Functions

2008·3.4 MB·English

by  Sunil Kumar Singh

Functions
By:
Sunil Kumar Singh

Functions
By:
Sunil Kumar Singh
Online:
< http://cnx.org/content/col10464/1.64/ >
C O N N E X I O N S
RiceUniversity,Houston,Texas
This selection and arrangement of content as a collection is copyrighted by Sunil Kumar Singh. It is licensed under
the Creative Commons Attribution 2.0 license (http://creativecommons.org/licenses/by/2.0/).
Collection structure revised: September 23, 2008
PDF generated: May 6, 2011
For copyright and attribution information for the modules contained in this collection, see p. 484.
Table of Contents
1 Sets
1.1 Sets .........................................................................................1
1.2 Subsets ......................................................................................4
1.3 Union of sets ................................................................................9
1.4 Intersection of sets .........................................................................17
1.5 Di(cid:27)erence of sets ...........................................................................25
1.6 Working with two sets ......................................................................35
1.7 Working with three sets ....................................................................42
1.8 Cartesian product ..........................................................................49
1.9 Cartesian Product (exercise) ................................................................56
2 Relations
2.1 Relations ...................................................................................61
2.2 Relation types ..............................................................................67
2.3 Functions ..................................................................................72
2.4 Function types .............................................................................80
2.5 Composition of functions ...................................................................88
2.6 Inverse of a function ........................................................................95
3 Real functions
3.1 Domain and range .........................................................................103
3.2 Inequality .................................................................................112
3.3 Polynomial function .......................................................................116
3.4 Quadratic polynomial function ............................................................124
3.5 Rational function ..........................................................................136
3.6 Rational inequality ........................................................................153
3.7 Function operations .......................................................................167
3.8 Modulus function ..........................................................................173
3.9 Greatest and least integer functions .......................................................179
3.10 Exponential and logarithmic functions ....................................................188
3.11 Domain and range of exponential and logarithmic function ...............................198
3.12 Trigonometric functions ..................................................................205
3.13 Trigonometric values, equations and identities ............................................221
3.14 Trigonometric inequalities ................................................................232
3.15 Inverse trigonometric functions ...........................................................240
3.16 Composition of trigonometric function and its inverse ....................................256
3.17 Function operations (exercise) ............................................................270
3.18 Composition of functions (exercise) .......................................................274
3.19 Value of a function .......................................................................280
Solutions .......................................................................................285
4 Transformation of graphs
4.1 Transformation of graphs ..................................................................301
4.2 Transformation of graphs using output ....................................................313
4.3 Transformation of graphs by modulus function .............................................322
4.4 Transformation of graphs by greatest integer function ......................................335
4.5 Transformation applied by fraction part function ..........................................341
Solutions .......................................................................................349
5 Function properties
5.1 Equal functions ...........................................................................351
iv
5.2 Even and odd functions ...................................................................353
5.3 Periodic functions .........................................................................361
5.4 Periodicity and periods ....................................................................372
5.5 Monotonic functions .......................................................................377
5.6 Increasing and decreasing intervals ........................................................385
5.7 Minimum and maximum values ............................................................391
5.8 Least and greatest function values .........................................................405
5.9 One-one and many-one functions (exercise) ................................................413
5.10 Inverse functions (exercise) ...............................................................419
5.11 Limits ...................................................................................430
5.12 Limits of algebraic functions .............................................................443
5.13 Continuous function ......................................................................451
Solutions .......................................................................................467
Glossary ............................................................................................477
Index ...............................................................................................479
Attributions ........................................................................................484
Chapter 1
Sets
1.1 Sets1
Settheoryisaboutstudyingcollectionofobjects. Thecollectionmaycompriseanythingoranyabstraction.
It can be purely abstract thing like numbers or abstraction of real thing like students studying in class XI
in a school. The members of collection can be numbers, letters, titles of books, people, teachers, provinces (cid:21)
virtually anything - even other collections. Further, it need not be (cid:28)nite. For example, a set of integers has
in(cid:28)nite members. For a set, only requirement is that the members of a collection are properly de(cid:28)ned.
De(cid:28)nition 1.1: Set
A set is a collection of well de(cid:28)ned objects.
Inotherwords,thememberofsetisclearlyidenti(cid:28)able. Theterms(cid:16)object(cid:17),(cid:16)member(cid:17) or(cid:16)element(cid:17) mean
same thing and are used interchangeably.
1.1.1 How to represent a set?
A set is denoted by capital letters like (cid:16)A(cid:17), (cid:16)B(cid:17), (cid:16)C(cid:17) etc. In choosing a symbol for a set, it is generally
convenient to use a capital letter that identi(cid:28)es with the set. For example, it is appropriate to use symbol
(cid:16)V(cid:17) to represent collection of vowels in English alphabet.
On the other hand, the members or elements of a set are denoted by small letters like (cid:16)a(cid:17),(cid:17)b(cid:17),(cid:17)c(cid:17) etc.
Membership of a set is denoted by the symbol (cid:16) ∈(cid:17) . Its literal meaning is (cid:16)belongs to(cid:17). If an object does
not belong to a set, then we convey the same, using symbol (cid:16) ∈/ (cid:17).
a∈A : we read this as (cid:16)a(cid:17) belongs to set "A".
a∈/ A : we read this as (cid:16)a(cid:17) does not belong to set "A".
The set is represented in two ways :
• Roaster form
• Set builder form
1.1.1.1 Roaster form
Allelementsofthesetarelistedwithacomma((cid:16),(cid:17))inbetweenandthelistingitselfisenclosedwithinbraces
(cid:16){(cid:16) and (cid:16)}(cid:17). The order or sequence of elements within the set is not important (cid:21) though desirable.
The set of numbers, which divide 12, is written as :
A={1,2,3,4,6,12}
1Thiscontentisavailableonlineat<http://cnx.org/content/m15194/1.3/>.
1
2 CHAPTER 1. SETS
If a pattern or sequence is easily made out, then we can use ellipsis ("...") to represent continuity of such
pattern. This type of representation is particularly useful to represent an in(cid:28)nite set. Clearly, sequence of
members in this type of representation is important.
The set of even numbers is written as,
B ={2,4,6,8.........}
Theroasterformislimitedincertaincircumstance. Forexample,wecannotrepresentsetofrealnumbers
in roaster form. Real numbers is an in(cid:28)nite set, but the elements of this set do not follow a pattern or have
a particular sequence. As such, we can not de(cid:28)ne same with the help of ellipsis.
Every member of the set is unique and distinct. However, we encounter situations in which collection
can have repeated elements. For example, the set representing scores of three students can be a set of three
numbers one of which is repeated :
S ={80,80,70}
We need to reduce such collection as :
⇒S ={80,80,70}={80,70}
1.1.1.2 Set builder form
Collections are often characterized by a common property. We can, therefore, de(cid:28)ne members of a set in
terms of the common property. However, we need to ensure that objects outside the collection do not have
the same common property.
The construction of quali(cid:28)cation for the common property is quite (cid:29)exible. Only thing is that we need
to be explicit in what we mean. Generally, we denote the member by a symbol like (cid:16)x(cid:17) and then de(cid:28)ne the
membership. Consider the examples :
A={x: x is a vowel in English alphabet}
B ={x: x is an integer and0< x< 10}
The roaster equivalents of two sets are :
A={a,e,i,o,u}
B ={1,2,3,4,5,6,7,8,9}
Canwewriteset(cid:16)B(cid:17) astheonewhichcomprisessingledigitnaturalnumber? Yes. Thus,wecanseethat
thereareindeeddi(cid:27)erentwaystode(cid:28)neandidentifymembersandhencethe(cid:29)exibilityinde(cid:28)ningcollection.
We should be careful in using words like (cid:16)and(cid:17) and (cid:16)or(cid:17) in writing quali(cid:28)cation for the set. Consider the
example here :
C ={x:x∈Z and 2< x< 4}
Both conditional quali(cid:28)cations are used to determine the collection. The elements of the set as de(cid:28)ned
above are integers. Thus, the only member of the set is (cid:16)3(cid:17).
Now, let us consider an example, which involves (cid:16)or(cid:17) in the quali(cid:28)cation,
C ={x: x∈A or x∈B}
The member of this set can be elements belonging to either of two sets "A" and "B". The set consists of
elements(i)belongingexclusivelytoset"A",(ii)elementsbelongingexclusivelytoset"B"and(iii)elements
common to sets "A" and "B".
3
1.1.1.3 Example
Problem 1 : A set in roaster form is given as :
52 62 72
A={ , , }
6 7 8
Write the set in (cid:16)set builder form(cid:17).
Solution : We see here that we are dealing with natural numbers. The numerators are square of
natural numbers in sequence. The number in denominator is one more than numerator for each member.
Wecandenotenaturalnumberby(cid:16)n(cid:17). Clearly, ifnumeratoris(cid:16) n2 (cid:17), thendenominatoris(cid:16)n+1(cid:17). Therefore,
the expression that represent a member of the set is :
n2
x=
n+1
However, this set is not an in(cid:28)nite set. It has exactly three members. Therefore, we need to specify (cid:16)n(cid:17)
so that only members of the set are exclusively denoted by the above expression. We see here that (cid:16)n(cid:17) is
greater than 4, but (cid:16)n(cid:17) is less than 8. For representing three elements of the set,
5≤n≤7
We can write the set, now, in the builder form as :
n2
A={x: x= ,where "n" is a natural number and 5≤n≤7}
n+1
In set builder form, the sequence within the range is implied. It means that we start with the (cid:28)rst valid
natural number and proceed sequentially till the last valid natural number.
1.1.2 Some important sets representing numbers
Fewkeynumbersetsareusedregularlyinmathematicalcontext. Asweusethesesetsoften, itisconvenient
to have prede(cid:28)ned symbols :
• P(prime numbers)
• N (natural numbers)
• Z (integers)
• Q (rational numbers)
• R (real numbers)
Weputasuperscript(cid:16)+(cid:17),ifwewanttospecifymembershipofonlypositivenumbers,whereappropriate.
" Z+ ", for example, means set of positive integers.
1.1.3 Empty set
An empty set has no member or object. It is denoted by symbol (cid:16)φ(cid:17) and is represented by a pair of braces
without any member or object.
φ={}
The empty set is also called (cid:16)null(cid:17) or (cid:16)void(cid:17) set. For example, consider a de(cid:28)nition : (cid:16)the set of integer
between 1 and 2(cid:17). There is no integer within this range. Hence, the set corresponding to this de(cid:28)nition is
an empty set. Consider another example :
B ={x: x2 =4 and x is odd}
4 CHAPTER 1. SETS
An odd integer squared can not be even. Hence, set (cid:16)B(cid:17) also does not have any element in it.
Thereisabitofparadoxhere. Ifthede(cid:28)nitiondoesnotyieldanelement,thenthesetisnotwellde(cid:28)ned.
We may be tempted to say that empty set is not a set in the (cid:28)rst place. However, there is a practical
reason to have an empty set. It enables mathematical operations. We shall (cid:28)nd many examples as we study
operations on sets.
1.1.4 Equal sets
The members of two equal sets are exactly same. There is nothing more to it. However, we need to know
two special aspects of this equality. We mentioned about repetition of elements in a set. We observed that
repetition of elements does not change the set. Consider example here :
A={1,5,5,8,7}={1,5,8,7}
Another point is that sequence does not change the set. Therefore,
A={1,5,8,7}={5,7,8,1}
In the nutshell, when we have to compare two sets we look for distinct elements only. If they are same,
then two sets in question are equal.
1.1.5 Cardinality
Cardinality is the numbers of elements in a set. It is denoted by modulus of set like |A|.
De(cid:28)nition 1.2: Cardinality
The cardinality of a set (cid:16)A(cid:17) is equal to numbers of elements in the set.
The cardinality of an empty set is zero. The cardinality of a (cid:28)nite set is some positive integers. The
cardinality of a number system like integers is in(cid:28)nity. Curiously, the cardinality of some in(cid:28)nite set can be
compared. For example, the cardinality of natural numbers is less than that of integers. However, we can
not make such deduction for the most case of in(cid:28)nite sets.
1.2 Subsets2
The collections are generally linked in a given context. If we think of ourselves, then we belong to a certain
society, which in turn belongs to a province, which in turn belongs to a country and so on. In the context
of a school, all students of a school belong to school. Some of them belong to a certain class. If there are
sections within a class, then some of these belong to a section.
Weneedtohaveamathematicalrelationshipbetweendi(cid:27)erentcollectionsofsimilartypes. Insettheory,
we denote this relationship with the concept of (cid:16)subset(cid:17).
De(cid:28)nition 1.3: Subset
A set, (cid:16)A(cid:17) is a subset of set (cid:16)B(cid:17), if each member of set (cid:16)A(cid:17) is also a member of set (cid:16)B(cid:17).
We use symbol (cid:16) ⊂ (cid:17) to denote this relationship between a (cid:16)subset(cid:17) and a (cid:16)set(cid:17). Hence,
A⊂B
Wereadthissymbolicrepresentationas: set(cid:16)A(cid:17) isasubsetofset(cid:16)B(cid:17).Weexpresstheintentofrelationship
as :
A⊂B if x∈A, then x∈B
2Thiscontentisavailableonlineat<http://cnx.org/content/m15193/1.6/>.