How to learn calculus of one variable vol i

If x be a symbol representing an unspecified element of a set D, then x is said to vary over the set D i.e., x can stand for any element of the set D, i.e., x can take any value of the s

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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 as

a 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 range

of 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 advantages

by 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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10 Differentiation of Inverse Trigonometric Functions 424

16 Evaluation of Derivatives for Particular Arguments 615

19 Tangent and Normal to a Curve 692

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

Question: What is a quantity?

Answer: In fact, anything which can be measured or

which can be divided into parts is called a quantity

But in the language of mathematics, its definition is

put in the following manner

Definition: Anything to which operations of

mathematics (mathematical process) such as addition,

subtraction, multiplication, division or measurement

etc are applicable is called a quantity

Numbers of arithmetic, algebraic or analytic

expressions, distance, area, volume, angle, time,

weight, space, velocity and force etc are all examples

of quantities

Any quantity may be either a variable or a constant

Note: Mathematics deals with quantities which have

values expressed in numbers Number may be real or

imaginary 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 a

variable

Definition: 2 (Set theoritic): In the language of set

theory, a variable is symbol used to represent anunspecified (not fixed, i.e arbitrary) member (element

or point) of a set, i.e., by a variable, we mean an elementwhich can be any one element of a set or which can

be in turn different elements of a set or which can be

a particular unknown element of a set or successivelydifferent unknown elements of a set We may think of

a 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 unspecified element of a set D, then x is said to vary over the set

D (i.e., x can stand for any element of the set D, i.e., x

can take any value of the set D) and is called a variable

on (over) the set D whereas the set D over which the variable x varies is called domain or range of x.

Example: Let D be the set of positive integers and x

Î D = {1, 2, 3, 4, …}, then x may be 1, 2, 3, 4, … etc.

Note: A variable may be either (1) an independent

variable (2) dependent variable These two terms havebeen explained while defining a function

Question: What is a constant?

Answer:

Definition 1 (General): If in a mathematical

discussion, a quantity cannot assume more than one

vale, then the quantity is called a constant or a

constant quantity and is denoted by a symbol

Examples: 1 The weights of men are different for

different individuals and therefore weight is a variable

But the numbers of hands is the same for men of

different weights and is therefore a constant

2 The position of a point moving in a circle is a

variable but the distance of the point from the centre

of the circle is a constant

3 The expression x + a denotes the sum of two

quantities The first of which is variable while the

second is a constant because it has the same value

whatever values are given to the first one

Definition: 2 (Set theoritic): In the language of set

theory, a constant is a symbol used to represent a

member of the set which consists of only one member,

i.e if there is a variable ‘c’ which varies over a set

consisting of only one element, then the variable ‘c’

is called a constant, i.e., if ‘c’ is a symbol used to

represent precisely one element of a set namely D,

then ‘c’ is called a constant.

Example: Let the set D has only the number 3; then

c = 3 is a constant.

Note: Also, by a constant, we mean a fixed element

of a set whose proper name is given We often refer to

the proper name of an element in a set as a constant

Moreover by a relative constant, we mean a fixed

element of a set whose proper name is not given We

often refer to the “alias” of an element in a set as a

relative constant

Remark: The reader is warned to be very careful

about the use of the terms namely variable and

constant These two terms apply to symbols only not

to numbers or quantities in the set theory Thus it is

meaningless to speak of a variable number (or a

variable quantity) in the language of set theory for

the simple reason that no number is known to human

beings which is a variable in any sense of the term

Hence the ‘usual’ text book definition of a variable as

a quantity which varies or changes is completely

misleading 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 the

same 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 equations

takes rest This is why 1 = 1, 2 = 2, 3 = 3, … x = x and

a = a and so on).

(ii) p and logarithm of positive numbers (as log2, log

3, log 4, … etc) are also included in absolute constants

2 Arbitrary constants: Any arbitrary constant is

one 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 speaking

of any letter, variable or constant, other than thecoordinate variables in an equation of a curve defined

by 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 a circle, are variables while ‘a’ the radius of a circle may

have any constant value and is therefore an arbitraryconstant or parameter

(ii) The general form of the equation of a straight line

put in the form y = mx + c contains two parameters namely 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 represent constants while the letters from “t to z” of the English

alphabet are taken to represent variables

Question: What is increment?

Answer: An increment is any change (increase or

growth) in (or, of) a variable (dependent or

independent) It is the difference which is found by

subtracting the first value (or, critical value) of the

variable 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 making

addition, positive or negative, to a given value (initial

value) of a variable

(ii) The increments may be positive or negative, in

both cases, the word “increment” is used so that a

negative increment is an algebraic decrease

Examples on Increment in a Variable

1 Let x1 increase to x2 by the amount ∆ x Then we

can set out the algebraic equation x1 + ∆ x = x2 which

⇒ ∆ x = x2 – x1

2 Let y1 decrease to y2 by the amount ∆ y Then we

can 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.

F

HG I KJ.

Question: What is the symbol used to represent (or,

denote) an increment?

Answer: The symbols we use to represent small

increment 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 as

it increases or, decreases from the initial value toanother value, i.e., the notation ∆ x is used to denote

a fixed non zero, number that is added to a given

number x0 to produce another number x = x0 + ∆ x if

y = f (x) then ∆ y = f (x0 + ∆ x) – f (x0)

Notes: If x, y, u v are variables, then increments in

them are denoted by ∆ x, ∆ y, ∆ u, ∆ v respectively signifying how much x, y, u, v increase or decrease,

i.e., an increment in a variable (dependent orindependent) tells how much that variable increases

or decreases

Let us consider y = x2

When x = 2, y = 4

x = 3, y = 9

∴ ∆ 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 is

defined 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 element

of D is associated (or, related, or, paired) with a unique

(i.e., a single, or, one and only one, or, not more than

one) element of R The set D is called the domain of

the function while the set R is called the range of the

function Moreover, elements of the domain (or, the

set D) are called the independent variables and the

elements of the range (or, range set or, simply the set

R) are called the dependent variables If x is the

element of D, then a unique element in R which the

rule (or, rules) symbolised as f assigns to x is termed

“the value of f at x” or “the image of x under the rule

f” which is generally read as “the f-function of x” or, “f

of x” Further one should note that the range R is the

set of all values of the function f whereas the domain

D is the set of all elements (or, points) whose each

element is associated with a unique elements of the

range set R.

Functions are represented pictorially as in the

accompanying diagram

One must think of x as an arbitrary element of the

domain D or, an independent variable because a value

f of x can be selected arbitrarily from the domain D as

well as y as the corresponding value of f at x, a

dependent variable because the value of y depends

upon the value of x selected It is customary to write

y = f (x) which is read as “y is a function of x” or, “y is

f of x” although to be very correct one should say

that y is the value assigned by the function f

corresponding to the value of x.

Highlight on the Term “The Rule or the Law”.

1 The term “rule” means the procedure (or

procedures) or, method (or, methods) or, operation(or, operations) that should be performed over the

independent variable (denoted by x) to obtain the value 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 = e x, … 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 independent

variable x which produces the value for the dependent variable 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 set

signifies any specified member (or, element or point)

of that set)

2 The precise relationship between two sets of

corresponding values of dependent and independentvariables is usually called a law or rule Often the rule

is 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 x

x

a f=+

1

1 2 tells that one should

square the independent variable x, add unity and then

divide unity by the obtained result to get the value of

the function f at the point x, i.e., to square the independent variable x, to add unity and lastly to

divide unity by the whole obtained result (i.e., square

of the independent variable x plus unity).

D

x y = ( )f x

R

(ii) f (x) = x2 + 2, where the rule f signifies to square

the number x and to add 2 to it.

(iii) f (x) = 3x – 2, where the rule f signifies to multiply

x by 3 and to subtract 2 from 3x.

(iv) C = 2πr an equation involving the variables C

(the circumference of the circle) and r (the radius of

the circle) which means that C= 2 πr = a function

of r.

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

means that y= 64s 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 mathematical

operation is to be performed over (or, on) the elements

of the domain) which takes the elements of the domain

D, processes them and produces the elements of the

range R.

Example of a function of functions:

Integration of a continuous function defined on some

closed interval [a b] is an example of a function of

functions, namely the rule (or, the correspondence)

that associates with each object f (x) in the given set

of objects, the real number f x dx

a

b

a f

Notes: (i) We shall study functions which are given

by simple formulas One should think of a formula as

a 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 in

terms of x say y = f (x), to find the value of f at a

number a, we substitute that number a for x wherever

x occurs in the given formula and then simplify it.

(ii) For x∈D f x , a f∈R should be unique means

that f can not have two or more values at a given

point (or, number) x.

(iii) f (x) always signifies the effect or the result of

applying the rule f to x.

(iv) Image, functional value and value of the function

“f is a function from D to R”.

2. f x: → y or, x y

f

→ or, x→ f xa f for “a

function f from x to y” or “f maps (or, transforms) x into y or f (x)”.

3. f D: → R defined by y = f (x) or, f D: →R by

y = 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 D

signifies “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 of

the domain, we use the notation (1)

(ii) When we want to indicate only the images of

elements of the domain, we use the notation (2)

(iii) When we want to indicate the range and the rule

of a function together with a functional value f (x), we

use the notation (3)

(iv) In the language of set theory, the domain of a

function is defined in the following style:

D (f): kx x: ∈D1p where, D1 = the set ofindependent variables (or, arguments) = the set of all

those members upon which the rule ‘f ’ is performed

to find the images (or, values or, functional values)

(v) In the language of set theory, the range of a

function is defined in the following way:

R fa f a f=lf x x: ∈D f x , a f∈Rq= 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 both

x and y are real numbers consisting of.

(a) Zero

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

–3, –17, … etc

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

5

172

,− ,

… etc

(d) Irrational numbers e.g.: 7,− 14, … etc

(viii) Generally the rule/process/method/law is not

given in the form of verbal statements (like, find the

square root, find the log, exponential, … etc.) but in

the form of a mathematical statement put in the form

of expression containing x (i.e in the form of a formula)

which may be translated into words (or, verbal

statements)

(ix) If it is known that the range R is a subset of some

set 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 f

is a function on the set D into the set C.”

N.B: To define some types of functions like “into

function and on to function”, it is a must to define a

function " f D: →C " where C = codomain and

hence we are required to grasp the notion of

co-domain Therefore, we can define a co-domain of a

function in the following way:

Definition of co-domain: A co-domain of a function

is a set which contains the range or range set (i.e., set

of all values of f) which means R ⊆ , where R = the C

set of all images of f and C = a set containing images

of 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 (or

formula,) 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 eachspecified 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 (corresponding

to a)and it is included in the range set R.

Question: Distinguish between the terms “a function

and a function of x”.

Answer: A function of x is a term used for “an image

of x under the rule f” or “the value of the function f at (or, for) x” or “the functional value of x” symbolised

as y = f (x) which signifies that an operation (or, operations) denoted by f has (or, have) been performed

on x to produce an other element f (x) whereas the

term “function” is used for “the rule (or, rules)” or

“operation (or, operations)” or “law (or, laws)” to be

performed upon x, x being an arbitrary element of a

set known as the domain of the function

Remarks: 1 By an abuse of language, it has been

customary to call f (x) as function instead of f when a particular (or, specifies) value of x is not given only

for convenience Hence, wherever we say a “function

f (x) what we actually mean to say is the function f

whose 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 transformation

are synonymes

4 If domain and range of a function are not known, it

is 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 and

independent variables”

Answer:

1 Independent variable: In general, an independent

variable is that variable whose value does not depend

f D

upon any other variable or variables, i.e., a variable in

a mathematical expression whose value determines

the value of the whole given expression is called an

independent variable: in y = f (x), x is the independent

variable

In set theoretic language, an independent variable

is the symbol which is used to denote an unspecified

member of the domain of a function

2 Dependent variable: In general a dependent

variable is that variable whose value depends upon

any other variable or variables, i.e., a variable (or, a

mathematical equation or statement) whose value is

determined by the value taken by the independent

variable is called a dependent variable: in y = f (x), y is

the dependent variable

In set theoretic language, a dependent variable is

the symbol which is used to denote an unspecified

member of the range of a function

e.g.: In A= f rb g= πr2

, r is an independent variable and A is a dependent variable.

Question: Explain the term “function or function of

x” in terms of dependency and independency.

Answer: When the values of a variable y are

determined by the values given to another variable x,

y is called a function of (depending on) x or we say

that y depends on (or, upon) x Thus, any expression

in x depends for its value on the value of x This is

why an expression in x is called a function of x put in

the form: y = f (x).

Question: What are the symbols for representing the

terms “a function and a function of a variable”?

Answer: Symbols such as f, F, φ etc are used to

denote a function whereas a function of a variable is

denoted by the symbols f (x), φ x f ta f a f, , F ta f,

φ ta f and can be put in the forms: y = f (x); y= φa fx ;

y = f (t); y = F (t); y= φa ft , that y is a function of

(depending on) the variable within the circular bracket

( ), i.e., y depends upon the variable within circular

2 The value of f /functional value of f corresponding

to x = a / the value of the dependent variable y for a

particular value of the independent variable is

symbolised as (f (x)) x = a = f (a) or [f (x)] x = a = f (a) while evaluating the value of the function f (x) at the point

x = 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

= 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 is

obtained directly by putting in the given value of the

independent variable x = a wherever x occurs in a

given mathematical equation representing a function,

we say that the function f or f (x) has the actual value

f (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 some

definite quantity is termed as the approaching (or,

limiting) value of the function y = f (x) at x = a This

value may be calculated when the actual value of the

function f (x) becomes indeterminate at a particular value ‘a’ of x.

5 When the actual value of a function y = f (x) is

anyone of the following forms: 0

, , imaginary, any real number

0

for a particular value ‘a’ of x, it is said that the function

f (x) is not defined or is indeterminate or is meaningless

at x = a.

6 To find the value of a function y = f (x) at x = a

means to find the actual value of the function y = f (x)

at x = a.

Pictorial Representation of a Function, its

Domain and Range.

1 Domain: A domain is generally represented by

any closed curve regular (i.e., circle, ellipse, rectangle,

square etc) or irregular (i.e not regular) whose

members are represented by numbers or alphabets or

dots

2 Range: A range is generally represented by

another closed curve regular or irregular or the some

closed curve regular or irregular as the domain

3 Rule: A rule is generally represented by an arrow

or arc (i.e., arc of the circle) drawn from each member

of the domain such that it reaches a single member or

more than one member of the codomain, the codomain

being a superset of the range (or, range set)

Remarks:

1 We should never draw two or more than two arrows

from a single member of the domain such that it reaches

more than one member of the codomain to show that

the venn-diagram represents a function Logic behind

it is given as follows

If the domain are chairs, then one student can not

sit on more than one chair at the same time (i.e., one

student can not sit on two or more than two chairs at

the 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 the

word “rule” means

(i) Every point/member/element in the domain D is

joined by an arrow a f→ or arc a f∩ to some point in

range 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 joined

to the same point in R⊆C (See Fig 1.4 where the

points 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 two

or 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 not

joined to any element in D (See Fig 1.4 where the points y4, y5 and y6 in C are not joined to any point in

D.

Precaution: It is not possible to represent any

function as an equation involving variables always

At such circumstances, we define a function as a set

of 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 element

is twice its corresponding first element

But f = {(0, 1), (0, 2), (0, 3), (0, 4)} does not define a

function since its first element is repeated

Note: When the elements of the domain and the range

are represented by points or English alphabet with

subscripts as x1, x2, … etc and y1, y2, … etc

respectively, we generally represent a function as a

set 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 xa f∈R} provided it is not possible to

represent the function as an equation y = f (x).

Question: What is meant whenever one says a

function 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 provided

the value of the function f (x) at x = a (i.e f (a)) is finite

which means that the value of the function f (x) at x =

a should not be anyone of the following forms

(i) A symbol in mathematics is said to have been

defined when a meaning has been given to it

(ii) A symbol in mathematics is said to be undefined

or non-existance when no meaning is attributed to

the 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,cos−15 5, ÷ 0, log (–3), 52 are

undefined or they are said not to exist

(iii) Whenever we say that something exists, we mean

that it has a definite finite value

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

(i) Algebraic function: A function which satisfies

the equation put in the form:

taking m = 1 and A0 = a constant in algebraic function.

2 The quotient of two polynomials termed as a

rational function of x put in the form:

If f1 (x) and f2 (x) are two polynomials, then general

rational functions may be denoted by R x f x

f x

a f a faf= 1 2

where R signifies “a rational function of” In case f2(x) reduces itself to unity or any other constant (i.e., a term not containing x or its power), R (x) reduces

itself to a polynomial

3 Generally, there will be a certain number of values

of x for which the rational function is not defined and these are values of x for which the polynomial in

4 Rational integral functions: If a polynomial in x is

in a rational form only and the indices of the powers

of x are positive integers, then it is termed as a rational

integral function

5 A combination of polynomials under one or more

radicals termed as an irrational functions is also an

algebraic function Hence, y= x = f xa f; y=

x5 3 = f xa f; y x

x

=+24 serve as examples for

irrational algebraic functions

6 A polynomial or any algebraic function raised to

any power termed as a power function is also an

algebraic function Hence, y= x n ,a f a fn∈R = f x ;

y= x2 + 3 = f x

1

e j a f serve as examples for power

functions which are algebraic

Remarks:

1 All algebraic, transcendental, explicit or implicit

function or their combination raised to a fractional

power reduces to an irrational function Hence,

y= x5 3 = f xa f; y=asinx+ xf a f1 = f x

as examples for irrational functions

2 All algebraic, transcendental, explicit or implicit

function or their combination raised to any power is

always regarded as a power function Hence, y = sin2

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

functions

Transcendental function: A function which is not

algebraic is called a transcendental function Hence,

all trigonometric, inverse trigonometric, exponential

and logarithmic (symoblised as “TILE”) functions are

transcendental 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),

a x (for any a > 0), e x , [f (x)] g (x) etc serve as examples

for transcendental functions

Notes: (In the extended real number system)

(A)

(i) e x = ∞ when x= ∞

(ii) e x = 1 when x = 0

(iii) e x = 0 when x= − ∞

(B) One should remember that exponential functions

obeys the laws of indices, i.e.,

−

= 1

(C) (i) log 0= − ∞

(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 a

function put in the form y = f (x) which signifies that a relation between the dependent variable y and the independent variable x put in the form of an equation can be solved for y and we say that y is an explicit function 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) serve

as 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) is

called an explicit transcendental function otherwise it

is called an explicit algebraic function

(ii) Implicit function: An implicit function is a

function put in the form: f (x, y) = c, c being a constant, which signifies that a relation between the variables y and x exists such that y and x are in seperable in an equation and we say that y is an implicit function of x Hence, x3 + y2 = 4xy serves as an example for the implicit 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 ordered pain (x, y) signifies the combination of the variables x and y, then f (x, y) = c is called an implicit algebraic function of x, i.e., y is said to be an implicit algebraic function of x, if a relation of the form:

y m + R1 y m – 1 + … + R m = 0 exists, where R1, R2, …

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

Note: Discussion on “the explicit and the implicit

functions” has been given in detail in the chapter

“differentiation of implicit function”

On Some Important Functions

Some types of functions have been discussed in

previous sections such as algebraic, transcendental,

explicit and implicit functions In this section definition

of some function used most frequently are given

1 The constant function: A function f: R →R

defined 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 function

represents 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 = 3

are constant functions

Remarks:

(i) A polynomial a0 x n + a1 x n – 1 + … a m – 1 x + a m

(whose domain and range are sets of real numbers)

reduces to a constant function when degree of

polynomial is zero

(ii) In particular, if c = 0, then f (x) is called the “ zero

function” and its graph is the x-axis itself

2 The identity function: A function f: R →R

defined 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 of

45° with the x-axis, i.e., an identity function represents

straight lines passing through origin and making an

angle of 45° with the x-axis

3 The reciprocal of identity function: A function

x

a f= 1 is called the

reciprocal function of the identity function f (x) = x or

simply reciprocal function

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 fact

that 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 value

Notes: (A) A function put in the form | f (x) | is called

the “modulus of a function” or simply “modulus of afunction” which signifies that:

(i) | f (x) | = f (x), provided f xa f≥ 0 , i.e., if f (x) is positive or zero, then | f (x) | = f (x).

(ii) | f (x) | = –f (x), provided f (x) < 0, i.e., if f (x) is

negative, 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.

= –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 range

of absolute value function = R (f) = {non negative real numbers} = R+ ∪{0}

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

| x – a | = –(x – a) when a fx −a <0

(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., we

do not know whether f (x) is positive or negative),

then we generally use the following definition of the

absolute value of a function

f xa f = f xa f2 = f2a fx

3 Absolute means to have a magnitude but no sign.

4 Absolute value, norm and modulus of a function

are synonymes

5 Notation: The absolute value of a function is

denoted by writing two vertical bars (i.e straight lines)

within which the function is placed Thus the notation

to signify “the absolute value of” is “| |”

a f which is false which means this

equation has no solution and a f a fx−2 = − x+3 ⇒

11 | f (x) | n = (f (x) n , where n is a real number.

12. | f xa f|≥ 0 always means that the absolute value

of a functions is always non-negative (i.e., zero orpositive real numbers)

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

Geometric Interpretation of Absolute Value

of 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 the point corresponding to a (i.e x = a) i.e, | x | signifies the distance between the origin and the given point x

= a on the real line.

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 the

distance OP = | OP | = | –OP | = | –x | = x

Hence, | x | =

x, provided x > o means that the absolute value

of a positive number is the positive number

itself

o, provided x = o means the absolute value of

zero is taken to be equal to zero

–x, provided x < o means that the absolute value

of a negative number is the positive value of

that 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 on

the number line are equidistant from the origin We

call the distance of either from zero, the absolute value

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 is

symmetrical about the y-axis having the shape of

English alphabet 'V ' which opens (i) upwards if y =

| x | (ii) downwards if y = – | x | (iii) on the right side if

x = | 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 a

quantity 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 negative

square 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 positive

square root is the one used generally, unless there is

a remark to the contrary Hence, 100=10;

169 =13; x2 = x

2. x2 + y2 = ⇔1 x2 = −1 y2 ⇔ x2

=

1− y2 ⇔ x = 1− y2 ⇔ = ±x 1− y2

e.g.: cos2θ= −1 sin2θ⇔ cosθ =

1−sin2θ ⇔ cosθ= ± 1−sin2θone should note that the sign of cosθ isdetermined by the value of the angle ' 'θ and thevalue of the angle ' 'θ is determined by the quadrant

in which it lies Similarly for other trigonometricalfunctions of θ, such as, tan2 θ = sec2 θ– 1 ⇔ tan

θ= ± sec2θ− ⇔1 tanθ = sec2θ−1cot2θ= cosec2θ− ⇔1 cotθ=

± cosec2θ− ⇔1 cotθ = cosec2θ−1sec2θ= +1 tan2θ⇔ secθ=

± 1 + tan2θ ⇔ secθ = 1 + tan2θ ,w h e r ethe 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

On Greatest Integer Function

Firstly, we recall the definition of greatest integer

function

Definition: A greatest integer function is the function

defined on the domain of all real numbers such that

with any x in the domain, the function associates

algebraically the greatest (largest or highest) integer

which is less than or equal to x (i.e., not greater than

x) designated by writing square brackets around x as

[x].

The greatest integer function has the property of

being less than or equal to x, while the next integer is

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

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

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

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

N Q or [[ ]] in some books inspite of [ ]

3 The symbol [ ] denotes the process of finding the

greatest integer contained in a real number but not

greater than the real number put in [ ]

Thus, in general y = [f (x)] means that there is a

greatest integer in the value f (x) but not greater than

the value f (x) which it assumes for any x∈R

This is why in particular y = [x] means that for a particular value of x, y has a greatest integer which is not greater than the value given to x.

4 The function y = [x], where [x] denotes integral

part of the real number x, which satisfies the equality

x = [x] + q, where 0≤ <q 1 is discontinuous at everyinteger x=0,± ±1, 2, and at all other points, thisfunction is continuous

5 If x and y are two arbitrary real numbers satisfying

the inequality n≤ < + 1x n and n≤ < + 1y n ,

where n is an integer, then [x] = [y] = n.

6 y = [x] is meaningless for a non-real value of x

because its domain is the set of all real numbers and

the range is the set of all integers, i.e D [x] = R and R [x] = {n: n is an integer} = The set of all integers, …

–3, –2, –1, 0, 1, 2, 3, …, i.e., negative, zero or positiveinteger

7. f xa f = ⇔ ≤0 0 f xa f<1 Further the solution

of 0≤ f xa f<1 provides us one of the adjacent

intervals where x lies The next of the a adjacent intervals

is determined by adding 1 to the left and right endpoint of the solution of 0≤ f xa f<1 This process

of adding 1 to the left and right end point is continuedtill we get a finite set of horizontal line segments

representing the graph of the function y = [f (x)]

More on Properties of Greatest Integer Function.

x

L

NM O QP= L NM O QP, ∈ and ∈

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

of x,∀ ∈x R

(xi) x− <1 x ≤ x ,∀ ∈x R

(xii) x ≤ <x x + 1 for all real values of x.

Question: Define “logarithmic” function.

Answer: A function f : 0 ,b g∞ →R defined by f (x) =

loga x is called logarithmic function, where

a≠1, a>0 Its domain and range are b g0 ,∞ and R

respectively

Question: Define “Exponential function”.

Answer: A function f: R →R defined by f(x) = a x,

where a≠ 1, a > 0 Its domain and range are R and

0 ,∞

b g respectively

Question: Define the “piece wise function”.

Answer: A function y = f (x) is called the “piece wise

function” if the interval (open or closed) in which the

given function is defined can be divided into a finite

number of adjacent intervals (open or closed) over

each of which the given function is defined in different

1 Non-overlapping intervals: The intervals which

have no points in common except one of the end

points of adjacent intervals are called non overlapping

intervals whose union constitutes the domain of the

piece wise function e.g.: 0 1

3

13

23

whose union [0, 1] is the domain of the piece wise

function if it is defined as:

23

2 A function y = f (x) may not be necessarily defined

by a single equation for all values of x but the function

y = f (x) may be defined in different forms in different

parts of its domain

3 Piecewise function is termed also “Piecewise defined

function” because function is defined in each piece

If every function defined in adjacent intervals is linear,

it is termed as “Piecewise linerar function” and if everyfunction defined in adjacent intervals is continuous,

it is called “piecewise continuous function.”

Question: What do you mean by the “real variables”? Answer: If the values assumed by the independent

variable ‘x’ are real numbers, then the independent variable ‘x’ is called the “real variable”.

Question: What do you mean by the “real function

(or, real values of function) of a real variable”?

Answer: A function y = f (x) whose domain and range

are sets of a real numbers is said to be a real function(or more clearly, a real function of a real variable) whichsignifies that values assumed by the dependentvariable are real numbers for each real value assumed

by the independent variable x.

Note: The domain of a real function may not be

necessarily a subset of R which means that the domain

of a real function can be any set

Remarks:

1 In example (i) The domain of f is a class of sets and

in example (ii) The domain of f is R But in both examples, the ranges are necessarily subsets of R.

2 If the domain of a function f is any set other than

(i.e different from) a subset of real numbers and therange is necessarily a subset of the set of real

numbers, the function must be called a real function

(or real valued function) but not a real function of a

real variable because a function of a real variable

signifies that it is a function y = f (x) whose domain

and range are subsets of the set of real numbers

Question: What do you mean by a “single valued

function”?

Answer: When only one value of function y = f (x) is

achieved for a single value of the independent variable

x = a, we say that the given function y = f (x) is a

single valued function, i.e., when one value of the

independent variable x gives only one value of the

function y = f (x), then the function y = f (x) is said to

be single valued, e.g.:

1 y = 3x + 2

2 y = x2

3 y = sin –1 x, − ≤ ≤π π

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 valued

function”?

Answer: when two or more than two values of the

function y = f (x) are obtained for a single value of the

independent variable x = a, we say that the given

function y = f (x) is a multiple (or, many) valued

function, i.e if a function y = f (x) has more than one

value 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.:

Question: What do you mean by standard functions?

Answer: A form in which a function is usually written

is termed as a standard function

e.g.: y = x n , 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

a x , log e x , a x , e x, etc are standard functions

Question: What do you mean by the “inverse

f− 1a fy = ⇔x f xa f= y

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 inverse

can not exist from its domain to its range

3 If a function y = f (x) is such that for each value of

x, there is a unique values of y and conversely for

each value of y, there is a unique value of x, we say that the given function y = f (x) is one-one or we say

that there exists a one to one (or, one-one) relation

between x and y.

4 In the notation f–1, (–1) is a superscript written at

right hand side just above f This is why we should not consider it as an exponent of the base f which

means it can not be written as f

Pictorial Representation of Inverse

Function

To have an arrow diagram, one must follow the

following 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 R

under f).

2 Inter change the sets such that original range of f

is the domain of f–1 and original domain of f is the

1 Values and range of an independent variable x: If

x is a variable in (on/over) a set C, then members

(elements or points) of the set C are called the values

of the independent variable x and the set C is called

the range of the independent variable x, whereas x

itself signifies any unspecified (i.e., an arbitrary)

member of the set C.

2 Interval: The subsets of a real line are called

intervals There are two types (or, kinds) of an interval

namely (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 real

numbers known as the left and right and points is

said to be a finite interval A finite interval is classified

into two kinds namely (a) closed interval and (b) open

interval mainly

(a) Closed interval: The set of all real numbers x

subject to the condition a≤ ≤x b is called closed

interval and is denoted by [a, b] where a and b are

real numbers such that a < b.

In set theoretic language, [a, b] = {x: a≤ ≤x b , x

is real}, denotes a closed interval

Notes:

1 The notation [a, b] signifies the set of all real

numbers between a and b including the end points a and b, i.e., the set of all real from a to b.

2 The pharase “at the point x = a” signifies that x

assumes (or, takes) the value a.

3 A neighbourhood of the point x = a is a closed

interval put in the form [a – h, a + h] where h is a

positive number, i.e.,

[ a – h, a + h] = { x a: − ≤ ≤ + , h x a h h is a

small positive number}

4 All real numbers can be represented by points on

a 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 represent

the members of the interval concerned.(Notes 1 It is

a postulate that all the real numbers can be represents

by the points of a straight line 2 Neigbourhood

roughly means all points near about any specifiedpoint.)

(b) Open interval: The set of all real numbers x subject

to the condition a < x < b is called an open interval and is denoted by (a, b), where a and b are two real numbers 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 real

numbers between a and b excluding the end points a and b.

2 The number ‘a’ is called the left end point of the

interval (open or closed) if it is within the circular orsquare brackets on the left hand side and the number

b is called the right end point of the interval if it is

within the circular or square brackets on the right h

and side

3 Open and closed intervals are represented by the

circular and square brackets (i.e., ( ) and [ ] )

respectively within which end points are written

separated by a comma

(c) Half-open, half closed interval (or, semi-open, semi

closed 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 b

are 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 real

numbers between a and b excluding the left end point

a and including the right end point b.

(d) Half closed, half open (or, semi closed, semi open

interval): 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 be

two real numbers such that a < b.

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

is real}

Note: The notation [a, b) signifies the set of all real

numbers between a and b including the left end point

a and excluding the right end point b.

2 Infinite interval

(a) The interval a−∞ ∞, f: The set of all real numbers

x is an infinite interval and is denoted by a−∞ ∞, f or

R.

In set theoretic language,

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

(b) The interval a fa ,∞ : The set of all real numbers x

such that x > a is an infinite interval and is denoted

(c) The interval a ,∞f: The set of all real numbers

x such that x≥a is an infinite interval and is denoted

by a ,∞f

In set theoretic language,

a ,∞f = {x: x≥a , x is real}

or, a ,∞f = {x: a≥ > ∞x , x is real}

(d) The interval a f−∞ , a : The set of all real numbers

x such that x < a is an infinite interval and is denoted

by a f−∞ , a

−∞ , a

a f = {x: x < a, x is real}

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

(e) The interval a−∞ , a : The set of all real numbers

x such that x≤a is an infinite interval and is denoted

1 In any finite interval, if a and/b is (or, more) replaced

by ∞ and /− ∞, we get what is called an infiniteinterval

a

– ∞

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 specified

number in the indicated interval finite or infinite The

square bracket (i.e., [,)also (put before and/after any

specified number) signifies the inclusion of that

specified number in the indicated interval finite or

infinite

5 The sign of inequality without the sign of equality

(i.e > or <) signifies the exclusion of the specified

number in the indicated interval finite or infinite The

circular bracket (i.e ( , ) also (put before and/after any

specified number) signifies the exclusion of that

specified number indicated interval finite or infinite

7 Intervals expressed in terms of modulus: Many

intervals can be easily expressed in terms of absolute

values and conversely

(i) | x | < a ⇔ –a < x < a ⇔ ∈x (–a, a), where ‘a’ is

any positive real number and x∈R

(ii) x ≤ ⇔ − ≤ ≤ ⇔ ∈ − , a a x a x a a where

‘a’ is any positive real number and x∈R

(iii) | x | > a ⇔ ∉x [–a, a] ⇔ ∈ −∞ −x b , ag∪

a ,∞ ⇔

a f either x <–a or x> a, a being any positive

real number and x∈R

(iv) x ≥ ⇔a either x≤ −a or x≥ ⇔ ∈a x

−∞ −, a ∪ a ,∞

Evaluation of a Function at a Given Point

Evaluation: To determine the value of a function y =

f (x) at a given point x = a, is known as evaluation (or,

more clearly evaluation of the function y = f (x) at the given point x = a)

Notation: [f (x)] x = a = (f (x)) x = a = f (a) is a notation to signify the value of the function f at x = a.

Type 1: To evaluate a function f (x) at a point x = a

when the function f (x) is defined by a single expression,

equation or formula

Working rule: The method of finding the value of a

function f (x) at the given point x = a when the given function f (x) is defined by a single expression, equation or formula containing x consists of following

steps

Step 1: To substitute the given value of the

independent variable (or, argument) x wherever x

occurs in the given expression, equation, or formula

containg x for f (x)

Step 2: To simplify the given expression, equation or

formula containg x for f (x) after substitution of the

given value of the independent variable (or, argument)

f (1) = 12 – 1 + 1 = 1

and f 1

2

12

Type 2: (To evaluate a piecewise function f (x) at a

point belonging to different intervals in which different

expression for f (x) is defined) In general, a piece wise

function is put in the form

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

= f2 (x), when x = a

f3 (x), when x < a, ∀ ∈x R

and 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 > a

which denote the domains of different function f1 (x),

f2 (x) and f3 (x) etc for f (x).

Note: The domains over which different expression

f1 (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 and

represent 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 find the value f (a1), provided x = a1 > a and to and to put

x = a1 in f (x) = f1 (x) which will provide one the value

f (a1) after simplification

Step 2: To consider the function f (x) = f2 (x) to find the value f (a), provided x = a is the restriction against

f2 (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 the

inclusion 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) = given constant will be the required value of f (x) i.e f (x) = given constant when x = a signifies not to find the value other than f (a) which is equal to the given

1 f (x) = f1 (x), when (or, for, or, if) a ≤ <x a2 signifies

that one has to consider the function f (x) = f1 (x) to find the functional value f1 (x) for all values of x (given

or specified in the question) which lie in between a1and a2 including x = a1

2 f (x) = f2 (x), when (or, for, or, if) a2 < ≤x a3,

signifies that one has to consider the function f (x) =

f2 (x),to find the functional value f2 (x) for all values of

x (given or specified in the question) which lie in

between a2 and a3 including x = a3

3 f (x) = f3 (x), when (or, for, or, if) a4 < x < a5 signifies

that one has to consider the function f (x) = f3 (x) to find the functional value f3 (x) for all values of x (given

or specified in the question) which lie in between a4and a5 excluding a4 and a5

find the values of (i) f (4) (ii) f (2) (iii) f (0) (iv) f (–3)

12

342

2 a f0 2,

Refresh your memory:

1 If a function f (x) is defined by various expressions

f1 (x), f2 (x), f3 (x) etc, then f (a0) denotes the value of

the function f (x) for x = a0 which belongs to the

domain of the function f (x) represented by various

restrictions x > a, x < a, x≥ a, x≤a , a < x < b,

a≤ ≤x b, a≤ <x b, and a< ≤x b etc

2 Supposing that we are required to find the value

of the function f (x) for a point x = a0 which does not

belong to the given domain of the function f (x), then

f (a0) is undefined, i.e., we cannot find f (a0), i.e., f (a0)does not exist

3 Sometimes we are required to find the value of a

piecewise function f (x) for a0 ±h where h > 0, in such cases, we may put h = 0.0001 for easiness to

guess in which domain (or, interval) the pointrepresented by x= ±a h lies

e.g.: If a function is defined as under

f (x) = 1 + x, when − ≤ <1 x 0

= x2 – 1, when 0 < x < 2 2x, when x≥ 2

find f (2 – h) and f (–1 + h)

(Footnotes: 1 f (a) exists or f (a) is defined ⇔ ‘a’ lies in the domain of f 2 f (a) does not exist or f (a) is

undefined ⇔ ‘a’ does not lie in the domain of f.)

Solution: 1 Putting h = 0.001, we get 2– h = 2 – 0.001

Sometimes a function of an independent variable x is

described by a formula or an equation or an expression

in x and the domain of a function is not explicitly

stated In such circumstances, the domain of a function

is understood to be the largest possible set of realnumbers such that for each real number (of the largestpossible set), the rule (or, the function) gives a realnumber or for each of which the formula is meaningful

or defined

Definition: If f D: →R defined by y = f (x) be a

real valued function of a real variable, then the domain

of the function f represented by D (f) or dom (f) is

defined as the set consisting of all real numbers

representing the totality of the values of the

independent variable x such that for each real value

of x, the function or the equation or the expression in

x has a finite value but no imaginary or indeterminate

value

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

If f D: →R be real valued function of the real

variable x, then its domain is D or D (f) or dom (f)

= {x∈ :R f xa f has finite values }

= {x∈ :R f xa f has no imaginary or indeterminate

value.}

To remember:

1 Domain of sum or difference of two functions f (x)

and g (x) = dom f xa f a f± g x = dom (f (x)) ∩ dom

(g (x)).

2 Domain of product of two functions f (x) and g (x)

g (x) = dom f xa f a f⋅g x = dom (f (x)) ∩ dom (g (x)).

3 Domain of quotient of two functions f (x) and g (x)

=domb gf xa f ∩domb gg xa f − lx g x: a f≠ 0q i.e.,

the domain of a rational function or the quotient

function is the set of all real numbers with the exception

of those real numbers for which the function in

denominator becomes zero

Notes: 1 The domain of a function defined by a

formula y = f (x) consists of all the values of x but no

value 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 a−∞ ∞, f

(ii) The statements “f (x) is defined in an interval

finite or infinite” signifies that f (x) exists and is real

for 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 real values of x from a to b, a and b being real numbers such that a < b Similarly, the statement “f (x) is defined

in the open interval (a, b)” means that f (x) exists and

is real for all real values of x between a and b (excluding

MM MM

0

0000, or

MM MM

0

0000, or

MM MM

0

0000, or

MM MM

0

0000, or

4 (i) ex2 −a2j< ⇔ − < <0 a x a

(ii) ex2 −a2j≤ ⇔ − ≤ ≤0 a x a

(iii) ex2 −a2j> ⇔ < −0 x a or x>a

(iv) ex2 −a2j≥ ⇔ ≤ −0 x a or x≥a

Question: How to represent the union and

intersection on a number line?

Answer: Firstly, we recall the definitions of union and

intersection of two sets

Union: The union of two sets E and F is the set of

elements belonging to either E or F.

Intersection: The set of all elements belonging to

both sets E and F is called intersection of E and F.

Method of Representation of

Union and Intersection on Real Lines

If the set of the points on the line segment AB be the

set E and the set of the point on segment CD be the

set F, then the union of E and F is the segment AD =

AB + BD = sum or union and the intersection of E and

F is the segment CB = common segment.

Now some rules to find the domain of real valued

functions are given They are useful to find the

domain of any given real valued function

Finding the Domain of Algebraic Functions

Type 1: Problems based on finding the domains of

polynomial functions

Working rule: One must remember that a polynomial

in x has the domain R (i.e., the set of the real numbers) because any function f of x which does not become undefined or imaginary for any real value of x has the domain R Hence, the linear y = ax + b; the quadratic

y = ax2 + bx + c; and the square functions y = x2 have

the domain R.

Solved Examples

Find the domain of each of the following functions:

1 y = 11x – 7

Solution: y = 11x – 7 is a linear function and we know

that a linear function has the domain R.

Hence, domain of y (= 11x – 7) = R = a−∞ + ∞, f

2 y = x2 – 3x + 7

Solution: y = x2 – 3x + 7 is a quadratic function and

we know that a quadratic function has the domain R Hence, domain of y = (= x2 – 3x + 7) =

R = a−∞ + ∞, f

3 y = x2

Solution: y = x2 is a square function and we know

that a square function has the domain R.

Hence, domain of y = (= x2) = R = a−∞ + ∞, f

Type 2: Problems based on finding the domain of a

function put in the form:

Working rule: It consists of following steps:

1 To put the function (or, expression in x) in the

denominator = 0, i.e., g (x) = 0

2 To find the values of x from the equation g (x) = 0

3 To delete the valued of x from R to get the required

domain, i.e., domain of f x

a f

a f a for 1 = R – {roots of the equation g (x) = 0}, where f (x) and g (x) are polynomials in x.

common segment C A

Note: When the roots of the equation g (x) = 0 are

imaginary then the domain of the quotient function

put in the form: f x

=

−

112

Now, putting, x2 – 1 = 0

⇒x2 = ⇒ = ±1 x 1

∴ domain = R – {–1, 1}

7. y x

6

Type 3: Problems based on finding the domain of the

square root of a function put in the forms:

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 xa f

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 xa f, when f (x) = ax + b.

Working rule: It consists of following steps:

Step 1: To put a x+ ≥ 0b

Step 2: To find the values of x for which a x+ ≥ 0b

to 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 xa f consists of the values of x for which

∴ domain of y=doma fy1 ∩doma fy2

Now, domain of y1e j a f= x = D1 say = 0,+ ∞f

[from example 1.] again, we require to find the domain

of y2e= x−1j

Putting x− ≥ ⇒ ≥ ⇒1 0 x 1 domain y2

= x−1 = D2 = 1 + ∞

e j a fsay , f Hence, domain of

y = D (say) = dom (y1) ∩ dom ya f2

= D1 ∩D2

= 0,+ ∞ ∩f 1,+ ∞f

= 1,∞f

(ii) Problems based on finding the domain of a

function put in the form:

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

are the roots of ax2 + bx + c = 0 a fα β<

working rule: It consists of following steps:

Step 1: To put a x2 +b x+ ≥c 0

Step 2: To solve the in equation a x2 +b x + c≥ 0

for x by factorization or by completing the square.

Step 3: To write the domain of ax2 +bx + =c

α≤ ≤x β only when the coefficient of x2 = a = – ve

and ax2 + bx + c = a xa fa f−α x−β and to write the

domain of ax2 +bx + =c R − α βa f, only when

the coefficient of x2 = a +ve and ax2 + bx + c =

a xa fa f−α x −β

Notes: (i) a = coefficient of x2 = –ve (and, ax2 + bx +

c = a xa f a f−α x−β ≥0)⇒ x lies between α and

β ⇒ domain of ax2 +bx+ =c α ≤ ≤x β

α β<

a f

(ii) a = coefficient of x2 = + ve (and, ax2 + bx + c =

a xa fa f−α x−β ≥ 0)⇒x does not lie between αand β ⇒ domain of ax2 +bx+ =c R −a fα β, =

−∞,α ∪ β,+ ∞

(iii) a fa fx+α x+β should be written as

x− −αa f

b g, bx− −βa f g while finding the domain of

the square root of ax2 + bx + c = a xa fa f+α x+β

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

a function put in the form : y f x

g x

= a f

a f

While finding the domain of the square root

of a quotient function (i.e; y f x

g x

= a f

a f ) one mustremember the following facts:

+ ∞

x x

x≥ 2 ⇔ ∈ −∞ − ∪x L NM + ∞IK

23

Solution: y is defined for all those x for which

x x

Solution: y is defined for all those x for which

x x

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

function put in the form: y

3 To form the Domain with the help of the roots of

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

function put in the form: y f x

g x

a f

Working rule: The rule to find the domain of a

function of the form y f x

g x

a f is the same as for

the domain of a function of the form y

Finding the Domain

(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 a

function 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 obtained

values 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)

Solution: y is defined when (x + 6) (6 – x) > 0 ⇔ (x +

6) (x – 6) < 0 ⇔ x lies between –6 and 6 ⇔ –6 < x <

46

2

∴ D ya f= R = −∞ + ∞a , f

Notes: 1 Imaginary or a complex numbers as the

roots of an equation a x2 +b x + = ⇔c 0 domain

of log f xa f= R = −∞ + ∞a , f 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 greater

than any negative number

Method 2 This method consists of showing that

If x < –1, then f (x) < 0 as all the three factors are

< 0