Elementary Functions

Main subject categories: • Elementary functions ‒ Pre-Calculus • Properties or elementary functions ‒ pre-limit level • Mathematical reasoning • Pre-CalculusThis 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. 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 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.