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Unformatted text preview: University of the Philippines Diliman MATHEMATICS 20 Precalculus: Functions and Their Graphs Course Module Institute of Mathematics MATHEMATICS 20 Precalculus: Functions and Their Graphs Course Module Institute of Mathematics University of the Philippines Diliman ©2018 by the Institute of Mathematics, University of the Philippines Diliman. te of M at he m at ic s All rights reserved. No part of this document may be distributed in any way, shape, or form, without prior written permission from the Institute of Mathematics, University of the Philippines Diliman. U P Gari Lincoln C. Chua Odessa D. Consorte Russelle H. Guadalupe Marvin M. Olavides Thomas Herald M. Vergara In sti tu Mathematics 20 Module Writers and Editors: Reviewed by: Rovin B. Santos, Ph.D. ii Contents U P In sti tu te of M at he m at ic s 0 Review Topics in Algebra 0.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.1 Definitions and Notations . . . . . . . . . . . . . . . . . . . 0.1.2 Relations on Sets . . . . . . . . . . . . . . . . . . . . . . . . 0.1.3 Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . 0.1.4 The Set of Real Numbers . . . . . . . . . . . . . . . . . . . . 0.2 The Real Number System . . . . . . . . . . . . . . . . . . . . . . . . 0.2.1 Field Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2.2 Equality Axioms . . . . . . . . . . . . . . . . . . . . . . . . 0.2.3 Order Properties of R . . . . . . . . . . . . . . . . . . . . . . 0.3 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4.1 Special Products . . . . . . . . . . . . . . . . . . . . . . . . 0.4.2 Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . 0.5 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 0.5.1 Rational Expressions in Lowest Terms . . . . . . . . . . . . . 0.5.2 Operations Involving Rational Expressions . . . . . . . . . . 0.5.3 Complex Fractions . . . . . . . . . . . . . . . . . . . . . . . 0.6 Rational Exponents and Radicals . . . . . . . . . . . . . . . . . . . . 0.7 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.7.1 Definitions and Notations . . . . . . . . . . . . . . . . . . . 0.7.2 Operations Involving Complex Numbers in Rectangular Form 1 Equations and Inequalities 1.1 Review of Equations . . . . . . . . . . . . . . . . . . . . 1.1.1 Linear Equations . . . . . . . . . . . . . . . . . . 1.1.2 Quadratic Equations . . . . . . . . . . . . . . . . 1.1.3 Equations Involving Rational Expressions . . . . . 1.1.4 Equations Involving Radicals . . . . . . . . . . . . 1.1.5 Equations in Quadratic Form . . . . . . . . . . . . 1.1.6 Equations Involving Absolute Values . . . . . . . 1.2 Verbal Problems . . . . . . . . . . . . . . . . . . . . . . . 1.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Solving Linear Inequalities . . . . . . . . . . . . . 1.3.2 Inequalities Involving Higher Degree Polynomials iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 3 4 6 7 7 8 8 9 10 10 10 12 12 13 15 15 19 19 20 . . . . . . . . . . . 25 25 26 28 31 32 34 36 38 42 44 45 iv Contents 1.3.3 . . . . 55 55 58 60 63 3 Conic Sections 3.1 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 68 70 74 4 Systems of Equations and Inequalities 4.1 Review of Linear Systems . . . . . . . . . . . . . . . . . . . 4.1.1 Two Linear Equations in Two Variables . . . . . . . . 4.1.2 Systems of Three Linear Equations in Three Variables 4.1.3 Systems of Linear Inequalities . . . . . . . . . . . . . 4.2 Nonlinear Systems of Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . at ic s U P 6 . . . . Functions 5.1 Functions and Relations . . . . . . . . . . . . . . 5.1.1 Relations . . . . . . . . . . . . . . . . . 5.1.2 Functions . . . . . . . . . . . . . . . . . 5.1.3 Evaluating Functions . . . . . . . . . . . 5.1.4 Domain of a Function . . . . . . . . . . 5.1.5 Graph of a Function . . . . . . . . . . . 5.2 Some Basic Types of Functions and their Graphs 5.2.1 Polynomial Functions . . . . . . . . . . . 5.2.2 Rational Functions . . . . . . . . . . . . 5.2.3 Functions Involving Absolute Value . . . 5.2.4 Piecewise Functions . . . . . . . . . . . 5.3 Operations on Functions . . . . . . . . . . . . . In sti tu 5 The Two-Dimensional Coordinate System 2.1 Two-Dimensional Cartesian Coordinate System 2.2 Graphs of Equations . . . . . . . . . . . . . . 2.3 Lines . . . . . . . . . . . . . . . . . . . . . . 2.4 Circles . . . . . . . . . . . . . . . . . . . . . . te of M at he m 2 Inequalities Involving Absolute Values . . . . . . . . . . . . . . . . . . . . 47 . . . . . . . . . . . . Polynomial, Exponential, and Logarithmic Functions 6.1 Polynomial Functions and Their Zeros . . . . . . . 6.1.1 Synthetic Division . . . . . . . . . . . . . 6.1.2 Remainder Theorem . . . . . . . . . . . . 6.1.3 Factor Theorem . . . . . . . . . . . . . . . 6.1.4 Zeros of Polynomial Functions . . . . . . . 6.2 Inverse Functions . . . . . . . . . . . . . . . . . . 6.3 Exponential and Logarithmic Functions . . . . . . 6.3.1 Properties of Real Exponents . . . . . . . . 6.3.2 Exponential Functions . . . . . . . . . . . 6.3.3 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 81 81 85 87 88 . . . . . . . . . . . . 95 95 95 96 98 98 99 104 104 104 105 107 108 . . . . . . . . . . 115 115 116 118 119 120 124 128 128 129 131 Contents v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 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. . . . . . . . . . . . . . . . . . . . . . . 193 193 196 196 198 200 200 201 201 204 U P In sti tu 8 Trigonometric Identities 8.1 Basic Identities . . . . . . . . . . . . . . . . . 8.2 Proving Identities . . . . . . . . . . . . . . . . 8.3 Sum, Difference and Cofunction Identities . . . 8.4 Double-Angle, Half-Angle Identities . . . . . . 8.5 Product-to-Sum and Sum-to-Product Identities . . . . . . . . . . . . . . . at ic s . . . . . . . . . . . . . . te of M at he m 7 Trigonometric Functions 7.1 The Point Function . . . . . . . . . . . . . . . . . 7.2 Circular Functions . . . . . . . . . . . . . . . . . 7.2.1 Cosine and Sine Values of Special Numbers 7.2.2 The Other Circular Functions . . . . . . . 7.3 Graphs of Circular Functions . . . . . . . . . . . . 7.3.1 Graphs of the Sine and Cosine Functions . 7.3.2 Graphs of Sine Waves . . . . . . . . . . . 7.3.3 Graph of the Tangent Function . . . . . . . 7.3.4 Graph of the Cotangent Function . . . . . . 7.3.5 Graph of the Secant Function . . . . . . . 7.3.6 Graph of the Cosecant Function . . . . . . 7.4 Angles and Their Measure . . . . . . . . . . . . . 7.4.1 Measurement of Angles . . . . . . . . . . 7.5 Trigonometric Functions of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Solutions of Triangles 209 10.1 Solutions of Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 10.2 Solutions of Oblique Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 10.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 11 Polar Form of Complex Numbers 11.1 Polar Form of Complex Numbers . . . . . . . . . . . . . . . . . . . . 11.2 Products, Quotients, and Powers of Complex Numbers in Polar Form . . 11.2.1 Multiplication and Division of Complex Numbers in Polar Form 11.2.2 Integer Powers of Complex Numbers in Polar Form . . . . . . . 11.3 Roots of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 223 226 226 227 229 U P at ic s te of M at he m In sti tu vi Contents Unit 1 Module 1: Equations and Inequalities at ic s Module 0: Review Topics in Algebra Module 2: The Two-Dimensional Coordinate System U P In sti tu te of M at he m Module 3: Conic Sections Module 4: Systems of Equations and Inequalities 1 U P at ic s te of M at he m In sti tu 2 Contents Module 0 at ic s Review Topics in Algebra te of M at he m 0.1 Sets 0.1.1 Definitions and Notations Definition 0.1.1. A set is a well-defined collection of objects. Any object in the set is called an element or member of the set. A set A is finite if it is possible to list down all of its elements or has no elements. Otherwise, A is infinite. We use the following notations and terminology: • If a is an element of set A, then we denote this by a ∈ A. In sti tu • The empty or null set is a set without any elements and is denoted by ∅ or {}. • The universal set U is the set consisting of all elements under consideration. U P • The symbol n(A) denotes the cardinality of a finite set A; that is, the number of elements contained in A. There are two ways to describe the elements in a set: 1. The listing/roster method describes the set by listing all the elements in the set. 2. The rule method uses a descriptive phrase in describing the elements that are in the set. This method is usually used when there are too many elements to list down. 0.1.2 Relations on Sets Definition 0.1.2. Let A and B be sets. 1. A is a subset of B or A is contained in B, written as A ⊆ B, if and only if every element of A is an element of B; that is, if x ∈ A then x ∈ B. If A is not contained in B, we write this as A ̸⊆ B. 2. A is a proper subset of B or A is properly contained in B, written A ⊂ B, if and only if A is a subset of B and there exists an element of B which is not in A. 3 4 Module 0. Review Topics in Algebra 3. A and B are equal, written as A = B, if and only if they have precisely the same elements. This means x ∈ A if and only if x ∈ B. Equivalently, A = B if and only if every element of A is an element of B and every element of B is in A. If sets A and B are not equal, we write A ̸= B. 4. A is equivalent to B, written as A ∼ B, if and only if n(A) = n(B), that is, if sets A and B have the same number of elements. at ic s We can also define equivalence of two sets in terms of a one-to-one correspondence between their elements. Two sets A and B are in one-to-one correspondence if each element of A corresponds to a unique element of B and each element of B corresponds to a unique element of A. te of M at he m Definition 0.1.3. Sets A and B are equivalent if and only if sets A and B are in one-to-one correspondence. A Venn Diagram is used to visualize sets and their relations. The universal set U is usually represented by a rectangle and any set A ⊆ U is represented by some closed region inside the rectangle. An element of a set may be represented as a point inside the set. To illustrate, if we have x ∈ A ⊆ U and y ∈ U but y ∈ / A, then these may be represented by Venn diagram (a). On the other hand, the relation B ⊆ A ⊆ U may be represented by Venn diagram (b): U A U (b) A B y U P x In sti tu (a) 0.1.3 Operations on Sets Definition 0.1.4. Let A and B be sets in a universal set U . 1. The union of A and B, denoted A ∪ B, is the set of elements that belong to either A or B or both. In symbols, A ∪ B = {x|x ∈ A or x ∈ B}. 2. The intersection of A and B, denoted A ∩ B, is the set of elements that belong to both A and B. Equivalently, A ∩ B = {x|x ∈ A and x ∈ B}. A and B are said to be disjoint if A ∩ B = ∅. 0.1. Sets 5 3. The complement of A, denoted Ac or A′ , is the set of elements in U which are not in A. That is, Ac = A′ = {x ∈ U |x ∈ / A}. 4. The set difference A − B or A \ B (read as A minus B) is the set of elements in A which are not in B. In symbols, A − B = A \ B = {x ∈ U |x ∈ A and x ̸∈ B} Similarly, the set difference B − A is given by at ic s B − A = B \ A = {x ∈ U |x ∈ B and x ∈ / A}. U U A B B U P A In sti tu A∪B U te of M at he m The above operations are represented by the shaded regions in the following Venn diagrams: A∩B A Ac U A B A−B Definition 0.1.5. 1. An ordered pair (a, b) is a set with two elements in which we distinguish a first and second element. For equality of ordered pairs, (a, b) = (x, y) if and only if a = x and b = y. 2. The Cartesian product (or cross product) of sets A and B, denoted A × B, is the set of ordered pairs (a, b) where a ∈ A and b ∈ B. In symbols, A × B = {(a, b) | a ∈ A and b ∈ B}. Remark 0.1.6. Note that if a ̸= b, the ordered pair (a, b) is not equal to (b, a). In general, for any sets A and B, A × B ̸= B × A. Note that to write down an ordered pair, we use parentheses instead of braces. 6 Module 0. Review Topics in Algebra 0.1.4 The Set of Real Numbers Throughout the Math 20 module, the following sets of numbers and their corresponding notations are considered: Natural Numbers or Counting Numbers, N = {1, 2, 3, . . .} Whole Numbers, W = N ∪ {0} = {0, 1, 2, 3, . . .} Integers, Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .} te of M at he m Rational Numbers, Q = {x|x = p/q, p, q ∈ Z, q ̸= 0} at ic s Note that the set of integers consists of three disjoint sets: the set of natural numbers or the positive integers, {0}, and the set of negative integers. Zero is neither positive nor negative. The rational numbers are those numbers that can be expressed as ratios of integers. Among these are fractions, terminating decimals, and nonterminating repeating deci1 1 1 mals. e.g. = 0.5, = 0.25, = 0.3333 . . . 2 4 3 Irrational Numbers, Q′ In sti tu These numbers cannot be expressed as a ratio of two integers. These are the nonrepeating, nonterminating decimals. The numbers π = 3.1416 . . ., e = 2.718281 . . . , √ 2 are examples of irrational numbers. U P The above sets of numbers are subsets of the set of real numbers which is the union of the set of rational numbers and the set of irrational numbers. That is, R = Q ∪ Q′ . Note that the sets of rational and irrational numbers are disjoint. That is, Q ∩ Q′ = ∅. The set R can be represented by a Venn diagram as shown below: R Q Z W N Q′ We have the following relations: N ⊂ Z ⊂ Q ⊂ R. 0.2. The Real Number System 7 0.2 The Real Number System The real number system is the set of real numbers with two operations called addition and multiplication. If a, b ∈ R, then the sum of a and b is a + b (where a and b are called addends or terms) and the product of a and b is a · b or ab (where a and b are called factors). 0.2.1 Field Axioms 1. Closure Axiom for Addition and Multiplication For any real numbers a and b, a + b and a · b are unique real numbers, i.e. for all a, b ∈ R, a + b ∈ R and a · b ∈ R. te of M at he m 3. Commutative Axiom for Addition and Multiplication For all a, b ∈ R, a + b = b + a and a · b = b · a. at ic s 2. Associative Axiom for Addition and Multiplication For all a, b, c ∈ R, (a + b) + c = a + (b + c) and (a · b) · c = a · (b · c). 4. Distributive Axiom of Multiplication over Addition For all a, b, c ∈ R, c · (a + b) = c · a + c · b. 5. Identity Axioms • Existence of Additive Identity There exists a real number, zero (0), such that for any real number a, In sti tu a + 0 = 0 + a = a. U P 0 is called the identity element for addition. • Existence of Multiplicative Identity There exists a real number, 1, such that for any real number a, a · 1 = 1 · a = a. 1 is called the identity element for multiplication. 6. Inverse Axioms • Inverse Axiom for Addition For every real number a, there is a unique inverse element, −a, such that a + (−a) = 0. The number −a is called the additive inverse of a. • Inverse Axiom for Multiplication 1 If a ̸= 0, there is a unique inverse element, , such that a 1 1 a · = 1 = · a. a a 1 The number is called the multiplicative inverse or reciprocal of a. a 8 Module 0. Review Topics in Algebra 0.2.2 Equality Axioms 1. Reflexive Property of Equality For all a ∈ R, a = a. 2. Symmetric Property of Equality For all a, b ∈ R, if a = b then b = a. 3. Transitive Property of Equality For all a, b, c ∈ R, if a = b and b = c, then a = c. te of M at he m 5. Multiplicative Property of Equality For all a, b, c ∈ R, if a = b then a · c = b · c. at ic s 4. Additive Property of Equality For all a, b, c ∈ R, if a = b then a + c = b + c. Definition 0.2.1. If a, b ∈ R, then subtraction assigns to a and b a real number denoted a − b, called the difference of a and b, where a − b = a + (−b). Definition 0.2.2. If a, b ∈ R and b ̸= 0, then division assigns to a and b a real number, denoted a ÷ b = ab , called the quotient of a and b, where ab = a · 1b . In sti tu Remark 0.2.3. Note that division by zero is NOT defined. Operations on Quotients Let a, b, c, d ∈ R with b, d ̸= 0 ad ± bc a c ± = b d bd a c ac • Product: · = b d bd U P • Sum/Difference: • Quotient: a c a d ad ÷ = · = b d b c bc 0.2.3 Order Properties of R Definition 0.2.4. Let a, b ∈ R. 1. A real number a is a positive real number if a > 0. 2. A real number a is a negative real number if 0 > a. 3. We say a < b (a less than b) if b > a. Order Axioms 0.3. Integer Exponents 9 1. Trichotomy Axiom. For all a, b ∈ R, exactly one of the following holds: a > b, a = b, or b > a. 2. Transitive Axiom for Order. For all a, b, c ∈ R, if a > b and b > c, then a > c. 3. Addition Axiom for Order. For all a, b, c ∈ R, if a > b , then a + c > b + c. 4. Multip...
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