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Analysis

Attila Bérczes

Attila Gilányi

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Analysis Attila Bérczes Attila Gilányi

Debreceni Egyetem

Reviewer

Dr. Attila Házy

University of Miskolc, Institute of Mathematics, Department of Applied Mathematics

The curriculum granted / supported by the Hungarian Scientific Research Fund (OTKA) Grant NK-81402 and

by the TÁMOP-4.1.2.A/1-11/1-2011-0098 and TÁMOP 4.2.2.C-11/1/KONV-2012-0001 projects.

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Table of Contents

Introduction ........................................................................................................................................ v 1. The concept and basic properties of functions ................................................................................ 1

1. 1.1. Relations and functions .................................................................................................. 1 2. 1.2. A `friendly' introduction to functions ............................................................................. 3 3. 1.3. The graph of a function .................................................................................................. 6

3.1. 1.3.1. Cartesian coordinate system ........................................................................... 6 3.2. 1.3.2. The graph of a function ................................................................................... 7 3.3. 1.3.3. The graph of equations or relations ................................................................ 9 3.4. 1.3.4. The vertical line test ...................................................................................... 10 3.5. 1.3.5. The -intercept and the -intercepts of functions ....................................... 13

4. 1.4. New functions from old ones ....................................................................................... 14 5. 1.5. Further properties of functions ..................................................................................... 22

5.1. 1.5.1. Injectivity of functions .................................................................................. 22 5.2. 1.5.2. Surjectivity of functions ............................................................................... 27 5.3. 1.5.3. Bijectivity of functions ................................................................................. 33 5.4. 1.5.4. Inverse of a function ..................................................................................... 43

2. Types of functions ........................................................................................................................ 56 1. 2.1. Linear functions ........................................................................................................... 56 2. 2.2. Quadratic functions ...................................................................................................... 60 3. 2.3. Polynomial functions ................................................................................................... 65

3.1. 2.3.1. Definition of polynomial functions ............................................................... 65 3.2. 2.3.2. Euclidean division of polynomials in one variable ....................................... 66 3.3. 2.3.3. Computing the value of a polynomial function ............................................ 67

4. 2.4. Power functions ............................................................................................................ 69 5. 2.5. Exponential functions ................................................................................................... 70 6. 2.6. Logarithmic functions .................................................................................................. 72

3. Trigonometric functions ............................................................................................................... 75 1. 3.1. Measures for angles: Degrees and Radians .................................................................. 75 2. 3.2. Rotation angles ............................................................................................................. 76 3. 3.3. Definition of trigonometric functions for acute angles of right triangles ..................... 77

3.1. 3.3.1. The basic definitions ..................................................................................... 77 3.2. 3.3.2. Basic formulas for trigonometric functions defined for acute angles ........... 79 3.3. 3.3.3. Trigonometric functions of special angles .................................................... 80

4. 3.4. Definition of the trigonometric functions over ......................................................... 82 5. 3.5. The period of trigonometric functions .......................................................................... 83 6. 3.6. Symmetry properties of trigonometric functions ......................................................... 84 7. 3.7. The sign of trigonometric functions ............................................................................. 84 8. 3.8. The reference angle ...................................................................................................... 87 9. 3.9. Basic formulas for trigonometric functions defined for arbitrary angles ..................... 90

9.1. 3.9.1. Cofunction formulas ..................................................................................... 90 9.2. 3.9.2. Quotient identities ......................................................................................... 90 9.3. 3.9.3. The trigonometric theorem of Pythagoras .................................................... 91

10. 3.10. Trigonometric functions of special rotation angles .................................................. 91 11. 3.11. The graph of trigonometric functions ....................................................................... 91

11.1. 3.11.1. The graph of ..................................................................................... 91 11.2. 3.11.2. The graph of .................................................................................... 92 11.3. 3.11.3. The graph of .................................................................................... 92 11.4. 3.11.4. The graph of .................................................................................... 93

12. 3.12. Sum and difference identities ................................................................................... 94 12.1. 3.12.1. Sum and difference identities of the function and ........................ 94 12.2. 3.12.2. Sum and difference identities of the function and ........................ 96

13. 3.13. Multiple angle identities ........................................................................................... 97 13.1. 3.13.1. Double angle identities ............................................................................. 97 13.2. 3.13.2. Triple angle identities ............................................................................... 99

14. 3.14. Half-angle identities ............................................................................................... 100 15. 3.15. Product-to-sum identities ....................................................................................... 101

Analysis

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16. 3.16. Sum-to-product identities ....................................................................................... 102 4. Transformations of functions ...................................................................................................... 105

1. 4.1. Shifting graphs of functions ....................................................................................... 105 2. 4.2. Scaling graphs of functions ........................................................................................ 134

5. Results of the exercises ............................................................................................................... 175 1. 5.1. Results of the Exercises of Chapter 1 ......................................................................... 175 2. 5.2. Results of the Exercises of Chapter 2 ......................................................................... 227 3. 5.3. Results of the Exercises of Chapter 3 ......................................................................... 240

Bibliography ................................................................................................................................... 242

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Introduction

This textbook contains the knowledge expected to be acquired by the students at the "College Functions" class

of the Foundation Year and Intensive Foundation Semester at the University of Debrecen.

We have tried to compose the textbook such that it is a self-contained material, including many examples,

figures and graphs of functions which help the students in better understanding the topic, however, this book is

primarily meant to be used as a Lecture Notes connected to the "College Analysis" class. Thus we suggest the

students to attend the classes, and use this book as a supplementary mean of study.

In Chapter 1 we define the basic concepts connected to functions, and the most important properties of functions

which we shall analyze later in the case of many types of functions.

In Chapter 2 we present some basic types of functions, like linear functions, quadratic functions, polynomial

functions, power functions, exponential and logarithmic functions, along with their most important properties.

Chapter 3 is devoted to a detailed description of trigonometric functions.

The topic of Chapter 4 is the transformation of functions.

The last Chapter contains the results of the unsolved exercises of the book, which will help the students to check

if their solutions lead to the right answer or not.

This work has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK-81402 and by the

TÁMOP-4.1.2.A/1-11/1-2011-0098 and TÁMOP 4.2.2.C-11/1/KONV-2012-0001 projects.

These projects have been supported by the European Union, co-financed by the European Social Fund.

Attila Bérczes and Attila Gilányi

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Chapter 1. The concept and basic properties of functions

1. 1.1. Relations and functions

In this section we provide a precise definition of binary relations and functions, which is the mathematically

correct way to introduce these concepts. However, this section is recommended mostly for the interested reader,

and we give a less precise but simpler introduction of functions in the next section.

Definition 1.1. Let be a non-empty set and elements of . The ordered pair is defined by

Theorem 1.2. Let and be two ordered pairs. We have

Definition 1.3. Let be two sets. The Cartesian product of and is the set of all ordered pairs

with the first element being from and the second element from , i.e.

Definition 1.4. Let be two non-empty sets. A binary relation between and be is a subset of the

Cartesian product , i.e. . If then we also say that

• is in relation with , or

• is assigned , or

• assigns to .

If then we say that is a binary relation on .

Notation. For we also use the notation .

Examples. Here we give some examples of binary relations from our earlier mathematical experience:

1. The relation on the set of real numbers.

2. The realtion on the integers.

3. The congruence of triangles.

4. The parallelity of lines in the plane.-

We also give an abstract example to illustrate the definition above:

Let and , .

Definition 1.5. Let be a binary relation. Then

The concept and basic properties of

functions


Definition 1.6. The binary relation is called a function if

The domain and the range of the function are the sets which are defined as the domain and the range of

considered as a relation.

Notation. If a relation is a function, then

• instead of we write and we say that is a function from to ;

• instead of writing or we shall write and we say that maps the element to the

element , or is the image of by the function .

Remark. The above definition means that a function maps each element of its domain to a well-defined single

element of the range.

Example. First we give an example for a function , given by a diagram, where and

:

This in fact means that , , and . The domain of is and

the range of is .

Example. The following diagram defines a relation from to , where and .

However this relation is not a function.


functions


We have , , , . The domain of is and the range of is . We

emphasize that the relation defined by the diagram is not a function, since for the same we have both

and , and this is not allowed in the case of a function.

Example. The following diagram also defines a function from to , where and

.

We have , , , . The domain of is and the range of is

. We emphasize that the relation defined by the diagram is a function, although ,

so two different elements may have the same image by a function.

Remark. We wish to emphasize the difference between the two last examples. In the last example is a

function, even if is the image of two different elements of . On the other hand in the preceding example is

not a function since assigns two different elements to .

2. 1.2. A `friendly' introduction to functions

Definition 1.7. A relation is a correspondence between the elements of two sets.

Definition 1.8. A function is a correspondence between the elements of two sets, where to elements of the first

set there is assigned or associated only one element of the second set. We use the following notations:

• For a function from the set to the set , we use the notation .

• To say that maps an element of to the element of we write or .

Remark. A function is in fact a relation, where to elements of the first set there is assigned or associated only

one element of the second set.


functions


Definition 1.9. Let be a function.

Remark. A function is given only if along the correspondence, the domain of the function is also given.

Indeed, the functions

and

are clearly different, although the formula describing the correspondence is the same. Indeed, while ,

does not assign any value to , and instead of we could choose any element of .

Convention. Whenever a function is given by a formula, without specifying the domain

of the function we agree that and the domain of the function is the largest

subset of on which the formula has sense.

Examples. Let be a function.

1. Compute .

2. Determine the domain of .

3. Decide if and is in the range of or not.

4. Determine the range of .

Solution.

1. To compute the value of the function given by the formula taken at a number or expression we

substitute every occurrence of in the right hand side of the formula defining by the number or expression

.


functions


2. To determine the (maximal) domain of the function given by a formula we have to determine the maximal

subset of on which the formula defining has sense. Since has sense for every real number

the (maximal) domain of is the whole set of real numbers, i.e. .

3. To decide if a concrete number belongs to range of a function given by a formula we need to solve the

equation in the variable , where is a given number. If the equation has a solution in the domain

of then belongs to the range of , otherwise does not belong to the range of .

To decide if is in the range of or not we have to check if the equation

has a solution in or not. So we solve the equation

We obtain that

thus,

Since

we see that .

To decide if is in the range of or not we have to check if the equation

has a solution in or not. So we solve the equation

Since the discriminant the above equation has no solution and we see that

.

4. To determine the range of a function given by a formula we need to solve the equation in the

variable , where is a parameter. The range of of consists of those values , for which the equation

has a solution in the domain of .

Thus we shall check for which values of the parameter the equation

has a solution in . The equation takes the form

which has a solution if and only if its discriminant is non-negative, i.e.

This is equivalent to , so the range of is .

Exercise 1.1. Let be one of the functions below.


functions


a) Compute .

b) Determine the domain of .

c) Decide if and (given below next to the function ) is in the range of or not.

d) Determine the range of .

3. 1.3. The graph of a function

3.1. 1.3.1. Cartesian coordinate system

Definition 1.10. Consider two perpendicular lines on the plane, one of them being horizontal, the other one

vertical. The intersection point of the two lines is called the origin, the horizontal line is called the X-axis and

the vertical line is called the Y-axis. On each of these lines we represent the set of real numbers in the usual way,

being represented on both lines by the origin, and with the convention that on the horizontal line the positive

direction is to the left, and on the vertical line the positive direction is upwards. The plane together with the two

axis is called the Cartesian coordinate system.

Then any point of the plane corresponds to a pair of real numbers in the following way: is the real

number corresponding to the projection of the point to the -axis, and is the real number corresponding to the

projection of the point to the -axis. Conversely, to represent a pair as a point of the plane we draw a

parallel line to the -axis through the point corresponding to on the -axis, and a parallel line to the -axis

through the point corresponding to on the -axis, and the point corresponding to will be the intersection

point of these two lines.

Example. Represent the following pairs of real numbers in the Cartesian coordinate system:

Solution.


functions


3.2. 1.3.2. The graph of a function

Definition 1.11. Let be a function. The set

is called the graph of the function .

The graph of a function can be represented in the so-called Cartesian coordinate system.

Graphing a function:

Example. Draw the graph of the function .

1. We compute the pairs for :


functions


2. We plot the points ,

,

3. We connect the points by a smooth curve


functions


Exercise 1.2. Sketch the graph of the following functions:

3.3. 1.3.3. The graph of equations or relations

Similarly to the graph of functions one may represent relations or solutions of an equation in two unknowns in

the Cartesian coordinate system.

Example. Represent the set of solutions of the equation

in the Cartesian coordinate system.

Solution. We have to draw the curve corresponding to the set

in the Cartesian system, so we get


functions


Remark. Clearly, the graph of the function is the same as the graph of the equation . Thus, when

there is no danger of confusion we shall not make any distinction among them.

3.4. 1.3.4. The vertical line test

Example. The following two figures show the essential difference between the graph of a function and a curve

that is not the graph of a function:


functions


Remark. The vertical line test is a graphical method, which is very useful to understand well the concept of a

function, however, in itself it is not a rigorous proving method. However, if we "see" the vertical line showing

that a curve is not the graph of a function, then it is easy to write down the proof rigorously.

Exercise 1.3. Using the vertical line test decide if the following curves are graphs of a function or not:

a)


functions


b)

c)


functions


d)

3.5. 1.3.5. The -intercept and the -intercepts of functions


functions


When drawing the graph of a function we pay special interest to the intersection of the graph with the coordinate

axis.

Definition 1.12. (Intercepts of the graph of a function)

• The intersection point of the graph of a function with the -axis is called the -intercept of the graph.

• An intersection point of the graph of a function with the -axis is called an -intercept of the graph.

Remark. Note the difference on the two points of the above definition: the -intercept is a single point (if it

exists), however there might exist more -intercepts. Indeed, the vertical line test guarantees that the graph of a

function might intersect the -axis (which is a vertical line) in at most one point.

Theorem 1.13. (The -intercept of a function) The graph of a function has -intercept if and only if it is

defined in , and it is the point with .

Theorem 1.14. (The -intercepts of a function) The graph of a function given by a formula has -

intercepts if and only if the equation

has solutions in the domain of , and the -intercepts are the points for , where for

are the solutions of the above equation.

Example. Compute the -intercept and the -intercepts of the graph of the function .

Solution. The intercept is the point , i.e. the point (since ).

The -intercepts are the points having -coordinate zero, their -coordinates being the solutions of the

equation

Using the "Almighty formula" these solutions are and , so the -intercepts are and

.

Exercise 1.4. Compute the -intercept and the -intercepts of the following functions:

4. 1.4. New functions from old ones


functions


Definition 1.15. (The sum, difference, product and quotient of functions) Let be two functions

defined by the formulas and , and denote by and the domain of and , respectively. Put

and . Then

• the sum of and is defined by

• the difference of and is defined by

• the product of and is defined by

• the quotient of and is defined by

Remark. The above definition means that the sum, difference, product and quotient of two functions given by a

formula is the function defined by the sum, difference, product and quotient of the two formulas, however, the

domain is not the maximal domain implied by the resulting formula, but the intersection of the domains of the

original functions, or in case of the quotient, an even more restricted set.

For example if and then , and thus

even if the function defined simply by the formula would have maximal domain .

Example. Let be two functions given by and . Then we have

Definition 1.16. (Composition of functions) Let and two functions such that ,

where denotes the range of and the domain of . Then the composite function is defined as

for every in the domain of .

Remark. The composition of functions may also be regarded as connecting two machines. Namely, the first

machine transforms the input to the output , which will be the input of the machine , and this

transforms to . Then the "connected machine" is the machine which transforms the input to

, and we already cannot see which part of the "work" was done by the machines and , respectively.

Remark. In most cases we shall work with composite functions in the case , so the composite

functions and can be both defined. However, these functions in general are different, i.e. the

composition of functions is not commutative.

Example. Let , and . Then


functions


Exercise 1.5. Compute , , , , and , provided that

Example. Let and for . Let and for

and . Further, let , and . Build the following function

as a composite function of the above functions:

Solution.

Exercise 1.6. Let and for . Let and for

and . Further, let , and . Build the following function

as a composite function of the above functions:


functions


Definition 1.17. Let , and be sets, and let be a function. The function , for

which for each , is said to be the restriction of to the set .

The restriction of the function to the set is also denoted by .

Examples.

1. Let us consider the function , . The graph of is:

Graph of

the

function

,


functions


.

The restriction of to the set is , . The graph of is:

Graph of

the

function

,

.

2. Let , . The graph of is:


functions


Graph of

the

function

,

.



functions


Graph of

the

function

,

.

5. 1.5. Further properties of functions

5.1. 1.5.1. Injectivity of functions

Definition 1.18. Let and arbitrary sets. A function is called injective if, for all ,

for which and , we have .

Remark. Injective functions are also called invertible.

Examples.

1. Let , and let us define such that , , ,

. It is easy to see that satisfies the requirements given in Definition 1.6 [2], therefore, it is a

function. It can be illustrated in the following form:

def_fct


functions


It is visible, that there are no elements and such that , which implies that is

injective.

2. Let now , and let , , , ,

. This function can be illustrated as

It is easy to see that is a function, but, since , it is not injective.

3. Let us consider the function , . The graph of this function has the following form.


functions


Graph of

the

function

,

.

It is easy to see, that there are no elements and such that , which yields the

injectivity of .



functions


Graph of

the

function

,

.

Obviously, there no no elements and such that , thus is also

injective.

5. Let , . The graph of :


functions


Graph of

the

function

,

.

Since there are elements , such that (for example ), the

function is not injective.

6. Let us consider the function , now. The graph of is:


functions


Graph of

the

function

,

.

Obviously, there does not exist elements and such that , therefore is

an injective function.

Remark. The last two examples point out that if we investigate the injectivity of a function then the domain of

the function plays an important role.

5.2. 1.5.2. Surjectivity of functions

Definition 1.19. Let and arbitrary sets. A function is said to be surjective if the range of is .

Remark. The definition above requires that, for each , there exists an such that .

Examples.

1. Let , and let such that , , ,

, i.e.,


functions


It is visible that is a function, furthermore, it is also clear that, for each element , there exists an

such that , therefore is surjective.

2. Let now , and let , , , , . A

diagram corresponding to is:

It is easy to see that is a function. However, there exists an element such that there is no for

which . (The element has this property.) This implies that is not surjective.

3. Let us consider the function , . The graph of this function has the following form.


functions


Graph of

the

function

,

.

It is easy to verify that, for each there exists an for which . (Solving the equation

for , we obtain such an for each .) This means that the function is surjective.



functions


Graph of

the

function

,

.

Since there are elements in the range of such that there is no element in the domain of for

which . (For example fulfills this property.) Therefore is not surjective.



functions


Graph of

the

function

,

.

It is easy to see that, for each there exists an such that Thus, is surjective.



functions


Graph of

the

function

,

.

It is easy to check that is not surjective.



functions


Graph of

the

function

,

.

It is obvious that is surjective.

Remark. The last four examples show that concerning the injectivity of a function the range of the function

plays an important role.

5.3. 1.5.3. Bijectivity of functions

Definition 1.20. Let and arbitrary sets. A function is called bijective if it is injective and

surjective.

Examples.

1. Let , and let , , , ,

. It is easy to see that is a function, furthermore, it is injective and surjective, therefore,

it is bijective. The function has the diagram


functions


2. Let , and let , , , , . This

function is injective but not surjective, thus, it is not bijective. An illustration of is:

3. Let , and let , , , ,

. The function is surjective but not injective, which yields that it is not bijective. The

function can be illustrated in the following form:


functions


4. Let , and let , , , ,

. The function is neither injective nor surjective, therefore, it is not bijective. The

function has the form:

5. Let us consider the function , . The graph of this function has the form:


functions


Graph of

the

function

,

.

It is easy to see that this function is injective and surjective, therefore, it is bijective.

6. Let us consider the function , . The graph of g is:


functions


Graph of

the

function

,

.

This function is also injective and surjective, therefore, it is bijective.



functions


Graph of

the

function

,

.

The function is injective but not surjective, thus, it is not bijective.



functions


Graph of

the

function

,

.

The function is injective and surjective, therefore, it is bijective.



functions


Graph of

the

function

,

.

This function is neither injective nor surjective, thus, it is not bijective.



functions


Graph of

the

function

,

.

The function is injective but not surjective, therefore, it is not bijective.



functions


Graph of

the

function

,

.

The function is injective and surjective, thus, it is bijective.

Remark. The examples above show that concerning the bijectivity of a function its domain and its range is also

important.

Exercise 1.7. Illustrate the following functions and decide whether they are bijective, injective or surjective:

a) , ,

, , , , , ;

b) , ,

, , , , , ;

c) , ,

, , , , ;

d) , ,

, , , , , ;


functions


e) , ,

, , , , , ;

f) , ,

, , , , , .

Exercise 1.8. Draw the graph of the following functions in a coordinate system and decide whether they are

bijective, injective or surjective:

a) , ;

b) , ;

c) , ;

d) , ;

e) , ;

f) , ;

g) , ;

h) , ;

i) , ;

j) , ;

k) , ;

l) , .

5.4. 1.5.4. Inverse of a function

Definition 1.21. Let and arbitrary sets, let be a injective function and let us denote the range of

by . Interchanging the elements in the ordered pairs , we obtain a function , .

This function is said to be the inverse function of .

The inverse function of is denoted by .

Remark. It is important to note that the inverse function is defined in the case of injective functions only.

Examples.

1. Let , and let us define , , , ,

. This function can be illustrated as:


functions


It is easy to see that this function is injective, thus there exists its inverse. The inverse of has

the form

that is,


functions


which means that , , , and .

2. Let , and let us define , , , , .

This function can be illustrated as:

The function is injective, therefore, it is invertible. It is important to mention, that the range of is the set

, which means that the domain of will be this set, too. Thus, , ,

, , , that is,


functions


3. Let , and let , , , ,

, i. e.,

The function is not injective, thus, it does not have any inverse.

4. Let us consider the function , . The graph of this function has the form:


functions


Graph of

the

function

,

.

This function is injective, therefore, it is invertible. Its inverse can be determined in the following form: by

the definition of , we may write for all , which means that for each . Therefore,

the inverse of can be written as or, writing instead of ,

. The graph of is:


functions


Graph of

the

function

,

.

Let us draw the graphs of the functions and above in the same coordinate system.


functions


Graphs of

the

functions

,

and its

inverse

,

.

We may observe that the graph of is a reflection of the graph of about the line . (More

precisely the graph of is a reflection of the graph of about the graph of the identical function

, .) This important connection is always valid for the graph of a function and for the graph

of its inverse. We will formulate this fact in the next Remark.

Remark. It is easy to verify that the graph of the inverse of an invertible function is always a reflection of the

graph of about the graph of the identical function , .

Examples.

1. Let us consider the function , now.


functions


Graph of

the

function

,

.

This function is injective, thus, it is invertible. The equation implies that , therefore

the inverse of has the form .


functions


Graph of

the

function

,

.

Drawing the graphs of and in the same coordinate system, we obtain:


functions


Graphs of

,

and its

inverse

,

.

2. Let , . This function is injective, therefore, it is invertible. Its inverse is is

, . The graphs of these functions are:


functions


Graphs of

,

and its

inverse

,

.

3. Note that the function , is not injective, thus, its inverse does not exist.

4. Let , . This function is injective, thus, invertible. It inverse is ,

, that is the same function as . The graphs of the functions are:


functions


Graphs of

,

and its

inverse

,

.

Exercise 1.9. Illustrate the following functions with a diagram, investigate if they are invertible or not and, in

the case when they exist, determine their inverse functions and draw their diagrams:

a) , ,

, , , , , ;

b) Let , ,

, , , , ;

c) , ,

, , , , ;

d) , ,

, , , , , .


functions


Exercise 1.10. Draw the graphs of the following functions, determine their inverses (if they exist) and draw the

graph of the inverse functions, too:

a) , ;

b) , ;

c) , ;

d) , ;

e) , ;

f) , ;

g) , .


Chapter 2. Types of functions

1. 2.1. Linear functions

Definition 2.1. A function of the form

is called a linear function.

Remark. We remind the student that for all pairs of points and lying on the same (not

vertical) straight line in the plane (represented in a Cartesian coordinate system) the quantity

is constant (i.e. it is independent of the choice of and ). This number is called the slope of the line.

Theorem 2.2. Let be a linear function with .

1. The graph of is a straight line. More precisely, the graph of is the line with equation . Further,

the graph of is a horizontal line if and only if .

2. The -intercept of the graph of is the point .

3. Whenever the only -intercept of the graph of is the point . If then the graph

of has no -intercept. If then the graph of is the -axis itself.

The above theorem suggests the definition below:

Definition 2.3. The number is called the slope of the linear function .

Theorem 2.4. Fix a Cartesian coordinate system in the two dimensional plane. Except for the vertical lines

every straight line in the plane is the graph of a linear function. The function corresponding to a line may be

given by the following formulas:

1. The slope-intercept formula

2. The point-slope formula

3. The point-point formula

Types of functions


Example. Draw the graph of the following functions:

As the below figure shows the graph of these functions are all horizontal lines and the intersection with the -

axis is .


As the below figure shows the graph of these functions are parallel to each other and the intersection with the -

axis is .

Types of functions



As the below figure shows the graph of each of these functions goes through the origin, and if is positive, then

the line passes through the 1st and 3rd quadrant, if is negative, then it passes through the 2nd and 4th

quadrant. In the case of the graph is the -axis

Types of functions


Example. Determine the linear function whose slope is and whose graph has -intercept .

Solution. By the slope-intercept formula

Example. Determine the linear function whose slope is and whose graph passes through the point

.

Solution. By the point-slope formula the equation of the line is

and thus the linear function whose graph is this line is given by

Example. Determine the linear function whose graph passes through the points and .

Solution. By the point-point formula the equation of the line is

and thus the linear function whose graph is this line is given by

Types of functions


Exercise 2.1. Determine the linear function whose slope is and whose graph has -intercept where:

Sketch the graph of these functions.

Exercise 2.2. Determine the linear function whose slope is and whose graph passes through the point .

Exercise 2.3. Determine the linear function whose graph passes through the points and

2. 2.2. Quadratic functions

Definition 2.5. A function defined by

is called a quadratic function. If the quadratic function is given by a formula of the above shape, we say that it is

given in general form.

Theorem 2.6. (The canonical form of a quadratic function)

Let be a quadratic function with , . Put . Then can be

written in the form

with and .

Types of functions


Proof.

Example. Compute the canonical form of the following quadratic function:

Solution.

Another solution to the problem is to use directly Theorem 2.6 [60]. We have , and thus

and , and we get

Theorem 2.7. (The factorized form of a quadratic function)

Let be a quadratic function with , . Put and if then

put

Then can be written in the form

If then and the above factorization takes the form .

Theorem 2.8. Let be a quadratic function with , . Put and

if then put

The graph of is a parabola with the following properties:

thm_can_form

Types of functions


Example. Draw the graph of the functions for and for .

Types of functions


Example. Draw the graph of the functions for .

Types of functions




Types of functions


Exercise 2.4. Let be the quadratic function below.

1. Give the canonical form.

2. If give the factorized form.

3. Give the -intercept and -intercepts of the graph of .

4. Give the vertex of the graph of .

5. Draw the graph of .

3. 2.3. Polynomial functions

3.1. 2.3.1. Definition of polynomial functions

Definition 2.9. A function given by

Types of functions


with and given is called a polynomial function.

Remark. Linear and quadratic functions are special polynomial functions.

3.2. 2.3.2. Euclidean division of polynomials in one variable

In this section we present the division algorithm for univariate polynomials with complex, real or rational

coefficients. This procedure is a straightforward generalization of the long division of integers.

Example. Divide the polynomial by .

Remark. If in the dividend polynomial there are missing terms of lower degrees, than it is wise to include them

with coefficient in the scheme, so that for their like terms there is place below them.

Example. Divide the polynomial by .

Types of functions


Exercise 2.5. Divide the polynomial by the polynomial using the procedure of Euclidean division:

3.3. 2.3.3. Computing the value of a polynomial function

To compute the value of a polynomial function at it is clearly possible by the general method, i.e. by

replacing each the variable at each occurrence of it by , and computing the value of the resulting expression.

However in this case there is a better way to compute it, by using the following Theorem:

Theorem 2.10. (Horner's scheme) Let with , and

with be two polynomials. Build the following table:

where

Then the quotient of the polynomial division of by is the polynomial

and the remainder is the constant polynomial . Further, we also have

Remark. In other words the above theorem states that in Horner's scheme the numbers in the first line are the

coefficients of the polynomial , and the numbers computed in the second line of the scheme are the following:

• the first number is the zero of the polynomial

Types of functions


• the next numbers (except for the last one) are the coefficients of the quotient polynomial of the polynomial

division of by

• the last number is the remainder of the above division, but it is also the value of the polynomial at .

Example. Now we compute the value of the polynomial function in

using Horner's scheme. The first place in the first line is empty, then we list the coefficients of . The first

element in the second line of the Horner's scheme will be . Then we compute the consecutive elements of the

second line using 2.3 to get

This means that .

Remark. If in the dividend polynomial there are missing terms of lower degrees, than it is compulsory to

include the coefficients of these terms (i.e. -s) in the upper row of the Horner's scheme.

Example. We compute the value of the polynomial function at using

Horner's scheme. The first place in the first row is empty, then we list the coefficients of , including the

coefficient of . The first element in the second row of the Horner's scheme will be . Then we compute the

consecutive elements of the second line using 2.3 to get

This means that .

Exercise 2.6. Compute the value of the following polynomial function at the given value of the

indeterminate :

Types of functions


4. 2.4. Power functions


with given and is called a power function.

Example. Draw the graph of the functions , for and for

.

Types of functions


5. 2.5. Exponential functions


Types of functions


with a given , is called an exponential function.

Example. Draw the graph of the functions , for and for.

Definition 2.13. In the sequel we denote by the Euler constant

Definition 2.14. The function given by

is called the natural exponential function.

Example. Draw the graph of the functions .

Types of functions


Example. Draw the graph of the functions .

6. 2.6. Logarithmic functions

Types of functions



with a given , is called a logarithmic function.

Example. Draw the graph of the functions , for and for.

Definition 2.16. Recall that denotes the Euler constant

Definition 2.17. The function given by

where is the Euler-constant is called the natural logarithmic function. We denote the logarithm on base by

, and similarly the natural logarithmic function by .

Remark. The natural logarithmic function is sometimes also denoted by .

Example. Draw the graph of the functions , and ,

.

Types of functions


Remark. We also recall that the logarithmic function on base is denoted by .


Chapter 3. Trigonometric functions

1. 3.1. Measures for angles: Degrees and Radians

In geometry an angle is considered to be a figure which consists of two half-lines called the sides, sharing a

common endpoint, called the vertex. So basically we have the following kinds of angles:

By considering also the points of the plane between the two sides as parts of the angle, we may extend the

notion of angle to reflex angles (those larger than a straight angle) and full angles:

Definition 3.1. Divide the full angle into equal angles. The measure for one such part is defined to be one

degree, and the measure of any angle will be as many degrees as many parts of 1 degree fit between the sides of

the angle. The notation for degree is a circle "in the exponent" e.g. .

What is the reason for choosing the number ? In one hand, is divisible by many integers. So the half, the

third, the quoter, the one fifth, one sixth, one eighth, one tenth, one twelfth of a whole circle has an integer

measure, and this is very convenient. On the other hand is close to the number of days in the astronomical

years, which was convenient in astronomy. In geometry this measure for angles is also convenient, however, in

trigonometry and analysis it is inconvenient, and there we use radians to measure the angles:

Definition 3.2. The measure of an angle in radians is the length of the arc corresponding to the angle (as a

central angle) of a unit circle. This means that the measure of a full angle is . When measuring angles in

radians the size of the angle is a pure number without explicitly expressed unit.

Remark. In many cases the word "angle" will be used not only for the geometric figure, but also for its

measure.

Theorem 3.3. Let us have an angle of measure . Then its measure in radians is .

Trigonometric functions


Let us have an angle of measure radians. Then its measure in degrees is .

Example. The measure of the most important angles in degrees and radians:

2. 3.2. Rotation angles

To extend the notion of angles such that every real number corresponds to the measure of an angle we do the

following:

Remark. If is measured in degrees on the picture below, then the radius which corresponds to the angle is

the same which corresponds to for any .



Definition 3.4. Angles which have the same terminal side are called coterminal. Further, we say that an angle is

on the first circle if its measure in degrees is on the interval (or alternatively, if its measure in radians

is in the interval ).

Theorem 3.5. Every rotation angle is coterminal with an angle on the first circle.

The two coordinate axes divide the plane into four quadrants, as it is shown on the figure below.

We say that an angle belongs to one of the quadrants, if its terminal side is in that quadrant.

3. 3.3. Definition of trigonometric functions for acute angles of right triangles

3.1. 3.3.1. The basic definitions



The definitions above may also be formulated in the following way

Definition 3.6. Let be one acute angle of a right triangle.

• The sine of is defined by

• The cosine of is defined by

• The tangent of is defined by

• The cotangent of is defined by

Remark. The definition of trigonometric functions above are independent of the choice of the triangle. Indeed,

if we have two right triangles having an acute angle of measure , then these triangles are similar to each other.

Thus the quotient of the length of the two corresponding sides in the two triangles is the same.

Example. Let the lengths of the sides of a right triangle be . Let be the angle opposite to the leg of

length . Compute , , , .

Solution.



3.2. 3.3.2. Basic formulas for trigonometric functions defined for acute angles

Theorem 3.7. (The cofunction identities) Let be an acute angle in a right triangle. Then we have:

Theorem 3.8. (The quotient identities) Let be an acute angle in a right triangle. Then we have:

Theorem 3.9. (The trigonometric theorem of Pythagoras in right triangles) Let be an acute angle in a right

triangle. Then we have:

Proof of Theorems 3.7 [79], 3.8 [79], 3.9 [79]Let be a right triangle, the right angle being at the vertex

and with the size of the angle at vertex being .

thm_cofunction_formulas_triangle

thm_quotient_identities_triangle

thm_pythagora_triangle



Finally, we prove Theorem 3.9 [79]. The Theorem of Pythagoras for the triangle states that

. Using this we get

which conclude the proof of Theorem 3.9 [79].

3.3. 3.3.3. Trigonometric functions of special angles

In applications the angles are of special interest. In this paragraph we compute the trigonometric

functions of these angles.

Theorem 3.10. The values of the trigonometric functions of are summarized in the following table.

Proof. To prove the statements of the theorem concerning the angles and we consider an equilateral triangle

, and we draw its bisector .





In this triangle the Theorem of Pythagoras gives:

so we get

Clearly, have , and since is the bisector of the angle we also have

. Thus, in the right triangle we have

and similarly,



Clearly, we also have , so from the triangle using its angle we get:

This concludes the proof of our theorem.

4. 3.4. Definition of the trigonometric functions over

Definition 3.11. Consider a unit circle with a rotating radius. Let be a rotation angle on this circle. Let the

intersection point of the line of the terminal side of by the tangent line drawn to the circle in the point , if

it exists at all, be denoted by . Similarly, let the intersection point of the line of the terminal side of by the

tangent line drawn to the circle in the point , if it exists at all, be denoted by . We remind that the point

does not exist if for , and does not exist if for .



Then for any we define the trigonometric functions of as follows:

Remark. For acute angles Definition 3.11 [82] coincides with the definition of trigonometric functions in right

triangles. Thus, Definition 3.11 [82] extends the former concept of , , and for any real number.

More precisely,

• and are defined for any real number ;

• is not defined for for , but it is defined for every other ;

• is not defined for for , but it is defined for every other .

5. 3.5. The period of trigonometric functions

Theorem 3.12. The trigonometric functions sine, cosine, tangent and cotangent are periodic. Further, for the

period of these functions we have:

def_d_trig_real

def_d_trig_real



Proof. If we investigate the figure in Definition 3.11 [82] we see that for any angle the terminal side of and

is the same, so for and the point will be the same, which proves that and . Further,

it is clear that is the smallest angle with this property holding for every . This proves statements 1) and

2) of Theorem 3.12 [83].

Next, we also see that the terminal side of and the terminal side of are on the same line, thus fixing the

same points and , which proves and . Further, it is clear again, that

is the smallest angle with this property holding for every . This proves statements 3) and 4) of Theorem

3.12 [83].

Corollary 3.13. Theorem 3.12 [83] proves that we have

6. 3.6. Symmetry properties of trigonometric functions

Theorem 3.14. The functions are odd functions, and the function is an even function. This

means that we have the following formulas:

7. 3.7. The sign of trigonometric functions

def_d_trig_real

thm_t_period

thm_t_period

thm_t_period



The sign of the trigonometric functions , , , is determined by the situation of the terminal

side of . Whenever belongs to one of the four quadrants the sign of the above functions is already

determined, and the same is true for the cases when the terminal side of is at border of two quadrants.

We may summarize the information about the sign of the trigonometric functions , , , as

follows:

1. If is in the quadrant then

2. If is at the border of the and quadrant, i.e. then









Remark. The sign of trigonometric functions can also be summarized by the following diagrams:



8. 3.8. The reference angle

Definition 3.15. Let be a rotation angle such that its terminal side is not on the coordinate axis. The acute

angle formed by the -axis and the terminal side of is called the reference angle of .

Theorem 3.16. Let be a rotation angle, and let be its reference angle. Then we have the following:

Theorem 3.17. Let be a rotation angle, and let be its reference angle. The we have the following:

• If is in the first quadrant then ;

• If is in the second quadrant then ;

• If is in the third quadrant then ;

• If is in the fourth quadrant then .

Remark. The above theorem can be summarized by the below diagram:



The theorem below is a variant of Theorem 3.16 [87]:

Theorem 3.18. Let be a rotation and be its reference angle. Let be the sign function on , i.e.

Then we have the following

Example. Compute the exact value of the following expressions:

1.

2.

3.

Solution.

1. We shall use Theorem 3.18 [88]. First we mention that is in the third quadrant, so and by

Theorem 3.17 [87] the reference angle of is . So we have

2. Since is in the fourth quadrant we have

3. Since is in the fourth quadrant we have

Example. Compute the exact value of the following expressions:

1.

2.

3.

Solution.

thm_t_ref_ang_prop

thm_t_ref_ang_use

thm_t_ref_ang_comp



1. In contrast to the previous example the angle is not on the first circle, so we cannot use Theorem 3.17

[87] directly to compute the reference angle of . First we have to compute the angle on the first circle

which corresponds to (i.e. has the same terminal side as ). This means that we have to subtract form

as many times as it is needed the result to be on the first circle. So we have

Now like in the previous exercise we shall use Theorem 3.18 [88]. Since is in the fourth quadrant we

have

2. Similarly to the previous example we have

3. Finally we use the same method to solve the last exercise:

We mention that instead of finding the angle on the first circle with the same terminal side as we also can

subtract from as many times that the result is between and . This is since the period of is .

The same applies for , however it is important that this latter is not true in the case of and ,

since their period is . This means that in the case of and we have to subtract an integer multiple

of and it is not possible to replace by .

Exercise 3.1. Compute the exact value of the following expressions:

thm_t_ref_ang_use



9. 3.9. Basic formulas for trigonometric functions defined for arbitrary angles

The formulas presented in Section 3.3.2 generalize automatically for trigonometric functions defined for

arbitrary angles.

9.1. 3.9.1. Cofunction formulas

Theorem 3.19. Let be a rotation angle for which the trigonometric functions in the below formulas are

defined. Then we have the following formulas which connect trigonometric functions of to cofunctions of

:

9.2. 3.9.2. Quotient identities

Theorem 3.20. Let be a rotation angle for which the trigonometric functions in the below formulas are

defined. Then we have:



9.3. 3.9.3. The trigonometric theorem of Pythagoras

Theorem 3.21. Let be a rotation angle. Then we have:

10. 3.10. Trigonometric functions of special rotation angles

Previously, we computed the values of the trigonometric functions for the acute angles , which are of

special interest for applications. In this paragraph we extend this for rotation angles with their reference angles

being :

Theorem 3.22. The values of the trigonometric functions of some important angles are summarized in the

following table.

where the sign ` ' means that the corresponding value is not defined.

11. 3.11. The graph of trigonometric functions

11.1. 3.11.1. The graph of



11.2. 3.11.2. The graph of

11.3. 3.11.3. The graph of



11.4. 3.11.4. The graph of



12. 3.12. Sum and difference identities

12.1. 3.12.1. Sum and difference identities of the function and

Theorem 3.23. Let be two arbitrary real numbers. Then we have the following formulas:

Proof. We start with the proof of formula 4). Since without loss of

generality we may assume that . Let us denote by the endpoint of terminal side of the angle and by

the endpoint of terminal side of the angle . Then the coordinates of these points are and

. The distance between and is



Further, the size of the angle is just , and if we rotate this angle in such a way that the side is

moved to the -axis (i.e. the image of lies on the -axis), then the side is moved to such that

is just the terminal side of the rotation angle . So we have

Further, by the isometric property of the rotations we have , which gives

Now rising to the square both sides, subtracting from both sides and dividing them by we get the required

identity

Now the proof of the third identity is straightforward:

The proof of the first identity is

Finally, the proof of the second identity is

Example. Compute and .

Solution. We do not know the values of the exact value of the trigonometric functions of from tables, since

is not an "important" angle, however since

and we know the exact values of the trigonometric functions on and from our tables, we are able to compute

and using Theorem 3.23 [94]:

thm_t_sumdiff_sin_cos



and

12.2. 3.12.2. Sum and difference identities of the function and

Theorem 3.24. Let be two arbitrary real numbers such that the functions in the below formulas are defined

for their arguments. Then we have:

Proof. First we prove formula 1) for :

Next let us prove 2):

Now we present the proof of 3)

We mention that 3) can also be proved using formula 1):



Finally we prove 4):


Solution. We use

and Theorem 3.24 [96]:

Clearly, we may proceed similarly to compute , but it is much easier to use the fact that is the

reciprocal of (whenever is non-zero):

13. 3.13. Multiple angle identities

13.1. 3.13.1. Double angle identities

Theorem 3.25. Let be an arbitrary real number such that the functions in the below formulas are defined for

their arguments. Then we have:

thm_t_sumdiff_tan_cot



Proof. For proving this theorem we use the corresponding formulas from Theorem 3.23 [94] and Theorem 3.24

[96]. First we prove formula 1):

Now we prove 2):

For proving the other two statements of formula 2) we also need to use the trigonometric theorem of Pythagora,

i.e. the formula

So we get

Now we prove formulas 3) and 4)


Solution. Recall that we already have computed and in Section 3.12.1, using and

Theorem 3.23 [94], and we got the result:

Now we wish to use identity 2) of Theorem 3.25 [97] to compute and . Indeed, we have

Expressing from this expression we get

which together with the fact that is in the first quadrant so is positive, means that

Now this result looks different of the previously obtained , so it seems that one of the solutions is wrong.

However, this is not the case. As the below computation sows, the two results represent exactly the same real

number:



thm_t_double_angle_formulas



Exercise 3.2. Applying sum, difference and double angle identities compute the exact values of the following

expressions:

13.2. 3.13.2. Triple angle identities



Proof. First we prove formula 1) we use formula 1) from Theorem 3.23 [94], formulas 1) and 2) from Theorem

3.25 [97], and the trigonometric theorem of Pythagora:

Similarly, to prove formula 1) we need formula 3) from Theorem 3.23 [94], formulas 1) and 2) from Theorem

3.25 [97], and the trigonometric theorem of Pythagora:

Now we prove formula 3) using formula 1) of Theorem 3.24 [96] and formula 3) of Theorem 3.25 [97]:









Finally, we prove formula 4) by using formula 3) of Theorem 3.24 [96] and formula 4) of Theorem 3.25 [97]:

14. 3.14. Half-angle identities



Proof. To prove the half-angle formulas we basically use identity 2) of Theorem 3.25 [97], which applied for

instead of gives

Expressing from the above equation we get

meanwhile expressing leads to the identity

These conclude the proof of identities 1) and 2). By dividing the two latter identities we just get 3) and 4). So it

remains to prove 5) and 6). To do so first we mention that by the trigonometric theorem of Pythagoras we have

, which means that , and thus we have

Thus, whenever , we get






and

However, we mention that, by , whenever and is defined we have .

So the second equality of both 5) and 6) is proved.

To conclude the proof of this theorem we write

and

This concludes the proof of the theorem.

15. 3.15. Product-to-sum identities



Proof. To prove the above `product-to-sum' identities 1)-4) we start with the right-hand sides, and we transform

them until we reach the formula on the left-hand side. We start with the proof of identity 1):



Similarly, we get identity 2) by:

The proof of identity 3) is

and for the proof of identity 4) we have

For the proof of identities 5)–8) we have to divide correspondingly two of the identities 1)–4). So we have

16. 3.16. Sum-to-product identities





Proof. Again we shall start from the right-hand side and we will do equivalent transformations of the

expressions until we reach the left-hand side of the corresponding formula. We start with identity 1):

Similarly, we prove identity 2):

The proof of identity 3) is the following:

Finally, the proof of identity 4) is:


Chapter 4. Transformations of functions

1. 4.1. Shifting graphs of functions

In this section, we will investigate how we can obtain the graph of the functions and from the

graph of , where is a real constant.

Since we already have some experiences with drawing graphs of functions, let us consider the graph of the

function , .

Graph of

the

function

,

.

Let us modify the function above and sketch the graph of , now.

Transformations of functions


Graph of

the

function

,

.

Drawing both graphs above in the same coordinate system, we obtain the following:



Graphs of

the

functions

,

and

,

.

It is easy to see, that, starting with a point of the graph of the function , we can obtain the element

of the graph of by . Since this property is valid for all pairs of points and of the graphs

of our functions, we may obtain the graph of if we shift (or move, or translate) the graph of upwards by

units. Obviously, this method works for all positive integers . It is also clear, that, in the case when is a

negative real number, we have to shift (or move, or translate) the graph of downwards by units (where

denotes the absolute value of ). Finally, it is trivial, that, if then the functions and coincide.

Let us sketch now the graph of the function , .



Graph of

the

function

,

.

We shall illustrate the graphs of and in the same coordinate system now.



Graphs of

the

functions

,

and

,

.

Similarly to the case above, we can see, that we may obtain the graph of if we shift (or move, or translate) the

graph of to the left by units. This method also works for each positive integer . Furthermore, if is a

negative real number, we have to shift (or move, or translate) the graph of to the right by units. (Here, it is

also true that, in the case when then the functions and are equal.)

It is easy to see that the argumentation above is independent from the concrete functions , and , it only

depends on the connection between the functions. Thus, we can give a general method for the types of function

transformations considered above. In order describe this method, let us denote by the translation of the set

by units. More precisely, if is a set and is a real number, denotes the set

.

Let be a real number, be a set and let us consider a function .

– In the case when is nonnegative, we can obtain the graph of the function , by

shifting (or moving) the graph of the function by units upwards.

– In the case when is negative, we can obtain the graph of the function , by

shifting (or moving) the graph of the function by units downwards.



– In the case when is nonnegative, we can obtain the graph of the function , by

shifting (or moving) the graph of the function by units to the left.

– In the case when is negative, we can obtain the graph of the function , by

shifting (or moving) the graph of the function by units to the right.

Examples.

1. Let us sketch the graph of the function , using the graph of , .

Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.

2. Let us sketch the graph of the function , using the graph of , .



Graphs of

the

functions

,

and

,

.

3. Let us sketch the graph of the function , using the graph of ,

.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.


.



Graphs of

the

functions

,

and

,

.

5. Let us sketch the graphs of the function , using the graph of ,

.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.


.



Graphs of

the

functions

,

and

,

.

7. Let us sketch the graphs of the function , using , .



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.

8. Let us sketch the graphs of the function , using , .



Graphs of

the

functions

,

and

,

.


.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.


.



Graphs of

the

functions

,

and

,

.

11. Let us sketch the graph of the function , using the graph of

, . It is important to note here, that the domain of the function is the interval

.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.

12. Let us sketch the graph of the function , using the graph of

, . We note that the domain of the function is the interval .



Graphs of

the

functions

,

and

,

.


.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.


.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.


.



Graphs of

the

functions

,

and

,

.

Exercise 4.1. Sketch the graph of the function function using the graph of in the case when

a) , , , ;

b) , , , ;

c) , , , ;

d) , , , ;

e) , , , ;

f) , , , ;

g) , , , ;

h) , , , ;



i) , , , ;

j) , , , ;

k) , , , ;

l) , , , ;

m) , , , ;

n) , , , ;

o) , , , ;

p) , , , .

2. 4.2. Scaling graphs of functions

In this section, we will study, how we can get the graph of the functions and from the graph of ,

where is a real constant.

Similarly to the previous part, we try to get some observations based on our knowledge of drawing graphs of

functions.

First, we consider the transformation type ` ', and we start again with the graph of the simple quadratic

function , .



Graph of

the

function

,

.

Let us consider the graph of , :

Graph of

the

function

,

.

Finally, we sketch both functions in the same coordinate system:



Graphs of

the

functions

,

and

,

.

It is easy to see, that, we can get each point of the graph of using from the corresponding point

of the graph . Since this property is valid for all pairs of points and of the graphs of our

functions, we may obtain the graph of by stretching (or magnifying) the graph of the function with the factor

vertically.

Let us investigate the function , now.



Graph of

the

function

,

.

Sketching and in the same coordinate system, we obtain:



Graphs of

the

functions

,

and

,

.

Here, each point of the graph of can be obtained by the calculation from the corresponding point

of the graph . Therefore, we get the graph of the function by compressing (or shrinking) the graph of

the function with the factor vertically.

It is important to note, that the sign of plays also here an important role.

Namely, the graph of , is:



Graph of

the

function

,

.

If we draw it together with the graph of , we get



Graphs of

the

functions

,

and

,

.

It is easy to see, that we can get the graph of by reflecting the graph of about the -axis.

If we would like to determine the graph of the function , , we have to combine the two

steps above: we have to stretch (or magnify) the graph of the function with the factor vertically, and we have

to reflect this (new) graph about the -axis.

This situation is illustrated in the following picture.



Graphs of

the

functions

,

and

,

.

Let us investigate now the transformations of the type ` '. In the case of the function ,

, we have , therefore, it does not give new information for us in this case By this reason, in

the following, we will consider the function , us our `basis function'. The graph of this

function is:



Graph of

the

function

,

.

Let us consider the graph of , .



Graph of

the

function

,

.

Sketching these two graphs in the same coordinate system, we get



Graphs of

the

functions

,

and

,

.

It is visible that we can get the graph of by stretching (or magnifying) the graph of by the factor

horizontally.

The situation is analogous, if our constant is positive, but less than . In this case we compress (or shrinking) the

graph instead of stretching (or magnifying) it.

We illustrate these situations with the functions and , . in the following pictures.



Graph of

the

function

,

.

Sketching these two graphs in the same coordinate system, we get



Graphs of

the

functions

,

and

,

.

Finally, we will investigate the case when our factor is negative. Let us consider the function ,

.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.

Similarly to the case of a `negative constant ' above, we easily get, that in this case, after stretching (or

magnifying) or compressing (or shrinking) the graph of , we have to reflect the result about the -

axis.

We illustrate this situation with the graphs of the functions , and ,

.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.

It is clear that the argumentations above is independent from the concrete functions, it only depends on the

connection between them. Thus, we can give a general method for these types of function transformations, too.

In order describe this method, let be a set, let be a real number and denote the set

.

Let be a real number, be a set and let us consider a function .

– In the case when , we can obtain the graph of the function , by stretching (or

magnifying) the graph of the function with the factor vertically.

– In the case when , we can obtain the graph of the function , by compressing

(or shrinking) the graph of the function with the factor vertically.


magnifying) the graph of the function with the factor vertically, and reflecting this graph about the -axis.



– In the case when , we can obtain the graph of the function , by

compressing (or shrinking) the graph of the function with the factor vertically, and reflecting this graph

about the -axis.


magnifying) the graph of the function with the factor horizontally.


compressing (or shrinking) the graph of the function with the factor horizontally.

– In the case when , we can obtain the graph of the function , by stretching

(or magnifying) the graph of the function with the factor horizontally, and reflecting this graph about the -

axis.


compressing (or shrinking) the graph of the function with the factor horizontally, and reflecting this graph

about the -axis.

Examples.


.

Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.


.



Graph of

the

function

,

.



Graphs of

the

functions

,

and

,

.


.



Graphs of

the

functions

,

and

,

.

Exercise 4.2. Sketch the graph of the function function using the graph of in the case when

a) , , , ;

b) , , , ;

c) , , , ;

d) , , , ;

e) , , , ;

f) , , , ;

g) , , , ;

h) , , , ;



i) , , , ;

j) , , , ;

k) , , , ;

l) , , , ;

m) , , , ;

n) , , , ;

o) , , , ;

p) , , , ;

q) , , , ;

r) , , , ;

s) , , , ;

t) , , , ;


Chapter 5. Results of the exercises

1. 5.1. Results of the Exercises of Chapter 1

Solution of Exercise 1.1 [5]:

1. This part of the exercise is trivial, so we do not list the results.

2. The domain of the corresponding function is:

3.

4. The range of the corresponding function is:

Solution of Exercise 1.2 [9]

a)

xca_domain_range

xca_draw_graph

Results of the exercises


b)

c)



d)

e)



f)

g)



h)

i)



j)

k)



l)

m)



n)

o)



p)

q)



r)



Solution of Exercise 1.3 [11]: In the cases a) and c) the curve is not the graph of a function, which is shown by

the below application of the vertical line test:

a)

xca_vertical_line_test



c)

In exercises b) and d) the corresponding curve is the graph of a function.


xca_intercepts



The -intercept of the corresponding functions is:

The -intercepts of the corresponding functions are:


xca_composite_fn





a) Let , and let , , , , ,

. We can illustrate in the following form:

xca_composite_fn_ii

xca_exercbij1_



It is easy to see that is a function. It is also visible that there are no elements and such that

, which implies that is injective. Since, for each element , there exists an such that

, the function is also surjective. These properties imply that it is also bijective.

b) Let , and let , , , , ,


It is obvious that is a function. Since , is not injective. Furthermore, there is no for

which , thus is not surjective. This means that is not bijective.

c) Let , and let , , , , . We can

illustrate in the following form:



It is easy to se that is a function and it is injective. However, since there is no for which , the

function is not surjective. This implies that is not bijective.

d) Let , and let , , , , ,


It is visible that is a function and it is surjective. However is not injective, therefore, it is not bijective.

e) Let , and let , , , , ,




It is easy to see that is a function, it is injective, surjective, which implies that it is also bijective.

f) Let , and let , , , , ,


It is visible that is a surjective function. However is not injective, therefore, it is not bijective.


a) Let , . The graph of is:

xca_exercbij2_



Graph of

the

function

,

.

It is easy to see that there are no elements and in the domain of the function such that ,

which implies that is injective. It is also clear that, for each element of the range of the function , there

exists an in the domain of such that , therefore, is surjective. These properties imply that is

bijective.

b) Let , . The graph of is:



Graph of

the

function

,

.

It is easy to see that, similarly to the previous exercise, there are no elements and in the domain of the

function such that , which yields that is injective. It is also clear that, for each element of

the range of the function , there exists an in the domain of with the property , therefore, is

surjective. Thus, is bijective.

c) Let , . The graph of has the form:



Graph of

the

function

,

.

Similarly to the problems above, it is easy to verify that the function is injective and surjective, therefore, it is

bijective.

d) Let , . The graph of has the form:



Graph of

the

function

,

.

Obviously, there are elements and in the domain of the function such that . (For

example, .) Therefore is not injective. Furthermore, there are elements in the range of

the function , for which there does not exist any in the domain of with the property . (For

example has this property.) This means that is not surjective. Since is neither injective nor surjective,

it is not bijective.

e) Let , . The graph of has the form:



Graph of

the

function

,

.

It is easy to see that, for each element of the range of , there exists an in the domain of such that

, therefore, this function is surjective. However, similarly to the case above, there are elements and

in the domain of the function such that , which implies that is not injective. Since is

not injective, it is not bijective.

f) Let , . The graph of has the form:



Graph of

the

function

,

.

Analogously to the exercise above, for each element of the range of , there exists an in the domain

of such that , therefore, is surjective. Furthermore, there are no elements and in the domain

of such that , which implies that is injective. Since is surjective and injective, it is

also bijective.

g) Let , . The graph of has the form:



Graph of

the

function

,

.

Similarly to the argumentation given in Exercise d above, it can be verified that the function is neither

injective nor surjective, therefore, it is not bijective.

h) Let , . The graph of has the form:



Graph of

the

function

,

.

Similarly to Exercise e, we obtain that the function is surjective, but it is not injective, thus, it is not bijective.

i) Let , . The graph of has the form:



Graph of

the

function

,

.

Analogously to Exercise f, we may see that the function is surjective and injective and this implies that it is

also bijective.

j) Let , . The graph of has the form:



Graph of

the

function

,

.

Similarly to the solution of the exercises above, we obtain that the function is neither injective nor surjective,

therefore, it is not bijective.

k) Let , . The graph of has the form:



Graph of

the

function

,

.

It is easy to verify that the function is surjective, but it is not injective, thus, it is not bijective.

l) Let , . The graph of has the form:



Graph of

the

function

,

.

It is easy to see that the function is surjective and injective, therefore it is also bijective.


a) Let , and let , , , , ,

. We may illustrate in the following form:

xca_exercinv1_



It is easy to see that satisfies the requirements given in Definition 1.6 [2], thus, it is a function. Furthermore,

is injective, thus, there exists its inverse. The inverse of has the form:

i. e.,

def_fct



which means that , , , , and .

b) Let , and let , , , , . We may

illustrate as:

It is easy to see that an injective function, thus, there exists its inverse. The inverse of has the

form:



that is,

which means that , , , , .

c) Let , , , , , , . A diagram of

is:



It is visible that is a function and it is injective, therefore, it is invertible. Drawing `mechanically' the

correspondence between the elements of the sets and , we obtain:

It is important to notify that the range of the function is the set , therefore, this set will be the

domain of the inverse function of .



which gives that and , , , and .

d) Let , , , , , , , . A

diagram of is:

It is easy to see that is a function, but it is not injective, since . Therefore, the inverse of

does not exists.


a) The graph of the function , is

xca_exercinv2_



Graph of

the

function

,

.

It is easy to see that this function is injective, therefore, it is invertible. Its inverse can be determined in the

following form: by the definition of , we have for all , which gives for each

. Therefore, the inverse of is or, writing instead of ,

. The graph of is:



Graph of

the

function

,

.

The graph of and its inverse in the same coordinate system



Graphs of

,

and its

inverse

,

.

b) The graph of , is



Graph of

the

function

,

.

This function is injective, therefore, it is invertible. Its inverse can be obtained in the following form:

for all , which yields for each . The inverse of reads as

or, writing instead of , . The graph of is:



Graph of

the

function

,

.

The graph of and its inverse is:



Graphs of

,

and its

inverse

,

.

c) The graph of the function , is:



Graph of

the

function

,

.

This function is not injective, therefore, it is not invertible.

d) The graph of , is

Graph of

the

function

,

.

This function is injective, thus, it is invertible. Its inverse can be determined by the following calculation. The

equation for all , gives for each . It is important to note that the range of

is the set , therefore, this will be the domain of the inverse of . The inverse

of is . Its graph is:



Graph of

the

function

,

.

The graphs of and its inverse is:



Graphs of

,

and its

inverse

,

.

e) The graph of , is:



Graph of

the

function

,

.

The function is injective, thus, it is invertible. The range of is the set , thus, this set will be the

domain of the inverse of . The of is . Its graph is:



Graph of

the

function

,

.

The graphs of and are:



Graphs of

,

and its

inverse

,

.

f) The graph of , is:



Graph of

the

function

,

.





Graphs of

,

and its

inverse

,

.

g) The graph of , is:



Graph of

the

function

,

.





Graphs of

,

and its

inverse

,

.



The formula defining the corresponding functions is:

The graphs of the corresponding functions are:

a)

xca_linear_i



b)

c)



d)


xca_linear_ii





1. The canonical form of the corresponding functions is:

2. The factorized form of the corresponding function (provided that ) is:

3. The -intercept is denoted by , and the -intercepts (if they exist at all) by and

xca_linear_iii

xca_quadratic_i



4. The vertex of the graph of the corresponding function is

5. The graph of is

a)

b)



c)

d)



e)

f)



g)

h)



i)

j)



k)

l)



m)

n)



o)

p)




The quotient and the remainder of the corresponding division is

xca_polinomosztas






xca_horner_behelyettesitesre

xca_compute_trig




xca_sum_diff_dub_alkalmazas


Bibliography

[1] K. A. Stroud and D. J. Booth. Engineering Mathematics, Sixth Edition. Industrial Press, Inc., New York.

2007.

[2] K. Sydsaeter and P. Hammond. Essential Mathematics for Economic Analysis, Third Edition. Prentice Hall,

Harlow, London, New York. 2008.

[3] K. Sydsaeter, P. Hammond, A. Seierstad and A. Strom. Further Mathematics for Economic Analysis, Second

Edition. Prentice Hall, Harlow, London, New York. 2008.

[4] G. B. Thomas, Jr., M. D. Weir and J. Hass. Thomas' Calculus, Early Transcendentals, Twelfth Edition.

Addison–Wesley, Boston, Columbus, Indianapolis. 2009.

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