Post on 18-Jun-2020
Created by XMLmind XSL-FO Converter.
Analysis
Attila Bérczes
Attila Gilányi
Created by XMLmind XSL-FO Converter.
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.
iii Created by XMLmind XSL-FO Converter.
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
iv Created by XMLmind XSL-FO Converter.
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
v Created by XMLmind XSL-FO Converter.
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
1 Created by XMLmind XSL-FO Converter.
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
2 Created by XMLmind XSL-FO Converter.
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.
The concept and basic properties of
functions
3 Created by XMLmind XSL-FO Converter.
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.
The concept and basic properties of
functions
4 Created by XMLmind XSL-FO Converter.
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
.
The concept and basic properties of
functions
5 Created by XMLmind XSL-FO Converter.
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.
The concept and basic properties of
functions
6 Created by XMLmind XSL-FO Converter.
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.
The concept and basic properties of
functions
7 Created by XMLmind XSL-FO Converter.
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 :
The concept and basic properties of
functions
8 Created by XMLmind XSL-FO Converter.
2. We plot the points ,
,
3. We connect the points by a smooth curve
The concept and basic properties of
functions
9 Created by XMLmind XSL-FO Converter.
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
The concept and basic properties of
functions
10 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
11 Created by XMLmind XSL-FO Converter.
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)
The concept and basic properties of
functions
12 Created by XMLmind XSL-FO Converter.
b)
c)
The concept and basic properties of
functions
13 Created by XMLmind XSL-FO Converter.
d)
3.5. 1.3.5. The -intercept and the -intercepts of functions
The concept and basic properties of
functions
14 Created by XMLmind XSL-FO Converter.
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
The concept and basic properties of
functions
15 Created by XMLmind XSL-FO Converter.
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
The concept and basic properties of
functions
16 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
17 Created by XMLmind XSL-FO Converter.
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
,
The concept and basic properties of
functions
18 Created by XMLmind XSL-FO Converter.
.
The restriction of to the set is , . The graph of is:
Graph of
the
function
,
.
2. Let , . The graph of is:
The concept and basic properties of
functions
19 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The restriction of to the set is , . The graph of is:
The concept and basic properties of
functions
20 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The restriction of to the set is , . The graph of is:
The concept and basic properties of
functions
21 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The restriction of to the set is , . The graph of is:
The concept and basic properties of
functions
22 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
23 Created by XMLmind XSL-FO Converter.
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.
The concept and basic properties of
functions
24 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
It is easy to see, that there are no elements and such that , which yields the
injectivity of .
4. Let , . The graph of is:
The concept and basic properties of
functions
25 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Obviously, there no no elements and such that , thus is also
injective.
5. Let , . The graph of :
The concept and basic properties of
functions
26 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
27 Created by XMLmind XSL-FO Converter.
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.,
The concept and basic properties of
functions
28 Created by XMLmind XSL-FO Converter.
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.
The concept and basic properties of
functions
29 Created by XMLmind XSL-FO Converter.
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.
4. Let , . The graph of is:
The concept and basic properties of
functions
30 Created by XMLmind XSL-FO Converter.
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.
5. Let , . The graph of is:
The concept and basic properties of
functions
31 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
It is easy to see that, for each there exists an such that Thus, is surjective.
6. Let , . The graph of :
The concept and basic properties of
functions
32 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
It is easy to check that is not surjective.
7. Let , . The graph of :
The concept and basic properties of
functions
33 Created by XMLmind XSL-FO Converter.
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
The concept and basic properties of
functions
34 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
35 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
36 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
37 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
This function is also injective and surjective, therefore, it is bijective.
7. Let , . The graph of is:
The concept and basic properties of
functions
38 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The function is injective but not surjective, thus, it is not bijective.
8. Let , . The graph of is:
The concept and basic properties of
functions
39 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The function is injective and surjective, therefore, it is bijective.
9. Let , . The graph of :
The concept and basic properties of
functions
40 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
This function is neither injective nor surjective, thus, it is not bijective.
10. Let , . The graph of :
The concept and basic properties of
functions
41 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The function is injective but not surjective, therefore, it is not bijective.
11. Let , . The graph of :
The concept and basic properties of
functions
42 Created by XMLmind XSL-FO Converter.
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) , ,
, , , , , ;
The concept and basic properties of
functions
43 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
44 Created by XMLmind XSL-FO Converter.
It is easy to see that this function is injective, thus there exists its inverse. The inverse of has
the form
that is,
The concept and basic properties of
functions
45 Created by XMLmind XSL-FO Converter.
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,
The concept and basic properties of
functions
46 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
47 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
48 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Let us draw the graphs of the functions and above in the same coordinate system.
The concept and basic properties of
functions
49 Created by XMLmind XSL-FO Converter.
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.
The concept and basic properties of
functions
50 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
This function is injective, thus, it is invertible. The equation implies that , therefore
the inverse of has the form .
The concept and basic properties of
functions
51 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Drawing the graphs of and in the same coordinate system, we obtain:
The concept and basic properties of
functions
52 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
53 Created by XMLmind XSL-FO Converter.
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:
The concept and basic properties of
functions
54 Created by XMLmind XSL-FO Converter.
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) , ,
, , , , , .
The concept and basic properties of
functions
55 Created by XMLmind XSL-FO Converter.
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) , .
56 Created by XMLmind XSL-FO Converter.
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
57 Created by XMLmind XSL-FO Converter.
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 .
Example. Draw the graph of the following functions:
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
58 Created by XMLmind XSL-FO Converter.
Example. Draw the graph of the following 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
59 Created by XMLmind XSL-FO Converter.
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
60 Created by XMLmind XSL-FO Converter.
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
61 Created by XMLmind XSL-FO Converter.
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:
Types of functions
62 Created by XMLmind XSL-FO Converter.
Example. Draw the graph of the functions for and for .
Types of functions
63 Created by XMLmind XSL-FO Converter.
Example. Draw the graph of the functions for .
Types of functions
64 Created by XMLmind XSL-FO Converter.
Example. Draw the graph of the functions for .
Example. Draw the graph of the functions for .
Types of functions
65 Created by XMLmind XSL-FO Converter.
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
66 Created by XMLmind XSL-FO Converter.
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
67 Created by XMLmind XSL-FO Converter.
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
68 Created by XMLmind XSL-FO Converter.
• 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
69 Created by XMLmind XSL-FO Converter.
4. 2.4. Power functions
Definition 2.11. A function given by
with given and is called a power function.
Example. Draw the graph of the functions , for and for
.
Types of functions
70 Created by XMLmind XSL-FO Converter.
5. 2.5. Exponential functions
Definition 2.12. A function given by
Types of functions
71 Created by XMLmind XSL-FO Converter.
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
72 Created by XMLmind XSL-FO Converter.
Example. Draw the graph of the functions .
6. 2.6. Logarithmic functions
Types of functions
73 Created by XMLmind XSL-FO Converter.
Definition 2.15. A function given by
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
74 Created by XMLmind XSL-FO Converter.
Remark. We also recall that the logarithmic function on base is denoted by .
75 Created by XMLmind XSL-FO Converter.
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
76 Created by XMLmind XSL-FO Converter.
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 .
Trigonometric functions
77 Created by XMLmind XSL-FO Converter.
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
Trigonometric functions
78 Created by XMLmind XSL-FO Converter.
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.
Trigonometric functions
79 Created by XMLmind XSL-FO Converter.
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 .
Trigonometric functions
80 Created by XMLmind XSL-FO Converter.
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 .
Trigonometric functions
81 Created by XMLmind XSL-FO Converter.
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,
Trigonometric functions
82 Created by XMLmind XSL-FO Converter.
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 .
Trigonometric functions
83 Created by XMLmind XSL-FO Converter.
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:
Trigonometric functions
84 Created by XMLmind XSL-FO Converter.
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
Trigonometric functions
85 Created by XMLmind XSL-FO Converter.
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
3. If is in the quadrant then
4. If is at the border of the and quadrant, i.e. then
5. If is in the quadrant then
Trigonometric functions
86 Created by XMLmind XSL-FO Converter.
6. If is at the border of the and quadrant, i.e. then
7. If is in the quadrant then
8. 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:
Trigonometric functions
87 Created by XMLmind XSL-FO Converter.
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:
Trigonometric functions
88 Created by XMLmind XSL-FO Converter.
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.
Trigonometric functions
89 Created by XMLmind XSL-FO Converter.
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:
Trigonometric functions
90 Created by XMLmind XSL-FO Converter.
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:
Trigonometric functions
91 Created by XMLmind XSL-FO Converter.
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
Trigonometric functions
92 Created by XMLmind XSL-FO Converter.
11.2. 3.11.2. The graph of
11.3. 3.11.3. The graph of
Trigonometric functions
93 Created by XMLmind XSL-FO Converter.
11.4. 3.11.4. The graph of
Trigonometric functions
94 Created by XMLmind XSL-FO Converter.
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
Trigonometric functions
95 Created by XMLmind XSL-FO Converter.
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]:
Trigonometric functions
96 Created by XMLmind XSL-FO Converter.
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):
Trigonometric functions
97 Created by XMLmind XSL-FO Converter.
Finally we prove 4):
Example. Compute and .
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:
Trigonometric functions
98 Created by XMLmind XSL-FO Converter.
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)
Example. Compute and .
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:
Trigonometric functions
99 Created by XMLmind XSL-FO Converter.
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
Theorem 3.26. Let be an arbitrary real number such that the functions in the below formulas are defined for
their arguments. Then we have:
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]:
Trigonometric functions
100 Created by XMLmind XSL-FO Converter.
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
Theorem 3.27. Let be an arbitrary real number such that the functions in the below formulas are defined for
their arguments. Then we have:
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
Trigonometric functions
101 Created by XMLmind XSL-FO Converter.
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
Theorem 3.28. Let be an arbitrary real number such that the functions in the below formulas are defined for
their arguments. Then we have:
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):
Trigonometric functions
102 Created by XMLmind XSL-FO Converter.
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
Theorem 3.29. Let be an arbitrary real number such that the functions in the below formulas are defined for
their arguments. Then we have:
Trigonometric functions
103 Created by XMLmind XSL-FO Converter.
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:
Trigonometric functions
104 Created by XMLmind XSL-FO Converter.
105 Created by XMLmind XSL-FO Converter.
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
106 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Drawing both graphs above in the same coordinate system, we obtain the following:
Transformations of functions
107 Created by XMLmind XSL-FO Converter.
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 , .
Transformations of functions
108 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
We shall illustrate the graphs of and in the same coordinate system now.
Transformations of functions
109 Created by XMLmind XSL-FO Converter.
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.
Transformations of functions
110 Created by XMLmind XSL-FO Converter.
– 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
,
.
Transformations of functions
111 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
2. Let us sketch the graph of the function , using the graph of , .
Transformations of functions
112 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
3. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
113 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
114 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
4. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
115 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
5. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
116 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
117 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
6. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
118 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
7. Let us sketch the graphs of the function , using , .
Transformations of functions
119 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
120 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
8. Let us sketch the graphs of the function , using , .
Transformations of functions
121 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
9. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
122 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
123 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
10. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
124 Created by XMLmind XSL-FO Converter.
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
.
Transformations of functions
125 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
126 Created by XMLmind XSL-FO Converter.
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 .
Transformations of functions
127 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
13. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
128 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
129 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
14. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
130 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
15. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
131 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
132 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
16. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
133 Created by XMLmind XSL-FO Converter.
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) , , , ;
Transformations of functions
134 Created by XMLmind XSL-FO Converter.
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 , .
Transformations of functions
135 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Let us consider the graph of , :
Graph of
the
function
,
.
Finally, we sketch both functions in the same coordinate system:
Transformations of functions
136 Created by XMLmind XSL-FO Converter.
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.
Transformations of functions
137 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Sketching and in the same coordinate system, we obtain:
Transformations of functions
138 Created by XMLmind XSL-FO Converter.
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:
Transformations of functions
139 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
If we draw it together with the graph of , we get
Transformations of functions
140 Created by XMLmind XSL-FO Converter.
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.
Transformations of functions
141 Created by XMLmind XSL-FO Converter.
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:
Transformations of functions
142 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Let us consider the graph of , .
Transformations of functions
143 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Sketching these two graphs in the same coordinate system, we get
Transformations of functions
144 Created by XMLmind XSL-FO Converter.
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.
Transformations of functions
145 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Sketching these two graphs in the same coordinate system, we get
Transformations of functions
146 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
Finally, we will investigate the case when our factor is negative. Let us consider the function ,
.
Transformations of functions
147 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
148 Created by XMLmind XSL-FO Converter.
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 ,
.
Transformations of functions
149 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
150 Created by XMLmind XSL-FO Converter.
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.
– 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, and reflecting this graph about the -axis.
Transformations of functions
151 Created by XMLmind XSL-FO Converter.
– 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.
– 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.
– In the case when , we can obtain the graph of the function , by
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.
– In the case when , we can obtain the graph of the function , by
compressing (or shrinking) the graph of the function with the factor horizontally, and reflecting this graph
about the -axis.
Examples.
1. Let us sketch the graphs of the function , using the graph of ,
.
Graph of
the
function
,
.
Transformations of functions
152 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
2. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
153 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
3. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
154 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
4. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
155 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
5. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
156 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
6. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
157 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
7. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
158 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
8. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
159 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
9. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
160 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
10. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
161 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
11. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
162 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
12. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
163 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
13. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
164 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
Transformations of functions
165 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
14. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
166 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
15. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
167 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
16. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
168 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
17. Let us sketch the graph of the function , using the graph of ,
.
Transformations of functions
169 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
18. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
170 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
19. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
171 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
20. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
172 Created by XMLmind XSL-FO Converter.
Graphs of
the
functions
,
and
,
.
21. Let us sketch the graphs of the function , using the graph of ,
.
Transformations of functions
173 Created by XMLmind XSL-FO Converter.
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) , , , ;
Transformations of functions
174 Created by XMLmind XSL-FO Converter.
i) , , , ;
j) , , , ;
k) , , , ;
l) , , , ;
m) , , , ;
n) , , , ;
o) , , , ;
p) , , , ;
q) , , , ;
r) , , , ;
s) , , , ;
t) , , , ;
175 Created by XMLmind XSL-FO Converter.
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)
Results of the exercises
176 Created by XMLmind XSL-FO Converter.
b)
c)
Results of the exercises
177 Created by XMLmind XSL-FO Converter.
d)
e)
Results of the exercises
178 Created by XMLmind XSL-FO Converter.
f)
g)
Results of the exercises
179 Created by XMLmind XSL-FO Converter.
h)
i)
Results of the exercises
180 Created by XMLmind XSL-FO Converter.
j)
k)
Results of the exercises
181 Created by XMLmind XSL-FO Converter.
l)
m)
Results of the exercises
182 Created by XMLmind XSL-FO Converter.
n)
o)
Results of the exercises
183 Created by XMLmind XSL-FO Converter.
p)
q)
Results of the exercises
184 Created by XMLmind XSL-FO Converter.
r)
Results of the exercises
185 Created by XMLmind XSL-FO Converter.
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)
Results of the exercises
186 Created by XMLmind XSL-FO Converter.
c)
In exercises b) and d) the corresponding curve is the graph of a function.
Solution of Exercise 1.4 [14]
Results of the exercises
187 Created by XMLmind XSL-FO Converter.
The -intercept of the corresponding functions is:
The -intercepts of the corresponding functions are:
Solution of Exercise 1.5 [16]:
Results of the exercises
188 Created by XMLmind XSL-FO Converter.
Results of the exercises
189 Created by XMLmind XSL-FO Converter.
Solution of Exercise 1.6 [16]
Solution of Exercise 1.7 [42]
a) Let , and let , , , , ,
. We can illustrate in the following form:
Results of the exercises
190 Created by XMLmind XSL-FO Converter.
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 , , , , ,
. We can illustrate in the following form:
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:
Results of the exercises
191 Created by XMLmind XSL-FO Converter.
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 , , , , ,
. We can illustrate in the following form:
It is visible that is a function and it is surjective. However is not injective, therefore, it is not bijective.
e) Let , and let , , , , ,
. We can illustrate in the following form:
Results of the exercises
192 Created by XMLmind XSL-FO Converter.
It is easy to see that is a function, it is injective, surjective, which implies that it is also bijective.
f) Let , and let , , , , ,
. We can illustrate in the following form:
It is visible that is a surjective function. However is not injective, therefore, it is not bijective.
Solution of Exercise 1.8 [43]
a) Let , . The graph of is:
Results of the exercises
193 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
194 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
195 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
196 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
197 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
198 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
199 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
200 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
201 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
202 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
203 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
204 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
It is easy to see that the function is surjective and injective, therefore it is also bijective.
Solution of Exercise 1.9 [54]
a) Let , and let , , , , ,
. We may illustrate in the following form:
Results of the exercises
205 Created by XMLmind XSL-FO Converter.
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.,
Results of the exercises
206 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
207 Created by XMLmind XSL-FO Converter.
that is,
which means that , , , , .
c) Let , , , , , , . A diagram of
is:
Results of the exercises
208 Created by XMLmind XSL-FO Converter.
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 .
Results of the exercises
209 Created by XMLmind XSL-FO Converter.
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.
Solution of Exercise 1.10 [55]
a) The graph of the function , is
Results of the exercises
210 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
211 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The graph of and its inverse in the same coordinate system
Results of the exercises
212 Created by XMLmind XSL-FO Converter.
Graphs of
,
and its
inverse
,
.
b) The graph of , is
Results of the exercises
213 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
214 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The graph of and its inverse is:
Results of the exercises
215 Created by XMLmind XSL-FO Converter.
Graphs of
,
and its
inverse
,
.
c) The graph of the function , is:
Results of the exercises
216 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
217 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The graphs of and its inverse is:
Results of the exercises
218 Created by XMLmind XSL-FO Converter.
Graphs of
,
and its
inverse
,
.
e) The graph of , is:
Results of the exercises
219 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
220 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The graphs of and are:
Results of the exercises
221 Created by XMLmind XSL-FO Converter.
Graphs of
,
and its
inverse
,
.
f) The graph of , is:
Results of the exercises
222 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
223 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The graphs of and are:
Results of the exercises
224 Created by XMLmind XSL-FO Converter.
Graphs of
,
and its
inverse
,
.
g) The graph of , is:
Results of the exercises
225 Created by XMLmind XSL-FO Converter.
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:
Results of the exercises
226 Created by XMLmind XSL-FO Converter.
Graph of
the
function
,
.
The graphs of and are:
Results of the exercises
227 Created by XMLmind XSL-FO Converter.
Graphs of
,
and its
inverse
,
.
2. 5.2. Results of the Exercises of Chapter 2
Solution of Exercise 2.1 [60]:
The formula defining the corresponding functions is:
The graphs of the corresponding functions are:
a)
Results of the exercises
228 Created by XMLmind XSL-FO Converter.
b)
c)
Results of the exercises
229 Created by XMLmind XSL-FO Converter.
d)
Solution of Exercise 2.2 [60]:
Results of the exercises
230 Created by XMLmind XSL-FO Converter.
Solution of Exercise 2.3 [60]
Solution of Exercise 2.4 [65]:
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
Results of the exercises
231 Created by XMLmind XSL-FO Converter.
4. The vertex of the graph of the corresponding function is
5. The graph of is
a)
b)
Results of the exercises
232 Created by XMLmind XSL-FO Converter.
c)
d)
Results of the exercises
233 Created by XMLmind XSL-FO Converter.
e)
f)
Results of the exercises
234 Created by XMLmind XSL-FO Converter.
g)
h)
Results of the exercises
235 Created by XMLmind XSL-FO Converter.
i)
j)
Results of the exercises
236 Created by XMLmind XSL-FO Converter.
k)
l)
Results of the exercises
237 Created by XMLmind XSL-FO Converter.
m)
n)
Results of the exercises
238 Created by XMLmind XSL-FO Converter.
o)
p)
Results of the exercises
239 Created by XMLmind XSL-FO Converter.
Solution of Exercise 2.5 [67]:
The quotient and the remainder of the corresponding division is
Results of the exercises
240 Created by XMLmind XSL-FO Converter.
Solution of Exercise 2.6 [68]
3. 5.3. Results of the Exercises of Chapter 3
Solution of Exercise 3.1 [89]:
Results of the exercises
241 Created by XMLmind XSL-FO Converter.
Solution of Exercise 3.2 [99]:
242 Created by XMLmind XSL-FO Converter.
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.