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PRECALCULUS

KEN KUNIYUKI SAN DIEGO MESA COLLEGE

TABLE OF CONTENTS

Chapter 0: Preliminary Topics Chapter 1: Functions Chapter 2: Polynomial and Rational Functions Chapter 3: Exponential and Logarithmic Functions Chapter 4: Trigonometric Functions Chapter 5: Analytic Trigonometry Chapter 6: Topics in Trigonometry Chapter 7: Systems Chapter 8: Matrices and Determinants Chapter 9: Discrete Mathematics Chapter 10: Conic Sections, Polar Coordinates, and Plane Curves

COLOR CODING

Warnings are in red. Tips are in purple. Calculus comments are (sometimes) in green (clover).

TECHNOLOGY USED This work was produced on Macs with Microsoft Word, MathType, Adobe Illustrator, Adobe Acrobat, and Mathematica and Calculus WIZ.

CONTACT INFORMATION

Ken Kuniyuki: Email address: [email protected] or [email protected] Website: http://www.kkuniyuk.com

• You may download these and other course notes, exercises, and exams. Feel free to send emails with suggestions, improvements, tricks, etc.

LICENSING

This work may be freely copied and distributed without permission under the specifications of the Creative Commons License at: http://www.kkuniyuk.com/Math141

PARTIAL BIBLIOGRAPHY / SOURCES Algebra: Blitzer, Lial, Tussy and Gustafson Trigonometry: Lial, Smith Precalculus: Axler, Cohen, Larson, Stewart, Sullivan; REA Problem Solvers Calculus: Hamming, Larson, Stewart, Swokowski, Tan, [Peter D.] Taylor Complex Variables: Churchill and Brown, Schaum’s Outlines Discrete Mathematics: Rosen Online: Britannica Online Encyclopedia: http://www.britannica.com

Wikipedia: http://www.wikipedia.org Wolfram MathWorld: http://mathworld.wolfram.com/

Other: Harper Collins Dictionary of Mathematics People: Larry Foster, Laleh Howard, Terrie Teegarden, Tom Teegarden (especially for the Frame Method for graphing trigonometric functions), and many more.

(Assumptions and Notation) A.1

ASSUMPTIONS and NOTATION Unless otherwise specified, we assume that:

• f , g, and s denote functions. (See Section 1.1.)

•• g sometimes denotes Earth’s gravitational constant.

•• h may denote a function, or it may denote the “run” in some difference quotients in Sections 1.10 and 1.11.

•• s often denotes a position or height function.

• a, b, c, k, m, and n denote real numbers. (See Section 0.1.)

•• “Let a ∈ ” means “let a be an arbitrary real number.” By “arbitrary,” we mean that any one will do.

•• c sometimes denotes the speed of light in a vacuum.

•• d sometimes denotes a distance function.

•• e denotes a mathematical constant defined in Chapter 3. e ≈ 2.718 .

•• i sometimes denotes the imaginary unit defined in Chapter 2 i = −1( ) .

In Chapter 9, we will use i as a generic subscript (as in ai ) and as an index

of summation

as in aii=1

n

∑⎛

⎝⎜⎞

⎠⎟; we also use j and k for these purposes.

•• n might be restricted to be an integer n ∈( ) . (See Section 0.1.)

• x, y, t, and θ typically denote variables that take on real numbers as values.

•• z can denote a variable or a complex number (see Chapter 2).

• The domain of a function f , which we will denote by Dom f( ) , is the implied (or mathematical) domain of f . (See Section 1.1.)

•• This might not be the case in applied “word problems.”

•• We assume that the domain and the range of a function only consist of real numbers, as opposed to imaginary numbers. That is, Dom f( )⊆ , and

Range f( )⊆ . (⊆ means “is a subset of”; see Section 0.1.)

• Graphs extend beyond the scope of a figure as expected, unless endpoints are clearly shown. Arrowheads help to make this clearer. (See Section 1.2.)


MORE NOTATION

Sets of Numbers (Section 0.1)

Notation Meaning Comments

+ , Z+ the set of positive integers

This is the set (collection) 1, 2, 3, ...{ } .

“Zahlen” is a related German word.

is in blackboard bold typeface; it is more commonly used than Z.

, Z the set of integers This set consists of the positive integers, the negative integers (−1, −2, − 3,…), and 0.

, Q the set of rational numbers

This set includes the integers and numbers

such as 13

, −94

, 7.13, and 14.3587 .

comes from “Quotient.”

, R the set of real numbers

This set includes the rational numbers and irrational numbers such as 2 , π , e, and 0.1010010001…. Think: “all decimal numbers.”

, C the set of complex numbers This set includes the real numbers and imaginary numbers such as i and 2 + 3i .

The Venn diagram below indicates the (proper) subset relations: ⊂ ⊂ ⊂ . For example, every integer is a rational number, so ⊂ . (⊆ permits equality.) Each disk is contained within each larger disk.


Set Notation (Section 0.1)


∈ in, is in This denotes set membership.

Example: 7 ∈ . ∉ not in, is not in Example: 1.7 ∉ . ∍ such that

| or : such that (in set-builder form)

Example: x ∈ x > 3{ } , or x ∈ : x > 3{ } , is the set of all real numbers greater than 3.

∀ for all, for any This is called the universal quantifier. ∃ there is, there exists This is called the existential quantifier.

∃! there exists a unique, there is one and only one

This is called the unique quantifier.

Example: ∃!x ∈ ∍ x = 3, which states that there exists a unique real number equal to 3.

∀x ∈ for every real number (denoted by x)

More precisely: for any arbitrary member of the set of real numbers; this member will be denoted by x.

Example: ∀x ∈, x < x +1 ; that is, every real number is less than one added to itself.

∀x, y ∈ for every pair of real

numbers (denoted by x and y)

More precise notation: ∀ x, y( )∈2 .

∅ or {} empty set (or null set)

This is the set consisting of no members.

Example: The solution set of the equation x = x +1 is ∅ .

The symbol ∅ is not to be confused with the Greek letter phi (φ ).

∪ set union

Example: If f x( ) = csc x , then Dom f( ) = −∞, −1( ]∪ 1, ∞[ ) .

∪ is used to indicate that one or more real number(s) is/are being skipped over.

Think: “all members are invited.”

∩ set intersection Example: 4, 6[ ]∩ 5, 7[ ] = 5, 6[ ] .

Think: “overlap.”

\ or − set difference, set complement

Example: If f x( ) = 1x

, then Dom f( ) is

\ 0{ } , or − 0{ } . We exclude 0 from .


Logical Operators (Sections 0.1, 0.2)


∨ or, disjunction Example: If f x( ) = csc x , then

Dom f( ) = x ∈ x ≤ −1 ∨ x ≥ 1{ } .

∧ and, conjunction Example: If f x( ) = x − 3

x − 4, then

Dom f( ) = x ∈ x ≥ 3 ∧ x ≠ 4{ } .

or ¬ not, negation Example: The statement x = 3( ) is equivalent to the statement x ≠ 3 .

⇒ implies Example: x = 2 ⇒ x2 = 4 . ⇔ if and only if (iff) Example: x +1= 3 ⇔ x = 2 .

Greek Letters

The lowercase Greek letters below (especially θ ) often denote angle measures.

Notation Name Comments α alpha This is the first letter of the Greek alphabet.

β beta This is the second letter of the Greek alphabet.

γ gamma This is the third letter of the Greek alphabet.

θ theta This is frequently used to denote angle measures.

φ or ϕ phi

This is not to be confused with ∅ , which denotes the empty set (or null set).

φ also denotes the golden ratio, 1+ 52

,

which is about 1.618. Tau τ( ) is also used.

The lowercase Greek letters below often denote (perhaps infinitesimally) small positive quantities in calculus, particularly when defining limits.

Notation Name Comments

δ delta This is the fourth letter of the Greek alphabet.

ε epsilon This is the fifth letter of the Greek alphabet.

This is not be confused with ∈, which denotes set membership.

(Assumptions and Notation) A.5 Some other Greek letters of interest:

Notation Name Comments

Δ (uppercase) delta

This denotes “change in” or increment.

Example: slope is often written as ΔyΔx

.

It also denotes the discriminant, b2 − 4ac , from the Quadratic Formula.

κ (lowercase) kappa This denotes the curvature of a curve.

λ (lowercase) lambda

This denotes an eigenvalue (in linear algebra), a Lagrange multiplier (in multivariable optimization), and a wavelength (in physics).

π (lowercase) pi

This is a famous mathematical constant.

It is the ratio of a circle’s circumference to its diameter.

π ≈ 3.14159 . It is irrational. Π (uppercase) pi This is the product operator.

ρ (lowercase) rho This denotes mass density and also the distance between a point in 3-space and the origin (ρ is a spherical coordinate).

Σ (uppercase) sigma This is the summation operator. See Chapter 9.

τ (lowercase) tau This denotes the golden ratio, though phi φ( ) is more commonly used.

ω (lowercase) omega This is the last letter of the Greek alphabet.

It denotes angular velocity.

Ω (uppercase) omega This denotes ohm, a unit of electrical resistance.

More lowercase Greek letters:

zeta ζ( ) , eta η( ) , iota ι( ) , mu µ( ) , nu ν( ) , xi ξ( ) , omicron ο( ) , sigma σ( ) , upsilon υ( ) , chi χ( ) , psi ψ( )


Geometry

Notation Meaning Comments ∠ angle See Section 4.1. is parallel to See Section 0.14 and Chapter 6.

⊥ is perpendicular to,

is orthogonal to, is normal to

See Section 0.14 and Chapter 6.

Vector Operators


• dot product, Euclidean inner product See Chapter 6.

× cross product, vector product See Chapter 8.

Other Notations


∴ therefore This is placed before a concluding statement.

Q.E.D., or

end of proof

Q.E.D. stands for “quod erat demonstrandum,” which is Latin for “which was to be demonstrated / proven / shown.”

≈ , ≅ is approximately, is about

⎢⎣ ⎥⎦ or

floor, greatest integer

Think: “round down.”

Examples: 2.9⎢⎣ ⎥⎦ = 2 , −2.9⎢⎣ ⎥⎦ = − 3 ∞ infinity See Section 0.1.

min minimum The least of … max maximum The greatest of …

Dom f( ) domain of a function f The set of legal (real) input values for f

deg f x( )( ) degree of a polynomial f x( ) See Section 0.6.

composition of functions Example: f g( ) x( ) = f g x( )( ) . See Section 1.6.

CHAPTER 0:

Preliminary Topics

0.1: Sets of Numbers 0.2: Logic 0.3: Rounding 0.4: Absolute Value and Distance 0.5: Exponents and Radicals: Laws and Forms 0.6: Polynomial, Rational, and Algebraic Expressions 0.7: Factoring Polynomials 0.8: Factoring Rational and Algebraic Expressions 0.9: Simplifying Algebraic Expressions 0.10: More Algebraic Manipulations 0.11: Solving Equations 0.12: Solving Inequalities 0.13: The Cartesian Plane and Circles 0.14: Lines 0.15: Plane and Solid Geometry 0.16: Variation

• This chapter will review notation, concepts, skills, techniques, and formulas needed in precalculus and calculus. Sections 0.2, 0.8, 0.9, 0.10, and 0.16 may be largely unfamiliar to incoming precalculus students.

(Section 0.1: Sets of Numbers) 0.1.1

SECTION 0.1: SETS OF NUMBERS

LEARNING OBJECTIVES

• Be able to identify different sets of numbers. • Know how to write sets of real numbers using set-difference, set-builder, graphical, and interval forms. • Be able to take the union and intersection of intervals of real numbers.

PART A: DISCUSSION

• Theorems and formulas require constants (denoted by c, a1 , a2 , etc.) to be from a particular set of numbers, usually the set of real numbers (denoted by ).

• Sets of real numbers can correspond to solutions of equations (in Section 0.11), solutions of inequalities (in Section 0.12), and domains and ranges of functions (in Section 1.1). There are several ways to describe these sets.

PART B: SETS OF NUMBERS

A set is a collection of objects, called the members (or elements) of the set.

• Two sets A and B are equal (that is, A = B ) when they consist of the same members. Typically, order is irrelevant, and members are not repeated.

∅ denotes the empty set (or null set), the set consisting of no members. Let A and B be sets. A is a subset of B, denoted by A⊆ B , when every member of A is also a member of B.

If A⊆ B , but A ≠ B , then A is a proper subset of B, denoted by A⊂ B . This means that B contains all of the members of A, as well as at least one other member not in A. Some important sets of numbers are:

+ , or Z+ , the set of positive integers.

• This set consists of 1, 2, 3, etc.

• comes from the German word “Zahlen.”

• is in blackboard bold typeface. It is more commonly used than Z.

• WARNING 1: Sources differ as to whether counting numbers, whole numbers, and natural numbers also include 0.


, or Z, the set of integers.

• This set consists of 1, 2, 3, etc.; their opposites, −1, −2 , −3, etc. (making up − , the set of negative integers); and 0.

, or Q, the set of rational numbers.

• This is the set of all numbers that can be written in the form:

integernonzero integer

• It is the set of numbers that can be written as finite (or terminating) decimals or repeating decimals.

• Examples include:

Fraction form Decimal form

12

0.5

13

0.3, or 0.3333…

−

8239900

−0.831, or −0.8313131…

71

, or 7 7

• As demonstrated by the last example, every integer is a rational number. That is, ⊆ .

• comes from “Quotient.”

, or R, the set of real numbers.

• This is the set of all numbers that can be written as decimals.

• This set can be represented by the real number line below. There should be tick marks at 0 and at least one other number to indicate scale.

• This set consists of rational numbers and irrational numbers, such as π and 2 .


, or C, the set of complex numbers.

• This set consists of real numbers and imaginary numbers, such as i and 2 + 3i. (See Chapter 2.)

The Venn diagram below indicates the (proper) subset relations: ⊂ ⊂ ⊂ . Each disk is contained within each larger disk.

PART C: SET NOTATION

Set Notation

∈ denotes “in,” or “is in.” It denotes set membership.

∉ denotes “is not in.”

∍ denotes “such that.”

∀ (the universal quantifier) denotes “for all” or “for every.”

∃ (the existential quantifier) denotes “there exists” or “there is.”

Example Set 1 (Using Set Notation)

Plain English translation 7 ∈ 7 is an integer.

12∉

12

is not an integer.

∀x ∈ , … For all real numbers x, … ∀a, b ∈ , … For all integers a and b, …

∀x ∈, x < x +1 Every real number is less than one added to itself.

∀x ∈, ∃y ∈ ∍ x = 2y Every real number is twice some real number. §

(Section 0.1: Sets of Numbers) 0.1.4 PART D: INFINITY

Infinity, denoted by the lemniscate ∞ , is a quantity that is greater than any real number.

Negative infinity, denoted by −∞ , is a quantity that is lesser than any real number.

WARNING 2: ∞ and −∞ are not real numbers, though they may be handled differently in higher math.

PART E: REPRESENTING SETS OF REAL NUMBERS

There are various ways to represent a set of real numbers.

We sometimes list the members of a set and surround them with braces. For example, the set 1,π{ } consists of the two members 1 and π .

An interval corresponds to a connected (“unbroken”) piece of the real number line.

A bounded interval has finite length on the real number line.

An unbounded interval has infinite length.

An open interval excludes its endpoints.

A closed interval includes its endpoints.

Example 2 (A Bounded, Open Interval)

The set of all real numbers x such that 3< x < 5 …

… in set-builder form is: x ∈ 3< x < 5{ } , or

x ∈ : 3< x < 5{ }

… in graphical form is:

… in interval form is: 3, 5( )

The set is an interval with 3 and 5 as its endpoints.

The set is an open interval, because it excludes its endpoints.

The exclusion of 3 and 5 is indicated by the use of:

• strict inequality signs (<) in set-builder form, • hollow circles (or parentheses) in graphical form, and • parentheses in interval form. §


Example 3 (A Bounded, Closed Interval)

The set of all real numbers x such that 3≤ x ≤ 5 …

… in set-builder form is: x ∈ 3≤ x ≤ 5{ } , or

x ∈ : 3≤ x ≤ 5{ }


… in interval form is: 3, 5⎡⎣ ⎤⎦

The set is a closed interval, because it includes its endpoints.

The inclusion of 3 and 5 is indicated by the use of:

• weak inequality signs (≤ ) in set-builder form, • filled-in circles (or brackets) in graphical form, and • brackets in interval form. §

Example Set 4 (Various Types of Intervals)

Set-builder Form

Graphical Form

Interval Form

Type of Interval

x ∈ 3≤ x < 5{ } 3, 5⎡⎣ ) Bounded;

Half-Open

x ∈ 3< x ≤ 5{ } 3, 5( ⎤⎦ Bounded;

Half-Open

x ∈ x > 3{ } 3, ∞( ) Unbounded

x ∈ x ≥ 3{ } 3, ∞⎡⎣ ) Unbounded

x ∈ x < 3{ } −∞, 3( ) Unbounded

x ∈ x ≤ 3{ } −∞, 3( ⎤⎦ Unbounded

x x ∈{ } ,

which is −∞, ∞( ) Unbounded

§

WARNING 3: In interval form, parentheses are always placed next to ∞ and −∞ , because they are not real numbers and are therefore excluded from the set. However, may be considered to be both an open interval and a closed interval.

(Section 0.1: Sets of Numbers) 0.1.6.

A∪ B , the union of set A and set B, consists of all elements in one or both sets. Think: “All members are invited.”

• A∪ B = x x ∈A or x ∈B{ } , where “or” is taken to be “inclusive or,” or

“and/or.” The disjunctive symbol ∨ denotes “or.”

A∩ B , the intersection (“overlap”) of set A and set B, consists of all elements in both sets.

• A∩ B = x x ∈A and x ∈B{ }. The conjunctive symbol ∧ denotes “and.”

Example 5 (A Union of Intervals)

The set \ 1,π{ } , which is written in set-difference form, consists of all real numbers except 1 and π . The set …

… in set-builder form is: x ∈ x ≠ 1 and x ≠ π{ } , or

x ∈ : x ≠ 1 and x ≠ π{ }


… in interval form is: −∞, 1( )∪ 1,π( )∪ π , ∞( )

TIP 1: The graphical form can help us write the interval form.

TIP 2: The union symbol ∪ separates intervals in interval form whenever a real number must be “skipped over.” §

Example 6 (Union and Intersection of Intervals)

a) Simplify 1, 5( )∪ 3, ∞⎡⎣ ) . (Think: “All members are invited.”)

b) Simplify 1, 5( )∩ 3, ∞⎡⎣ ) . (Think: “Overlap.”)

§ Solution

Interval Form

Graphical Form

1, 5( )

3, ∞⎡⎣ )

a) 1, 5( )∪ 3, ∞⎡⎣ ) = 1, ∞( )

b)

1, 5( )∩ 3, ∞⎡⎣ ) = 3, 5⎡⎣ )

§

(Section 0.2: Logic) 0.2.1

SECTION 0.2: LOGIC

LEARNING OBJECTIVES

• Be able to identify and use logical notation and terminology. • Understand the structure of “if-then” and “if and only if” statements. • Understand counterexamples and logical equivalence. • Know how to find the converse, inverse, and contrapositive of an “if-then” statement. • Understand necessary conditions and sufficient conditions. PART A: DISCUSSION

• Although logic is a subject that is often relegated to courses in discrete mathematics, computer science, and electrical engineering, its fundamentals are essential for clear and precise mathematical thought.

• Many theorems are “if-then” or “if and only if” statements. PART B: PROPOSITIONS AND “IF-THEN” STATEMENTS

A proposition is a statement that is either true or false. “If-Then” Statements

The statement “If p, then q” can be written as “ p → q .”

• The proposition p is called the hypothesis, assumption, or condition.

• The proposition q is called the conclusion.

• If there are no cases where p is true and q is false, we say that the statement is true.

• Otherwise, the statement is false, and any case where p is true and q is false is called a counterexample.

If the statement is known to be true, we can write “ p ⇒ q .” “

�

⇒” may be read as “implies.”

• WARNING 1: “→ ” denotes “approaches” when we discuss limits in calculus.


Example 1 (An “If-Then” Statement)

Consider the statement: “If I get an A, then I pass.”

This statement is of the form “If p, then q,” where:

p is the hypothesis “I get an A,” and q is the conclusion “I pass.”

The statement is true, because there is no case where a student gets an A but does not pass. §

The converse of p → q is q → p .

Example 2 (An “If-Then” Statement and Its Converse)

Consider the statement:

“If a number is an integer, then it is a rational number.”

• That is, x ∈( )→ x ∈( ) .

The statement is true, because every integer is also a rational number. We can write: x ∈( )⇒ x ∈( ) .

Now, consider the converse of the above statement:

“If a number is a rational number, then it is an integer.”

• That is, x ∈( )→ x ∈( ) .

This second statement is false, because a counterexample exists.

Observe that 12

is a rational number, yet it is not an integer.

WARNING 2: The discovery of even one counterexample can be used to disprove a statement (that is, to prove that a statement is false). However, a single example is usually not enough to prove that a statement is true. To prove that a statement is true, one often needs to present a rigorous and general argument that applies to all cases where the hypotheses hold.

Because the converse is false, we can write: x ∈( )⇒ x ∈( ) ,

where “ ⇒ ” denotes “does not imply.” §


PART C: “IF AND ONLY IF” (or “IFF”) STATEMENTS

“If and Only If ” (or “Iff”) Statements

The statement “p if and only if q,” or “p iff q,” can be written as “ p ↔ q .”

If the statement is known to be true, we can write “ p ⇔ q ,” or “ p ≡ q .”

• If p ⇒ q , and if q ⇒ p , then p ⇔ q .

• Then, the propositions p and q are logically equivalent; either both are true, or both are false.

• For example, 2x = 6( )⇔ x = 3( ) . (See Footnote 1.)

• Definitions are essentially “iff” statements. For example, a number is a rational

number iff it can be written in the form

integernonzero integer

.

• True “iff” statements arise when an “if-then” statement and its converse are true.

PART D: CONVERSE, INVERSE, AND CONTRAPOSITIVE

The converse of p → q is: q → p The inverse of p → q is: ~ p → ~ q

• “~” and “¬ ” are used to denote “not.” They are negation operators.

The contrapositive of p → q is: ~ q → ~ p

• TIP 1: Take the original statement, switch the propositions, and negate them.

Contrapositive Theorem

If an “if-then” statement is true, then its contrapositive must be true, and vice-versa. In other words, they are logically equivalent.

That is, p → q( )⇔ ~ q → ~ p( ) .

• This can be proven using truth tables in a discrete mathematics class.

As a result, any “if-then” statement you know to be true has a contrapositive associated with it that will automatically be true.

WARNING 3: An “if-then” statement may or may not be logically equivalent to its converse or its inverse. (However, the converse and the inverse must be logically equivalent to each other. Why? How are they related?)


Example 3 (Converse, Inverse, and Contrapositive of an “If-Then” Statement; Revisiting Example 1)

Consider the true “if-then” statement:

If I get an Ap

, then I passq

.

What are its converse, inverse, and contrapositive? Which of these are true?

§ Solution

Converse:

If I passq

, then I get an Ap

.

This statement is false, because we can find a counterexample. A student can pass the class with a grade of B or C.

Inverse:

If I do not get an A~ p

, then I do not pass~ q

.

This statement is false, because we can find a counterexample. A student can have a grade of B or C, and the student can pass.

Contrapositive:

If I do not pass~ q

, then I do not get an A~ p

.

By the Contrapositive Theorem, this must be true, since the original statement is.

The Venn diagram below may clarify matters. (We ignore “+” and “− ” modifiers, as well as pass/fail grading.)

§

(Section 0.2: Logic) 0.2.5.

Example 4 (An “Iff” Statement)

Consider the (true) statement:

If I get an A, B, or Cp

, then I passq

.

Its converse is true:

If I passq

, then I get an A, B, or Cp

.

Therefore, this is true:

I get an A, B, or Cp

iff I passq

. §

PART E: NECESSARY CONDITIONS and SUFFICIENT CONDITIONS

p is a necessary condition for q ⇔ q ⇒ p( ) • That is, for q to be true, it is required that p must be true.

p is a sufficient condition for q ⇔ p ⇒ q( ) • That is, if p is true, then it is guaranteed that q must be true.

Example 5 (Necessary Conditions and Sufficient Conditions; Revisiting Example 1)

Consider the “if-then” statement:

If I get an Ap

, then I passq

.

Is p sufficient for q? Is p necessary for q?

§ Solution

p is sufficient for q, because p ⇒ q . (Equivalently, q is necessary for p.) Think: If I get an A, then I am guaranteed to pass.

p is not necessary for q, because q⇒ p . Think: I don’t need to get an A to pass. §

Example 6 (“Necessary and Sufficient” Conditions; Revisiting Example 4)

Consider the “iff ” statement:

I get an A, B, or Cp

iff I passq

.

This is true, so p ⇔ q , and we say that p is necessary and sufficient for q. Therefore, p and q are logically equivalent. §

FOOTNOTES

1. Propositions. Unlike some sources such as Rosen’s Discrete Mathematics and Its Applications, we call “ 2x = 6 ” a proposition, even though its truth value (“true” or “false”) depends on the value of x.

(Section 0.3: Rounding) 0.3.1

SECTION 0.3: ROUNDING

LEARNING OBJECTIVES

• Be able to round off decimals to a specified decimal place, a specified number of decimal places, or a specified number of significant digits (or figures). • Understand scientific notation.

PART A: DISCUSSION (WARNINGS and TIPS)

• Using Calculators

Memory buttons on calculators can help preserve decimal accuracy. Taking too few decimal places may ruin the accuracy of final answers.

A calculator might approximate π as 3.14159265. Though seemingly impressive, this is not an exact representation of π . Calculators cannot give exact decimal representations of such numbers as 2 , ln5 , and sin37 .

• Exact vs. Approximate Answers

Math instructors typically expect exact answers, though decimal approximations may be required for “word problems.”

For example, if an answer is π , you should write π . You may also be asked to give a decimal approximation such as 3.14.

• “Word Problems”

Depending on context, decimal answers might need to be rounded down, rounded up, or rounded off.

Write any appropriate units such as feet, pounds, etc. as part of your answer.

PART B: ROUNDING

“≈” means “is approximately” or “is about.”

Example 1 (Rounding Off to a Specified Decimal Place)

π ≈ 3.14159 . π rounded off to the nearest …

… integer is: 3

… tenth (that is, to one decimal place) is: 3.1

… hundredth (that is, to two decimal places) is: 3.14

… thousandth (that is, to three decimal places) is: 3.142

• In the U.S., the “1” in the third decimal place of π is rounded up to “2.” This is because the digit in the next decimal place is 5 or higher. (Different countries have different rules if that digit is a “5.”) §

(Section 0.3: Rounding) 0.3.2.

Consider a number written in decimal form.

To count decimal places, count all digits to the right of the decimal point.

To count significant digits (or figures), locate the leftmost nonzero digit, and count it and all digits to the right of it.

• We may want numbers written out to the same number of decimal places when adding and subtracting them, while we may focus on significant digits when multiplying and dividing them. We can also use significant digits to roughly retain the same level of accuracy when handling different units such as inches, feet, and miles.

Example 2 (Counting Decimal Places and Significant Digits)

Consider: 70.1230

• This is written out to four decimal places. We count the four digits after the decimal point.

• This is written out to six significant digits. We include the two digits to the left of the decimal point.

• The “0” at the end indicates that we claim accuracy to four decimal places. Writing “70.123” might not have had that effect.

• We call 70 the integer part of this decimal. §

Example 3 (Counting Decimal Places and Significant Digits; Scientific Notation)

Consider: 0.001020, or .001020 (the integer part is 0; writing it is optional).

• These are written out to six decimal places. We count all six digits after the decimal point, including the two zeros after the decimal point and before the “1.”

• These are written out to four significant digits. We do not include the aforementioned zeros, but we include the “1” and all digits after it.

• These can be written as 1.020×10−3 using scientific notation. The “ −3” exponent tells us to move the decimal point three places to the left to obtain .001020 in standard form. Note that: 1.020×103 = 1020.= 1020 , since the “3” exponent tells us to move the decimal point three places to the right. §

(Section 0.4: Absolute Value and Distance) 0.4.1

SECTION 0.4: ABSOLUTE VALUE AND DISTANCE

LEARNING OBJECTIVES

• Be able to formally define absolute value. • Know how to take the absolute value of a real number. • Know properties of the absolute value operation. • Relate absolute value and distances on the real number line.

PART A: DISCUSSION

• Understanding the definition of absolute value is crucial to understanding the absolute value function and its graph. (See Section 1.3.)

PART B: ABSOLUTE VALUE

Definition of Absolute Value

The absolute value of a is denoted by a , where

a =a, if a ≥ 0−a, if a < 0

⎧⎨⎩

• That is, the absolute value of a nonnegative real number is itself, while the absolute value of a negative real number is its opposite.

• This is an example of a piecewise definition because of the different rules for different values of a. (See Section 1.5.)

TIP 1: The absolute value of a real number is never negative. WARNING 1: −a does not necessarily represent a negative value. In fact, if a < 0 , then −a > 0 . That is, the opposite of a negative real number is positive.

Example 1 (Finding Absolute Value)

a) Find 7 . b) Find 0 . c) Find − 3 .

§ Solution

a) 7 ≥ 0 , so we use the top rule in the definition. 7 = 7 .

b) 0 ≥ 0 , so we again use the top rule in the definition. 0 = 0 .

c) − 3 < 0 , so we use the bottom rule in the definition: −3 = − −3( ) = 3 . §

(Section 0.4: Absolute Value and Distance) 0.4.2

PART C: PROPERTIES OF ABSOLUTE VALUE

Properties of Absolute Value

1) a ≥ 0 Absolute values are never negative. (TIP 1) 2) −a = a Opposites have the same absolute value.

3) ab = a b The absolute value of a product equals the product of the absolute values.

4) ab

=ab

, if b ≠ 0 The absolute value of a quotient equals the quotient of the absolute values, provided the denominators are nonzero.

PART D: ABSOLUTE VALUE, DISTANCE, and OPPOSITES

Interpreting Absolute Value as a Distance

a = the distance between a and 0 on the real number line.

TIP 2: Just as for absolute value, a distance cannot be negative. However, we do use “signed distances” and “signed lengths” in trigonometry and motion problems.

Example 2 (Interpreting Absolute Value as a Distance)

−3 = 3 and 3 = 3 , because both −3 and 3 are at a distance of 3 units away from 0 on the real number line.

§

The Distance Between a and b

b − a = a − b = the distance between a and b on the real number line.

• In particular, a − 0 = a = the distance between a and 0 on the real number line.

• Absolute value notation allows us to denote such distances without worrying about whether a is greater than b or vice versa.

• In calculus, this notation helps us rigorously define limits and analyze sequences and series.

The following (important!) statement explains why b − a = a − b .

(Section 0.4: Absolute Value and Distance) 0.4.3.

“Opposite Differences” Rule

b− a( ) = − a − b( ) . That is, b− a( ) and a − b( ) are opposites.

• Prove this by rewriting − a − b( ) .

• For example, 7 − 3( ) = 4 , while 3− 7( ) = −4 , its opposite.

• Because they are opposites, b− a( ) and a − b( ) must have the same

absolute value by Property 2) in Part C. That is, b − a = a − b .

Example 3 (The Distance Between a and b)

x − 3 and 3− x both represent the distance between x and 3 on the real number line, regardless of whether x is greater than, less than, or equal to 3.

• For example, if x = 7 , its distance from 3 is given by: 7 − 3 = 4 = 4 ,

or by: 3− 7 = −4 = 4 .

• For example, if x = 1, its distance from 3 is given by: 1− 3 = −2 = 2 ,

or by: 3−1 = 2 = 2 . §

(Section 0.5: Exponents and Radicals: Laws and Forms) 0.5.1

SECTION 0.5: EXPONENTS AND RADICALS: LAWS AND FORMS

LEARNING OBJECTIVES

• Understand radicals and how they are related to powers. • Know laws of exponents and radicals and how to apply them, particularly when simplifying expressions. • Be able to rewrite expressions in different forms. • Recognize how expressions are restricted. PART A: DISCUSSION

• Rewriting expressions can help us solve problems.

PART B: RADICALS

is a radical symbol. In xn , n is the index, and x is the radicand.

• We assume n is an integer such that n ≥ 2 .

x , also written as x1/2 , is the principal square root of x.

• Here, the index n = 2 . • If x ≥ 0 , x is the unique nonnegative real number whose square is x. • If x < 0 , x cannot be defined as a real value.

(We will define the imaginary unit i as −1 in Chapter 2.)

• For example, 9 = 3 and 91/2 = 3, because 3( )2= 9 , and 3 is nonnegative.

Although 9 has two square roots, 3 and −3, we take the nonnegative square root (3) as our principal square root.

• TIP 1: Remember that 0 = 0 .

xn , also written as x1/ n , is the principal nth root of x.

• If n is even and x ≥ 0 , xn is the unique nonnegative real number whose

nth power is x. (If x < 0 , then xn cannot be defined as a real value.)

• If n is odd, xn is the unique real number whose nth power is x.

• For example, 164 = 2 and 161/4 = 2 , because 2( )4= 16 , and 2 is nonnegative.

• TIP 2: A principal odd root of a negative real number is negative. For example, −83 = −2 and −8( )1/3

= −2 , because −2( )3= −8. In Chapter 6, we

will see that −8 has three complex cube roots, but only one of them −2( ) is real.


xm/ n can be rewritten as xmn or as

xn( )m

.

• We assume that m and n are positive integers (not both even; see Footnote 1). • Think: take the mth power and the nth root, in either order.

• For example, x3/5 can be written as x35 or as

x5( )3.

PART C: LAWS OF EXPONENTS AND RADICALS

The term power usually refers to an exponent. However, we often refer to xn as a power of x.

Laws of Exponents

If the expressions involved are real-valued, then the following laws apply.

# Law In Plain English / Comments

1 xmxn = xm+n To multiply powers of x, add the exponents. For example, x

2 x3 = xx( ) xxx( ) = x5 .

2

xm

xn= xm−n To divide powers of x, subtract the exponents.

3

xm( )n= xmn To raise a power to a power, multiply the exponents.

4

xy( )n= xn yn The power of a product is the product of the powers.

5

xy

⎛⎝⎜

⎞⎠⎟

n

=xn

yn The power of a quotient is the quotient of the powers.

6 x0 = 1 This is consistent with Law 2, with m = n . (We define 00 to be 1 here. See Section 1.3, Part H.)

7

1xn

= x− n This is consistent with Laws 2 and 6.

8

xy

⎛⎝⎜

⎞⎠⎟

− n

=yx

⎛⎝⎜

⎞⎠⎟

n

xy

⎛⎝⎜

⎞⎠⎟

−1

=1xy

⎛⎝⎜

⎞⎠⎟

=yx

y ≠ 0( ) . Use Law 3 with m = −1.

WARNING 1: These laws may or may not apply when the expressions

involved are not real-valued. −2( )1/2, for example, cannot be defined as a

real value. Also, we require nonzero denominators in Laws 2, 5, 7, and 8.


Laws of Radicals

Assume that m and n are positive integers.

If the expressions involved are real-valued, then the following laws apply.

• Law 13 fundamentally distinguishes between even and odd roots.

• For Laws 9 through 12, the square root laws extend to even roots. They also extend to odd roots, but we allow cases where x < 0 or y < 0 .

For example,

xy3 = x3( ) y3( ) for all real values of x and y.

# Law In Plain English / Comments

9 xy = x y , if x ≥ 0 and y ≥ 0

The root of a product equals the product of

the roots. See Law 4 with n =

12

, 13

, etc.

10

xy=

x

y,

if x ≥ 0 and y > 0

The root of a quotient equals the quotient of

the roots. See Law 5 with n =

12

, 13

, etc.

11 xmn = xmn (if m or n is even, we require x ≥ 0 )

For example, x3 = x6 , if x ≥ 0 . This is related to Law 3, with m and n there being the reciprocals of m and n here.

12 x( )2

= x ,

if x ≥ 0

More generally,

xn( )n

= x , if n = 2, 3, 4, etc.

If n is even, we require x ≥ 0 .

13

x2 = x

x33 = x

xnn = x , if n is even; see Warning 4.

xnn = x, if n is odd.

• We simplify 18 , for example, using Law 9: 18 = 9 ⋅2 = 9 2 = 3 2 . (The greatest perfect square that divides 18 is 9. Then, the 9 “comes out” as a 3.)

• WARNING 2: Although Laws 9 and 10 cover the square root of a product or a quotient, there is no similar law for the square root of a sum or a difference. In particular, x + y is not equivalent to x + y . (See the Exercises.)

• WARNING 3: Do not apply Laws 9 through 12 to even roots if x < 0 or y < 0 .

As we will see in Chapter 2, −2 −3 ≠ 6 .

• WARNING 4: See Law 13. The statement x2 = x is incorrect if x < 0.

For example, if x = −3 , − 3( )2 = 9 = 3 , not −3.

(Section 0.5: Exponents and Radicals: Laws and Forms) 0.5.4 PART D: FRACTIONAL, RADICAL, AND EXPONENTIAL FORMS

Example Set 1 (Equivalent Expressions)

The following tables list pairs of equivalent expressions, together with relevant restrictions (we will discuss domains in Section 1.1). The restrictions guarantee that we never (do the equivalent of):

1) divide by zero, or

2) take the even root of a negative value.

WARNING 5: Do not confuse exponential form with exponential functions (see Chapter 3). “Power form” may be more appropriate.

Fractional Form Restrictions Exponential Form 1x

x ≠ 0 x−1

1x2

x ≠ 0 x− 2

17x− 4 /3

x ≠ 0 17x4/3 x ≠ 0( )

(See Warnings below.)

WARNING 6: In the last example, we need to state the “hidden”

restriction x ≠ 0( ) , because the expression 1

7x− 4 /3 implies it, yet the

expression 17x4 /3 does not. (See Warnings 9 and 10.)

WARNING 7: It is incorrect to re-express 1

7x− 4 /3 as 7x4 /3 , a common

error. Think of 1

7x− 4 /3 as

17⋅1

x− 4 /3. The negative exponent only applies to x.

Radical Form Restrictions Exponential Form x x ≥ 0 x1/2 x3 None x1/3

3 x57( ) or 3 x7( )5 None 3x5/7

TIP 3: The parentheses in 3 x57( ) prevent us from misreading the

expression as 37( ) x5( ) .

WARNING 8: The taking of odd roots (such as cube roots) does not impose any new restrictions. See Tip 2 in Part B.


Combined Form Restrictions Exponential Form 1x

x > 0 x−1/2

1x3

x ≠ 0 x−1/3

7x34

x > 0 7x− 3/4

PART E: SIMPLIFYING EXPRESSIONS

Example 2 (Simplifying an Expression)

Simplify 3

−2x2⎛⎝⎜

⎞⎠⎟

− 4

. Nonpositive exponents (that is, exponents that are

negative or zero) are not allowed in the final expression.

§ Solution

3−2x2

⎛⎝⎜

⎞⎠⎟

−4

= −2x2

3⎛⎝⎜

⎞⎠⎟

4

x ≠ 0( ) by Law 8

WARNING 9: The given expression should be examined for restrictions. Here, since division by 0 is forbidden, we require that x ≠ 0( ) .

WARNING 10: Whenever a step “hides” a restriction, we should write the restriction, although we sometimes wait until the final expression. (See Warning 6 and Example 3.) Restrictions can be hidden when we apply laws of exponents and radicals, when we take reciprocals (see the Comment for Law 8), and when we cancel (see Section 0.9).

=−2x2( )4

34 x ≠ 0( ) by Law 5

=−2( )4 x2( )4

81x ≠ 0( ) by Law 4

= 16x8

81x ≠ 0( ) by Law 3

§

(Section 0.5: Exponents and Radicals: Laws and Forms) 0.5.6.

Example 3 (Simplifying an Expression)

Simplify x xx7/2x0

. Nonpositive exponents are not allowed in the final

expression.

§ Solution

The given expression has the restriction x > 0( ) , because x and x7/2

require x ≥ 0( ) , and the powers of x x7/2 and x0( ) that are factors of the

denominator require x ≠ 0( ) .

x xx7/2x0 =

x1x1/2

x7/2 ⋅1by Law 6

=x

1+ 12

x7/2 by Law 1

=x3/2

x7/2

= x32− 7

2 x > 0( ) by Law 2= x− 2 x > 0( )

=1x2 x > 0( ) by Law 7

It might be unclear which step “hides” the restriction x > 0( ) , so we might

wait until the end to write x > 0( ) . § FOOTNOTES

1. Interpreting xm / n . xm / n , where m and n are positive, even integers, is interpreted differently by different sources.

• For example, if x2/6 is interpreted as x26 , there are no restrictions, and

x26 = x23 = x3 = x

1/3.

• On the other hand, if x2/6 is interpreted as

x6( )2, we need the restriction x ≥ 0 .

2. Irrational exponents. See Section 3.1 on how to interpret something like 2π .

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