This textbook focuses on the study of different kinds of elementary functions ubiquitous both in high school Algebra and Calculus. To analyze the functions ranging from polynomial to trigonometric ones, it uses rudimentary techniques available to high school students, and at the same time follows the mathematical rigor appropriate for university level courses.
Contrary to other books of Pre-Calculus, this textbook emphasizes the study of elementary functions with rigor appropriate for university level courses in mathematics, although the exposition is confined to the pre-limit topics and techniques. This makes the book useful, on the one hand, as an introduction to mathematical reasoning and methods of proofs in mathematical analysis, and on the other hand, as a preparatory course on the properties of different kinds of elementary functions.
The textbook is aimed at university freshmen and high-school students interested in learning strict mathematical reasoning and in preparing a solid base for subsequent study of elementary functions at advanced level of Calculus and Analysis. The required prerequisites correspond to the level of the high school Algebra. All the preliminary concepts and results related to the elementary functions are covered in the initial part of the text. This makes the textbook suitable for both classroom use and self-study.
Author(s): Andrei Bourchtein, Ludmila Bourchtein
Edition: 1
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 495
City: Cham
Tags: Graphs of Functions; Extrema; Elementary Functions; Concavity and Inflection; Monotonicity
Preface
Contents
1 Sets of Numbers and Cartesian Coordinates
1 General Sets and Operations
1.1 Description of a Set
1.2 Elementary Operations with Sets
1.3 Elementary Properties of Sets
2 Rational Numbers and Their Properties
3 Real Numbers and Their Properties
3.1 Decimal and Real Numbers
3.2 Properties of the Real Numbers
3.3 Absolute Value
4 Coordinate Line and its Equivalence with the Set of Real Numbers
5 Some Sets of Numbers and Their Properties
5.1 Distance Between Two Points
5.2 Interval, Midpoint, Symmetric Point, Neighborhood
6 Cartesian Coordinates on the Plane
6.1 Definition of the Coordinates
6.2 Coordinate Lines
6.3 Projections on the Coordinate Lines
7 Some Relations on the Cartesian Plane: Distance, Midpoint, Symmetry
7.1 Distance Between Two Points
7.2 Distance from a Point to a Coordinate Line
7.3 Midpoint of an Interval
7.4 Symmetry with Respect to a Point
7.5 Symmetry with Respect to a Line
Problems
General Sets
Rational Numbers
Real Numbers
Coordinate Line
Some Sets of Numbers
Cartesian Coordinates on the Plane
Some Relations on the Cartesian Plane
2 Functions and Their Analytic Properties
1 Function: Definition, Domain, Range
2 Modes of the Definition of a Function
2.1 Analytic (Algebraic) Mode
Graph of a Function
Analytic Forms of the Definition of a Function
2.2 Geometric Mode
2.3 Numerical Mode
2.4 Descriptive Mode
2.5 Relationship Between Different Forms of the Definition of a Function
3 Bounded Functions
4 Properties of Symmetry
4.1 Even Functions
Properties
4.2 Odd Functions
Properties
4.3 Arithmetic Operations with Even and Odd Functions
4.4 Periodic Functions
Properties
4.5 Elementary Operations with Periodic Functions
Arithmetic Operations with Periodic Functions
Operations with the Argument of Periodic Functions
5 Monotonicity of a Function
5.1 Definitions and Examples
Increase and Decrease on a Set
Increasing and Decreasing at a Point
5.2 Arithmetic Operations with Monotonic Functions
6 Extrema of a Function
6.1 Global Extrema
Relationship Between Extrema and Monotonicity
6.2 Local Extrema
Relationship Between Local Extrema and Monotonicity
Relationship Between Local and Global Extrema
6.3 Monotonicity, Extrema and Symmetry
7 Concavity and Inflection
7.1 General Concavity
7.2 Midpoint Concavity
7.3 Elementary Properties of Concave Functions
7.4 Inflection Points
7.5 Concavity, Inflection and Symmetry
8 Complimentary Properties of Functions
8.1 Behavior at Infinity and Convergence to Infinity
8.2 Horizontal and Vertical Asymptotes
9 Composite Functions
9.1 Composition of Functions
9.2 Decomposition of Complicated Functions
9.3 Compositions of Specific Types of Functions
Composition of Even/Odd Functions
Composition of Periodic Functions
Composition of Monotonic Functions
Composition of Concave Functions
10 Elementary Transformations of Functions and Their Graphs
10.1 Vertical and Horizontal Translations
Vertical Translation
Horizontal Translation
10.2 Vertical and Horizontal Reflection
Vertical Reflection
Horizontal Reflection
10.3 Vertical and Horizontal Stretching/Shrinking
Vertical Stretching/Shrinking
Horizontal Stretching/Shrinking
11 Injective, Surjective and Bijective Functions
12 Inverse Function
12.1 One-Sided Inverses
12.2 General Inverse. Definition and Elementary Examples
12.3 Conditions of the Existence of the Inverse
12.4 Analytic Properties of the Inverse
12.5 Geometric Property of the Inverse
13 Classification of Elementary Functions
14 Solved Exercises
14.1 Domain and Range
14.2 Bounded Functions
14.3 Even, Odd and Periodic Functions
14.4 Monotonicity
14.5 Global and Local Extrema
14.6 Concavity and Inflection
14.7 Composite Functions
14.8 Injection, Surjection, Bijection, Inverse
14.9 Study of Functions
Problems
3 Algebraic Functions: Polynomial, Rational and Irrational
1 Polynomial Functions
1.1 Linear Function
Study of y=b (a=0)
Study of y=x (a=1, b=0)
Study of y=ax+b, a=0
1.2 Quadratic Function
Study of y=x2
Study of y=-x2
Study of y=x2-6x+2
Study of y=ax2+bx+c
1.3 Monomials
Study of y=x3
Study of y=x2k+1, kN
Study of y=x2k, kN
Study of y=-2x7
2 Rational Functions
2.1 Functions y=1xn, nN
Study of y=1x
Study of y=1x2
Study of y=1x2k+1
Study of y=1x2k
2.2 Fractional Linear Functions
Study of fractional linear functions y=ax+bcx+d
2.3 Some General Properties of Rational Functions
3 Irrational Functions
3.1 Roots y=[n]x, nN
Study of y=x
Study of y=[3]x
Study of y=[2k]x, kN
Study of y=[2k+1]x, kN
3.2 Absolute Value Function
Study of y=|x|
4 Examples of the Study of Algebraic Functions
4.1 Preliminary Considerations
4.2 Study of Polynomial Functions
A. Study of f(x)=x3-x2-x+1
B. Study of f(x)=x44+x3-4x-1
4.3 Study of Rational Functions
A. Study of f(x)=x2x2-1
B. Study of f(x)=2x2+3x+1x2+4x+3
C. Study of f(x)=x2-1x2-3x
4.4 Study of Irrational Functions
A. Study of f(x)=2x+6
B. Study of f(x)=1[3]4-2x
C. Study of f(x)=|x2-4x|
5 Solved Exercises
5.1 Study of Polynomial Functions
A. Study of f(x)=x3+2x-3
B. Study of f(x)=x3-3x+2
C. Study of f(x)=4-3x2-x3
D. Study of f(x)=x4+x2-2
E. Study of f(x)=x4-5x2+4
5.2 Study of Rational Functions
A. Study of f(x)=1-x2x
B. Study of f(x)=x1-x2
C. Study of f(x)=1x2+1
D. Study of f(x)=1x2-1
E. Study of f(x)=1x3+2
F. Study of f(x)=2x+31-x
5.3 Study of Irrational Functions
A. Study of f(x)=x2+1
B. Study of f(x)=x2+1-|x|
C. Study of f(x)=x2-1
D. Study of f(x)=x+x2-1
E. Study of f(x)=x+4-x
Problems
4 Transcendental Functions: Exponential, Logarithmic, Trigonometric
1 Exponential and Logarithmic Functions
1.1 Preliminary Notions: Power with Real Exponent and its Properties
1.2 Exponential Function
Study of y=ax
1.3 Logarithmic Function
Study of y=loga x
2 Trigonometric Functions
2.1 Preliminary Notions and Results of Trigonometry
2.2 Function sinx
Study of y=sinx
2.3 Function cosx
Study of y=cosx
2.4 Function arcsinx
Study of y=arcsinx
2.5 Function arccosx
Study of y=arccosx
2.6 Function tanx
Study of y=tanx
2.7 Function cotx
Study of y=cotx
2.8 Function arctanx
Study of y=arctanx
2.9 Function arccot x
Study of y=arccot x
3 Examples of Study of Transcendental Functions
3.1 Study of Exponential Functions
A. Study of f(x)=2·103x
B. Study of f(x)=2-5 e-x/3=2-5(1e )x/3
3.2 Study of Logarithmic Functions
A. Study of f(x)=13log2 (5x-1)
B. Study of f(x)=1-log1/10 (4-2x)
3.3 Study of Trigonometric Functions
A. Study of y=2sin(3x+π6)
B. Study of y=12arccos(2-4x)-π4
C. Study of y=2cot(π4-x3)
4 Solved Exercises
4.1 Study of Exponential Functions
A. Study of f(x)=1-23x
B. Study of f(x)=1-2-3|x|
C. Study of f(x)=23x2-4
4.2 Study of Logarithmic Functions
A. Study of f(x)=1-log5 (x2+1)
B. Study of f(x)=log3 (|x|-1)-1
C. Study of f(x)=log2 (4-|x|)-1
4.3 Study of Trigonometric Functions
A. Study of f(x)=tanx + cotx
B. Study of f(x)=tan2 x
C. Study of f(x)=arctanx2
D. Study of f(x)=sin2x -2sinx
Problems
5 Epilogue: A Bridge to Calculus
1 Monotonicity and Extrema: First Derivative
1.1 Preliminary Considerations
1.2 Definition of Differentiability and Derivative
1.3 Relationship Between Derivative and Monotonicity
1.4 Rules of Differentiation and Derivatives of Rational Functions
1.5 Differentiability of Irrational Functions
1.6 Differentiability of Transcendental Functions
2 Concavity and Inflection: Second Derivative
2.1 Definition of Second Derivative
2.2 Relationship Between Second Derivative and Concavity
2.3 Applications of the Second Derivative
3 Epilogue to Epilogue
Remarks on Bibliography
Index
