Similarly, the statement “f (x) is definedin the open interval (a, b)” means that f (x) exists andis real for all real values of x between a and b (excludinga and b)

3. (i) f x g xf x

g x� � � � � �� �⋅ = ⇔

==

���

00

0, or

(ii) f x g x

f x

g xf x

g x

� � � �

� �� �� �� �

⋅ ≥ ⇔

≥≥

���

≤≤

���

�

�

����

0

0

00

0

, or

(iii) f x g x

f x

g xf x

g x

� � � �

� �� �� �� �

⋅ ≤ ⇔

≥≤

���

≤≥

���

�

�

����

0

0

00

0

, or

(iv)f x

g x

f x

g xf x

g x

� �� �

� �� �� �� �

≥ ⇔

≥<

���

≤>

���

�

�

����

0

0

00

0

, or

(v)f x

g x

f x

g xf x

g x

� �� �

� �� �� �� �

≤ ⇔

≥<

���

≤>

���

�

�

����

0

0

00

0

, or

4. (i) x a a x a2 2

0− < ⇔ − < ⇔ < − >� � or

(iv) x a x a x a2 2

0− ≥ ⇔ ≤ − ≥� � or

Function 23

5. (i) x a x b a x b a b− − < ⇔ < < < ⇔1 1 1 1 1 10� � � � � �x a b∈ 1 1,� �(ii) x a x b a x b a b− − ≤ ⇔ ≤ ≤ < ⇔2 2 2 2 2 20� � � � � �x a b∈ 2 2,

They mean the intersection of(a) x > a1 and x < b1(b) x a x b≥ ≤2 2and

(iii) x a x b x a x b a b− − > ⇔ < >

24 How to Learn Calculus of One Variable

Note: When the roots of the equation g (x) = 0 areimaginary then the domain of the quotient function

put in the form: f x

g x g x

� �� � � �or

1 = R

Solved ExamplesFind the domain of each of the following functions:

1. yx x

x x=

− +

+ −

2

2

3 2

6

Solution: yx x

x x= − +

+ −

2

2

3 2

6

Now, putting x2 + x – 6 = 0⇒ x2 + 3x – 2x – 6= 0⇒ x (x + 3) –2 (x + 3) = 0⇒ (x + 3) (x – 2) = 0⇒ x = 2, –3∴ domain = R – {2, 3}

2. yx x

x x=

− +

+ +

2

2

2 4

2 4

Solution: yx x

x x=

− +

+ +

2

2

2 4

2 4

Now, putting, x2 + 2x + 4 = 0⇒ x2 + 2x + 4 = 0⇒ (x + 1)2 + 3 = 0⇒ (x + 1)2 = –3

⇒ + = ± −x 1 3� �

⇒ = − ± −x 1 3 imaginary or complex numbers.

∴ domain = R

3. yx

x=

−5

Solution: yx

x=

−5Now, putting, 5 – x = 0⇒ x = 5

∴ domain = R – {5}

4. yx

x=

−+

2 4

2 4

Solution: yx

x=

−+

2 4

2 4

Now, putting, 2x + 4 = 0

⇒ − ⇒ =−

= −2 44

22x x

∴ domain R – {2}

5. f xx x

� � � � � �=

− −1

1 2

Solution: f xx x

� � � � � �=

− −1

1 2

Now, putting (x – 1) (x – 2) = 0⇒ x = 1, 2∴ domain = R – {1, 2}

6. yx

=−

1

12

Solution: yx

=−

1

12

Now, putting, x2 – 1 = 0

⇒ = ⇒ = ±x x2

1 1

∴ domain = R – {–1, 1}

7. yx

=1

Solution: yx

= 1

Now, putting x = 0 ⇒ x = 0 i.e. y is undefined at x= 0

∴ domain = R – {0}

8. yx x

x x=

− +

+ −

2

2

3 2

6

Solution: yx x

x x=

− +

+ −

2

2

3 2

6

Function 25

Now, putting, x2 + x – 6 = 0 ⇒ x2 + 3x – 2x – 6 = 0⇒ x (x + 3) –2(x + 3) = 0 ⇒ (x – 2) (x + 3) = 0 ⇒ x =2, –3

∴ domain = R – {2, –3}

9. y x=

−1

2 6

Solution: yx

=−1

2 6

Now, putting 2x – 6 = 0

⇒ = =x 62

3

∴ domain = R − = −∞ ∪ + ∞3 3 3 � � � � �, ,

10. yx x

=− +

1

5 62

Solution: yx x

=− +

1

5 62

⇒ x2 – 5x + 6 = 0⇒ x2 – 3x – 2x + 6 = 0⇒ x (x – 3) –2 (x –3) = 0⇒ (x – 3) (x – 2) = 0⇒ x = 2, 3

∴ domain = R − = −∞ ∪ ∪2 3 2 2 3, , , � � � � �3 , ∞� �

Type 3: Problems based on finding the domain of thesquare root of a function put in the forms:

(i) f x� �

(ii)f x

g x

� �� �

(iii)1

g x� �

(iv)f x

g x

� �� �

Now we tackle each type of problem one by one.

1. Problems based on finding the domain of a function

put in the form: f x� � .It consists of two types when:

(i) f (x) = ax + b = a linear in x.(ii) f (x) = ax2 + bx + c = a quadratic in x.(i) Problems based on finding the domain of a

function put in the form: f x� � , when f (x) = ax + b.

Working rule: It consists of following steps:Step 1: To put a x b+ ≥ 0Step 2: To find the values of x for which a x b+ ≥ 0to get the required domain.Step 3: To write the domain = [root of the inequation

a x b+ ≥ + ∞0 , )

Notes: 1. The domain of a function put in the form

f x� � consists of the values of x for which

f x� � ≥ 0 .

2. x c x c≥ ⇔ ∈ + ∞, � .Solved ExamplesFind the domain of each of the following fucntions:

1. y x=

Solution: y x=

Now, putting x x≥ ⇒ ≥0 0

∴ domain = + ∞0 , �2. y x= −2 4

Solution: y x= −2 4

Now, putting 2 4 04

22x x− ≥ ⇒ ≥ =

∴ domain = + ∞2 , �3. y x x= + − 1

Solution: y x x= + − 1

Putting y x1 = and y x2 = , we havey = y1 + y2

26 How to Learn Calculus of One Variable

∴ domain of y y y= ∩dom dom1 2� � � �

Now, domain of y x D1 1 0= = = + ∞� � � � �say ,[from example 1.] again, we require to find the domain

of y x2 1= −� � .

Putting x x− ≥ ⇒ ≥ ⇒1 0 1 domain y2

= − = = + ∞x D1 12� � � � �say , Hence, domain of

y = D (say) = dom (y1) ∩ dom y2� �= ∩D D1 2= + ∞ ∩ + ∞0 1, ,� �= ∞1, �

(ii) Problems based on finding the domain of afunction put in the form:

y f x= � � , when f (x) = ax2 + bx + c and α β,

are the roots of ax2 + bx + c = 0 α β

Function 27

Now, x x2

5 6 0− + ≥

⇒ − − ≥x x2 3 0� � � �⇒ ≤ ≥x x2 3or∴ domain = R – (2, 3)

4. y x x= − + −2

5 6

Solution: y x x= − + −2

5 6

Now, − + − ≥x x2 5 6 0

⇒ − + ≤x x2

5 6 0

⇒ − − ≤x x2 3 0� � � �⇒ x lies between 2 and 3⇒ ≤ ≤2 3x∴ domain = [2, 3]

5. y x x= − −16 242

Solution: y x x= − −16 242

Now, − − ≥16 24 02

x x

⇒ − − ≥2 3 02

x x

⇒ + ≤2 3 02

x x

⇒ + ≤x x2 3 0� �⇒ − − ≤x x2 3 0� �� �

⇒ x lies between −3

2 and 0

⇒ − ≤ ≤32

0x

∴ domain = −���

���

2

30,

6. y x x= − − −5 62

Solution: y x x= − − −5 62

Now, − − − ≥5 6 02

x x

⇒ + + ≤x x2

6 5 0

⇒ + + + ≤x x x2 5 5 0

⇒ + + + ≤x x x5 5 0� � � �⇒ + + ≤x x5 1 0� � � �⇒ − − − − ≤ ⇒x x1 5 0� �� � � �� � x lies between –5

and − ⇒ − ≤ ≤ −1 5 1x .∴ domain = [–5, –1]

7. y x x= − +1 3� � � �

Solution: y x x= − +1 3� � � �Now, 1 3 0− + ≥x x� � � �⇒ − − + ≥x x1 3 0� � � �⇒ − + ≤x x1 3 0� � � �⇒ − − − ≤x x1 3 0� � � �� �⇒ − ≤ ≤3 1x∴ domain = [–3, 1]

8. y x= −12

Solution: y x= −12

Now, 1 02− ≥x

⇒ − − ≤1 02

x� �

⇒ − ≤x2

1 0

⇒ − + ≤ ⇒ − − − ≤x x x x1 1 0 1 1� �� � � � � �� �0 ⇒ x lies between –1 and +1

⇒ − ≤ ≤1 1x∴ domain = [–1, 1]

9. y x= − −42

Solution: y x= − −42

28 How to Learn Calculus of One Variable

Now, putting 4 02− ≥x

⇒ − ≤x2

4 0

⇒ − + ≤x x2 2 0� � � �⇒ − − − ≤x x2 2 0� � � �� �⇒ x lies between –2 and 2 ⇒ − ≤ ≤2 2x∴ domain = [–2, 2]

10. y x= −1

24

2

Solution: y x= −1

24

2

Now, putting 4 02− ≥x

⇒ − ≤x2

4 0

⇒ − + ≤x x2 2 0� � � �⇒ − − − ≤x x2 2 0� � � �� �⇒ x lies between –2 and 2 ⇒ − ≤ ≤2 2x∴ domain = [–2, 2]

11. y x= − −1

24

2

Solution: y x= − −1

24

2

Now, putting 4 02− ≥x

⇒ − ≤x2

4 0

⇒ − + ≤x x2 2 0� � � �⇒ − − − ≤x x2 2 0� � � �� �⇒ − ≤ ≤2 2x∴ domain = [–2, 2]

12. y x x= − +2

4 3

Solution: y x x= − +2

4 3

Now, x x2 4 3 0− + ≥

⇒ − − + ≥x x x2

3 3 0

⇒ − − − ≥x x3 3 0� � � �⇒ − − ≥x x1 3 0� � � �⇒ x does not lie between 1 and 3⇒ ≤ ≥x x1 3or

∴ domain = R – [1, 3] = −∞ ∪ + ∞, ,1 3� �

13. y x x= − −2 3� � � �

Solution: y x x= − −2 3� � � �

Now, x x− − ≥2 3 0� � � �⇒ ≥ ≥x x2 3or⇒ x does not lie between 2 and 3∴ domain = R – [2, 3] = −∞ ∪ + ∞, ,2 3� �

14. y x x= + +2

2 3

Solution: � y x x= + +2

2 3

Now, x x2

2 3 0+ + ≥

⇒ + + ≥ ∀x x1 2 02� � ,⇒ + ≥ −x 1 22� � , which is true for all x R∈∴ domain = R = −∞ ∞,� �

Type (ii): Problems based on finding the domain of

a function put in the form : yf x

g x=

� �� �

While finding the domain of the square root

of a quotient function (i.e; yf x

g x=

� �� � ) one must

remember the following facts:

+ ∞+ ∞ 1 30

Function 29

1. The domain of yf x

g x( )=

� �� � consists of those

values of x for which f x

g x

� �� � ≥ 0

2.f x

g xf x g x

� �� � � � � �≥ ⇔ ≥ >0 0 0, , or f x� � ≥ 0 ,

g (x) < 0.

3.x

xx

−−

���

��� ≤ ⇔ ≤ <

αβ

α β0 or β α< ≤x ac-

cording as α β β α< β if α β< .

5. The function in the denominator ≠ 0 always.

Solved ExamplesFind the domain of each of the following functions:

1. yx

x=

−+

3 2

2 6

Solution: y is defined for those x for which

3 2

2 60

x

x

−+

≥

⇔ (1) 3 2 02 6 0

2

3

x

xx

− ≥+ >

���

≥� �� � , i.e;

or, (2) 3 2 0

2 6 03

x

xx

− ≤+ <

���

< −� �� � , i.e;

(1) and (2) ⇒ ≥x2

3, or x x< − ⇔ ∈3

−∞ − ∪ + ∞���

��, ,3

2

3� �

Hence, domain = R − −���

�� = −∞ − ∪3

2

33, ,� �

2

3, + ∞�

����

or, alternatively:

3 2

2 60

233

0 3x

x

x

xx

−+

≥ ⇔−

+≥ ⇔ < − or

x x≥ ⇔ ∈ −∞ − ∪ + ∞���

��

2

33

2

3, ,� �

Hence, domain = −∞ − ∪ + ∞���

��, ,3

2

3� �

2. yx

x=

−+

1

1

Solution: y is defined for all those x for which

x

x

x

xx

−+

≥ ⇔−

− −≥ ⇔ < −

1

10

1

10 1� � or x ≥ ⇔1

x ∈ −∞ − ∪ + ∞, ,1 1� � �Hence, domain = −∞ − ∪ + ∞, ,1 1� � �

3. yx

x=

−+

2

2

Solution: y is defined for all those x for which

x

x

x

xx

−+

≥ ⇔−

− −≥ ⇔ < −

2

20

2

20 2� �

or, x ≥ 2

⇔ ∈ −∞ − ∪ + ∞x , ,2 2� � �Hence, domain = −∞ − ∪ + ∞, ,2 2� � �

Type (iii): Problems on finding the domain of a

function put in the form: yg x

=1

� �

Working rule: It consists of following steps:1. To put g (x) > 02. To find the values of x for which g (x) > 0

0– 3 ∞−∞

α β

23

30 How to Learn Calculus of One Variable

3. To form the Domain with the help of the roots ofthe in equation g (x) > 0.

Note: The domain of a function put in the form

yg x

=1

� � consists of all those values of x for which

g (x) > 0.

Solved Examples1. Find the domain of each of the following functions:

yx x

=− +

1

2 3� � � �Solution: y is defined for all those values of x forwhich (2 – x) (x + 3) > 0 ⇔ (x – 2) (x + 3) < 0 ⇔ x liesbetween –3 and 2 ⇔ –3 < x < 2 ⇔ ∈ −x 3 2,� � hence,domain = (–3, 2)

2. yx x

=− +

1

1 2� � � �Solution: y is defined for all those values of x forwhich (1 – x) (x + 2) > 0 ⇔ (x – 1) (x – (–2)) < 0 ⇔ xlies between –2 and 1 ⇔ –2 < x < 1 ⇔ ∈ −x 2 1,� �hence, domain = (–2, 1).

3. yx x

=− +

1

5 62

Solution: y is defined for all those values of x forwhich x2 – 5x + 6 > 0 ⇔ x2 – 3x – 2x + 6 > 0 ⇔ (x –3) x – 2 (x – 3) > 0 ⇔ (x – 3) (x – 2) > 0 ⇔ x < 2 or x >

3 ⇔ ∈ −∞ ∪ + ∞x , ,2 3� � � �Hence, domain = R − = −∞ ∪2 3 2, ,� �

3 , + ∞� �

4. yx

=−1

Solution: y is defined for all those values of x for

which –x > 0 ⇔ x < 0 ⇔ ∈ −∞x , 0� �Hence, domain = −∞ , 0� � .

Type (iv): Problems on finding the domain of a

function put in the form: yf x

g x=

� �� �

.

Working rule: The rule to find the domain of a

function of the form yf x

g x=

� �� �

is the same as for

the domain of a function of the form yg x

=1

� �which means.1. To put g (x) > 0 and to find the values of x from thein equality g (x) > 0.2. To form the domain with the help of obtained valuesof x.

Solved ExamplesFind the domain of each of the following functions:

1. yx

x x=

− +2 3 2

Solution: y is defined when x2 – 3x + 2 > 0 ⇔ x2 – 2x– x + 2 > 0 ⇔ x (x – 2) – (x – 2) > 0 ⇔ (x – 1) (x – 2)> 0 ⇔ x < 1 or x > 2.

Hence, domain = R – [1, 2] = −∞ ∪ + ∞, ,1 2� � � �

2. yx

x x=

− −1 2� � � �Solution: y is defined when (1 –x) (x – 2) > 0 ⇔ (x –1) (x – 2) < 0 ⇔ x lies between 1 and 2 ⇔ 1 < x < 2

⇔ ∈x 1 2,� � .Hence, domain = (1, 2)

Finding the Domainof Logarithmic Functions

There are following types of logarithmic functionswhose domains are required to be determined.(i) y = log f (x)

(ii) y f x= log � �(iii) y = log | f (x) |

Function 31

(iv) yf x

g x=

���

���

log� �� �

(v) yf x

g x=

� �� �log

(vi) y = log log log f (x)Now we tackle each type of problem one by one.

Type 1: Problems based on finding the domain of afunction put in the form: y = log f (x).

Working rule: It consists of following steps:Step 1: To put f (x) > 0 and to solve the in equality f(x) > 0 for x.Step 2: To form the domain with the help of obtainedvalues of x.

Notes: 1. The domain of the logarithmic function y =log f (x) consists of all those values of x for which f (x)> 0.2. Log f (x) is defined only for positive f (x).

Solved ExamplesFind the Domain D of each of the following functions:1. y = log (4 – x)Solution: y is defined when 4 – x > 0 ⇔ –x > –4 ⇔x < 4.

∴ = −∞D y� � � �, 42. y = log (8 – 2x)Solution: y is defined when (8 – 2x) > 0 ⇔ –2x > –8⇔ x < 4.

∴ = −∞D y� � � �, 43. y = log (2x + 6)Solution: y is defined when (2x + 6) > 0 ⇔ 2x > –6

⇔ x > –3 ⇔ ∈ − + ∞x 3 ,� �Hence, D y� � � �= − + ∞3 ,

4. y = log {(x + 6) (6 – x)}Solution: y is defined when (x + 6) (6 – x) > 0 ⇔ (x +6) (x – 6) < 0 ⇔ x lies between –6 and 6 ⇔ –6 < x <

6 ⇔ ∈ −x 6 6,� �Hence, D (y) = (–6, 6)

5. y = log (3x2 – 4x + 5)

Solution: Method (1)y is defined when (3x2 – 4x + 5) > 0 ⇔

34

3

5

30

2x x– +��

�� >

⇔ + ���� −

���� +

���

���

�

���

�

���

>3 43

4

6

4

6

5

30

22 2

x x–

⇔ ���� + −

��

��

����

����

>3 23

5

3

4

90

2

x –

⇔ ���� +

���� >3

2

33

11

90

2

x – � �

⇔ −���� > −3

2

3

11

9

2

x which is true ∀ ∈x R

∴ = = −∞ + ∞D y R� � � �,Notes: 1. Imaginary or a complex numbers as the

roots of an equation a x b x c2

0+ + = ⇔ domain

of log f x R� � � �= = −∞ + ∞, as in the aboveexample roots are complex.2. The method adopted in the above example is called“if method”.3. A perfect square is always positive which is greaterthan any negative number.

Method 2. This method consists of showing that

a x b x c x2

0+ + > ∀, if a > 0 and discriminant = b2

– 4ac < 0 here 3 > 0, and discriminant = 16 – 60 = – 34< 0

∴ y is defined ∀ ∈x R

Therefore, D y R� � � �= = −∞ + ∞,6. y = log (x3 – x)Solution: y is defined when (x3 – x) > 0 ⇔ x (x2 – 1)>