College Algebra

About the Author.

Preface to the Instructor.

Acknowledgments.

Preface to the Student.

1 The Real Numbers.

1.1 The Real Line.

Construction of the Real Line.

Is Every Real Number Rational?

Problems.

1.2 Algebra of the Real Numbers.

Commutativity and Associativity.

The Order of Algebraic Operations.

The Distributive Property.

Additive Inverses and Subtraction.

Multiplicative Inverses and the Algebra of Fractions.

Symbolic Calculators.

Exercises, Problems, and Worked-out Solutions.

1.3 Inequalities.

Positive and Negative Numbers.

Lesser and Greater.

Intervals.

Absolute Value.

Exercises, Problems, and Worked-out Solutions.

Chapter Summary and Chapter Review Questions.

2 Combining Algebra and Geometry.

2.1 The Coordinate Plane.

Coordinates.

Graphs of Equations.

Distance Between Two Points.

Length, Perimeter, and Circumference.

Exercises, Problems, and Worked-out Solutions.

2.2 Lines.

Slope.

The Equation of a Line.

Parallel Lines.

Perpendicular Lines.

Midpoints.

Exercises, Problems, and Worked-out Solutions.

2.3 Quadratic Expressions and Conic Sections.

Completing the Square.

The Quadratic Formula.

Circles.

Ellipses.

Parabolas.

Hyperbolas.

Exercises, Problems, and Worked-out Solutions.

2.4 Area.

Squares, Rectangles, and Parallelograms.

Triangles and Trapezoids.

Stretching.

Circles and Ellipses.

Exercises, Problems, and Worked-out Solutions.

Chapter Summary and Chapter Review Questions.

3 Functions and Their Graphs.

3.1 Functions.

Definition and Examples.

The Graph of a Function.

The Domain of a Function.

The Range of a Function.

Functions via Tables.

Exercises, Problems, and Worked-out Solutions.

3.2 Function Transformations and Graphs.

Vertical Transformations: Shifting, Stretching, and Flipping.

Horizontal Transformations: Shifting, Stretching, Flipping.

Combinations of Vertical Function Transformations.

Even Functions.

Odd Functions.

Exercises, Problems, and Worked-out Solutions.

3.3 Composition of Functions.

Combining Two Functions.

Definition of Composition.

Order Matters in Composition.

Decomposing Functions.

Composing More than Two Functions.

Function Transformations as Compositions.

Exercises, Problems, and Worked-out Solutions.

3.4 Inverse Functions.

The Inverse Problem.

One-to-one Functions.

The Definition of an Inverse Function.

The Domain and Range of an Inverse Function.

The Composition of a Function and Its Inverse.

Comments about Notation.

Exercises, Problems, and Worked-out Solutions.

3.5 A Graphical Approach to Inverse Functions.

The Graph of an Inverse Function.

Graphical Interpretation of One-to-One.

Increasing and Decreasing Functions.

Inverse Functions via Tables.

Exercises, Problems, and Worked-out Solutions.

Chapter Summary and Chapter Review Questions.

4 Polynomial and Rational Functions.

4.1 Integer Exponents.

Positive Integer Exponents.

Properties of Exponents.

Defining x0.

Negative Integer Exponents.

Manipulations with Exponents.

Exercises, Problems, and Worked-out Solutions.

4.2 Polynomials.

The Degree of a Polynomial.

The Algebra of Polynomials.

Zeros and Factorization of Polynomials.

The Behavior of a Polynomial Near 1.

Graphs of Polynomials.

Exercises, Problems, and Worked-out Solutions.

4.3 Rational Functions.

Ratios of Polynomials.

The Algebra of Rational Functions.

Division of Polynomials.

The Behavior of a Rational Function Near 1.

Graphs of Rational Functions.

Exercises, Problems, and Worked-out Solutions.

4.4 Complex Numbers.

The Complex Number System.

Arithmetic with Complex Numbers.

Complex Conjugates and Division of Complex Numbers.

Zeros and Factorization of Polynomials, Revisited.

Exercises, Problems, and Worked-out Solutions.

Chapter Summary and Chapter Review Questions.

5 Exponents and Logarithms.

5.1 Exponents and Exponential Functions.

Roots.

Rational Exponents.

Real Exponents

Exponential Functions

Exercises, Problems, and Worked-out Solutions

5.2 Logarithms as Inverses of Exponential Functions.

Logarithms Base 2.

Logarithms with Any Base.

Common Logarithms and the Number of Digits.

Logarithm of a Power.

Radioactive Decay and Half-Life.

Exercises, Problems, and Worked-out Solutions.

5.3 Applications of Logarithms.

Logarithm of a Product.

Logarithm of a Quotient.

Earthquakes and the Richter Scale.

Sound Intensity and Decibels.

Star Brightness and Apparent Magnitude.

Change of Base.

Exercises, Problems, and Worked-out Solutions.

5.4 Exponential Growth.

Functions with Exponential Growth.

Population Growth.

Compound Interest

Exercises, Problems, and Worked-out Solutions

Chapter Summary and Chapter Review Questions.

6 e and the Natural Logarithm.

6.1 Defining e and ln.

Estimating Area Using Rectangles.

Defining e.

Defining the Natural Logarithm.

Properties of the Exponential Function and ln.

Exercises, Problems, and Worked-out Solutions.

6.2 Approximations with e and ln.

Approximation of the Natural Logarithm.

Inequalities with the Natural Logarithm.

Approximations with the Exponential Function.

An Area Formula.

Exercises, Problems, and Worked-out Solutions.

6.3 Exponential Growth Revisited.

Continuously Compounded Interest.

Continuous Growth Rates.

Doubling Your Money

Exercises, Problems, and Worked-out Solutions.

Chapter Summary and Chapter Review Questions.

7 Systems of Equations and Inequalities.

7.1 Equations and Systems of Equations.

Solving an Equation.

Solving a System of Equations.

Systems of Linear Equations.

Matrices.

Exercises, Problems, and Worked-out Solutions.

7.2 Solving Systems of Linear Equations.

Gaussian Elimination.

Gaussian Elimination with Matrices.

Special Cases—No Solutions.

Special Cases—Infinitely Many Solutions.

Exercises, Problems, and Worked-out Solutions.

7.3 Matrix Algebra.

Adding and Subtracting Matrices.

Multiplying Matrices.

The Inverse of a Matrix.

Exercises, Problems, and Worked-out Solutions.

Chapter Summary and Chapter Review Questions.

8 Sequences, Series, and Limits.

8.1 Sequences.

Introduction to Sequences.

Arithmetic Sequences.

Geometric Sequences.

Recursively-Defined Sequences.

Exercises, Problems, and Worked-out Solutions.

8.2 Series.

Sums of Sequences.

Arithmetic Series.

Geometric Series.

Summation Notation.

The Binomial Theorem.

Exercises, Problems, and Worked-out Solutions.

8.3 Limits.

Introduction to Limits.

Infinite Series.

Decimals as Infinite Series.

Special Infinite Series.

Exercises, Problems, and Worked-out Solutions.

Chapter Summary and Chapter Review Questions.