College Algebra

Implicit Multiplication In mathematics the absence of an operation symbol usually indicates multiplication: abmean a b Sometimes a dot is used to indicate multiplication and in computer languages an asterisk is often used ab:= ab:= ab:= a b Order of operations Parentheses are used to indicate the order of doing the operations: in

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Examples: Decimals on the Number Line Example 5 a) Plot 0 2 on the number line with a black dot b) Plot 0 43 with a green dot Solution: For 0 2 we split the segment from 0 to 1 on the number line into ten equal pieces between 0 and 1 and then count

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i occurs free in Θ or occurs free in Ψ; 6 Φ is (Θ ↔Ψ), Θ and Ψ are formulas and v i occurs free in Θ or occurs free in Ψ; 7 Φ is (∀v j)Ψ and Ψ is a formula and v i occurs free in Ψ and i 6=j; or, 8 Φ is (∃v j)Ψ and Ψ is a formula and v i occurs free in Ψ and i 6=j As in the example below, a variable can occur both

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Vedic Mathematics by the late Sankar~ciir~a (BhPrati Krsna Tirtha) of Govardhana Pitha is a monumental work In his deep-layer explorations of cryptic Vedic mysteries relat- ing specially to their calculus of shorthand formulae and their neat and ready application to practical problems, the late

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Versionp3 = 1:7320508075688772:::

Carl Stitz, Ph.D. Jeff Zeager, Ph.D.

Lakeland Community College Lorain County Community College

Modified by

Joel Robbin and Mike Schroeder

University of Wisconsin, Madison

June 29, 2010

Prefacev

0 Basic Algebra1

0.1 The Laws of Algebra

0.2 Kinds of Numbers

0.3 Exponents

0.4 Absolute Value

0.5 Solving Equations

0.6 Exercises

0.7 Answers

1 Coordinates15

1.1 The Cartesian Coordinate Plane

1.1.1 Distance in the Plane

1.1.2 Exercises

1.1.3 Answers

1.2 Relations

1.2.1 Exercises

1.2.2 Answers

1.3 Graphs of Equations

1.3.1 Exercises

1.3.2 Answers

1.4 Three Interesting Curves

1.4.1 Circles

1.4.2 Parabolas

1.4.3 Ellipses

1.4.4 Exercises

1.4.5 Answers

2 Functions77

2.1 Introduction to Functions

2.1.1 Exercises

83
iv Table of Contents

2.1.2 Answers

2.2 Function Notation

2.2.1 Exercises

2.2.2 Answers

2.3 Function Arithmetic

2.3.1 Exercises

103

2.3.2 Answers

105

2.4 Graphs of Functions

106

2.4.1 General Function Behavior

113

2.4.2 Exercises

120

2.4.3 Answers

123

2.5 Transformations

127

2.5.1 Exercises

149

2.5.2 Answers

151

3 Linear and Quadratic Functions

157

3.1 Linear Functions

157

3.1.1 Exercises

170

3.1.2 Answers

173

3.2 Dening Functions (Word Problems)

175

3.2.1 Exercises

181

3.2.2 Answers

181

3.3 Quadratic Functions

182

3.3.1 Exercises

192

3.3.2 Answers

194

3.4 Inequalities

198

3.4.1 Exercises

211

3.4.2 Answers

212

4 Polynomial Functions

215

4.1 Graphs of Polynomials

215

4.1.1 Exercises

228

4.1.2 Answers

231

4.2 The Factor Theorem and The Remainder Theorem

235

4.2.1 Exercises

244

4.2.2 Answers

245

5 Rational Functions

247

5.1 Introduction to Rational Functions

247

5.1.1 Exercises

259

5.1.2 Answers

261

5.2 Graphs of Rational Functions

263

5.2.1 Exercises

276

Table of Contents v

5.2.2 Answers

278

5.3 Rational Inequalities and Applications

284

5.3.1 Exercises

291

5.3.2 Answers

293

6 Further Topics in Functions

295

6.1 Function Composition

295

6.1.1 Exercises

305

6.1.2 Answers

307

6.2 Inverse Functions

309

6.2.1 Exercises

328

6.2.2 Answers

329

7 Exponential and Logarithmic Functions

331

7.1 Introduction to Exponential and Logarithmic Functions

331

7.1.1 Exercises

345

7.1.2 Answers

348

7.2 Properties of Logarithms

350

7.2.1 Exercises

359

7.2.2 Answers

361

7.3 Exponential Equations and Inequalities

362

7.3.1 Exercises

371

7.3.2 Answers

372

7.4 Logarithmic Equations and Inequalities

373

7.4.1 Exercises

381

7.4.2 Answers

383

7.5 Applications of Exponential and Logarithmic Functions

384

7.5.1 Applications of Exponential Functions

384

7.5.2 Applications of Logarithms

393

7.5.3 Exercises

395

7.5.4 Answers

397

8 Systems of Equations

399

8.1 Systems of Linear Equations: Gaussian Elimination

399

8.1.1 Exercises

414

8.1.2 Answers

416

8.2 Systems of Linear Equations: Augmented Matrices*

419

8.2.1 Exercises

428

8.2.2 Answers

431

8.3 Determinants and Cramer's Rule*

432

8.3.1 Denition and Properties of the Determinant

432

8.3.2 Cramer's Rule

434

8.3.3 Exercises

437
vi Table of Contents

8.3.4 Answers

438

8.4 Systems of Non-Linear Equations and Inequalities

438

8.4.1 Exercises

448

8.4.2 Answers

451

9 Sequences and Series

455

9.1 Sequences

455

9.1.1 Exercises

462

9.1.2 Answers

463

9.2 Series and Summation Notation

463

9.2.1 Exercises

469

9.2.2 Answers

469

9.3 IRAs and Mortgages

470

9.3.1 Exercises

473

9.3.2 Answers

473

9.4 Innite sums and Repeating Decimals*

473

9.4.1 Exercises

476

10 Complex Numbers and the Fundamental Theorem of Algebra

477

10.1 Complex Numbers

477

10.2 The Fundamental Theorem of Algebra

480

10.2.1 Exercises

487

10.2.2 Answers

488

A The Laws of Algebra Proved

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[PDF] College Algebra - Department of Mathematics