Precalculus Prerequisites a.k.a. ‘Chapter 0’

1 Precalculus Prerequisites Chapter 0'. by Carl Stitz, Jeff Zeager, Lakeland Community College Lorain County Community College August 16, 2013. Table of Contents 0 Prerequisites 1. Basic Set Theory and Interval Notation .. 3. Some Basic Set Theory Notions .. 3. Sets of Real Numbers .. 6. Exercises .. 13. Answers .. 15. Real Number Arithmetic .. 18. Exercises .. 36. Answers .. 37. Linear Equations and Inequalities .. 38. Linear Equations .. 38. Linear Inequalities .. 43. Exercises .. 47. Answers .. 49. Absolute Value Equations and Inequalities.

2 51. Absolute Value Equations .. 52. Absolute Value Inequalities .. 54. Exercises .. 58. Answers .. 59. Polynomial Arithmetic .. 60. Polynomial Addition, Subtraction and Multiplication.. 61. Polynomial Long Division.. 64. Exercises .. 68. Answers .. 69. Factoring .. 70. Solving Equations by Factoring .. 76. Exercises .. 81. Answers .. 82. Quadratic Equations .. 83. Exercises .. 94. Answers .. 95. Rational Expressions and Equations .. 96. iv Table of Contents Exercises .. 108. Answers .. 110. Radicals and Equations.

3 111. Rationalizing Denominators and Numerators .. 119. Exercises .. 124. Answers .. 125. Complex Numbers .. 126. Exercises .. 132. Answers .. 133. 1 Relations and Functions 137. Sets of Real Numbers and the Cartesian Coordinate Plane .. 137. The Cartesian Coordinate Plane .. 137. Distance in the Plane .. 142. Exercises .. 146. Answers .. 148. Relations .. 151. Graphs of Equations .. 154. Exercises .. 160. Answers .. 164. Introduction to Functions .. 174. Exercises .. 180. Answers .. 184. Function Notation.

4 186. Modeling with Functions .. 191. Exercises .. 194. Answers .. 201. Function Arithmetic .. 208. Exercises .. 216. Answers .. 219. Graphs of Functions .. 225. General Function Behavior .. 232. Exercises .. 239. Answers .. 246. Transformations .. 252. Exercises .. 272. Answers .. 277. 2 Linear and Quadratic Functions 283. Linear Functions .. 283. Exercises .. 295. Answers .. 301. Absolute Value Functions .. 305. Table of Contents v Exercises .. 316. Answers .. 317. Quadratic Functions .. 321. Exercises.

5 333. Answers .. 336. Inequalities with Absolute Value and Quadratic Functions .. 341. Exercises .. 353. Answers .. 355. Regression .. 358. Exercises .. 363. Answers .. 366. 3 Polynomial Functions 369. Graphs of Polynomials .. 369. Exercises .. 380. Answers .. 384. The Factor Theorem and the Remainder Theorem .. 391. Exercises .. 399. Answers .. 401. Real Zeros of Polynomials .. 403. For Those Wishing to use a Graphing Calculator .. 404. For Those Wishing NOT to use a Graphing Calculator .. 407. Exercises.

6 414. Answers .. 417. Complex Zeros and the Fundamental Theorem of Algebra .. 421. Exercises .. 429. Answers .. 431. 4 Rational Functions 437. Introduction to Rational Functions .. 437. Exercises .. 450. Answers .. 452. Graphs of Rational Functions .. 456. Exercises .. 469. Answers .. 471. Rational Inequalities and Applications .. 478. Variation .. 486. Exercises .. 490. Answers .. 493. vi Table of Contents 5 Further Topics in Functions 495. Function Composition .. 495. Exercises .. 505. Answers .. 508.

7 Inverse Functions .. 514. Exercises .. 530. Answers .. 532. Other Algebraic Functions .. 533. Exercises .. 543. Answers .. 547. 6 Exponential and Logarithmic Functions 553. Introduction to Exponential and Logarithmic Functions .. 553. Exercises .. 566. Answers .. 570. Properties of Logarithms .. 575. Exercises .. 583. Answers .. 585. Exponential Equations and Inequalities .. 586. Exercises .. 594. Answers .. 596. Logarithmic Equations and Inequalities .. 597. Exercises .. 604. Answers .. 606. Applications of Exponential and Logarithmic Functions.

8 607. Applications of Exponential Functions .. 607. Applications of Logarithms .. 615. Exercises .. 620. Answers .. 628. 7 Hooked on Conics 633. Introduction to Conics .. 633. Circles .. 636. Exercises .. 640. Answers .. 641. Parabolas .. 643. Exercises .. 651. Answers .. 652. Ellipses .. 655. Exercises .. 664. Answers .. 666. Hyperbolas .. 671. Exercises .. 681. Table of Contents vii Answers .. 684. 8 Systems of Equations and matrices 689. Systems of Linear Equations: Gaussian Elimination .. 689. Exercises.

9 702. Answers .. 704. Systems of Linear Equations: Augmented matrices .. 707. Exercises .. 714. Answers .. 716. Matrix Arithmetic .. 718. Exercises .. 731. Answers .. 735. Systems of Linear Equations: Matrix Inverses .. 738. Exercises .. 749. Answers .. 752. Determinants and Cramer's Rule .. 754. Definition and Properties of the Determinant .. 754. Cramer's Rule and Matrix Adjoints .. 758. Exercises .. 763. Answers .. 767. Partial Fraction Decomposition .. 768. Exercises .. 776. Answers .. 777. Systems of Non-Linear Equations and Inequalities.

10 778. Exercises .. 787. Answers .. 789. 9 Sequences and the Binomial Theorem 793. Sequences .. 793. Exercises .. 801. Answers .. 803. Summation Notation .. 804. Exercises .. 813. Answers .. 815. Mathematical Induction .. 816. Exercises .. 821. Selected Answers .. 822. The Binomial Theorem .. 824. Exercises .. 834. Answers .. 836. viii Table of Contents 10 Foundations of Trigonometry 837. Angles and their Measure .. 837. Applications of Radian Measure: Circular Motion .. 850. Exercises .. 853. Answers.