# Complete Solutions Manual

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Complete Solutions ManualPrecalculus Functions and GraphsTHIRTEENTH EDITIONEarl W. SwokowskiJeffery A. ColePrepared byJeffery A. ColeAustralia Brazil Mexico Singapore United Kingdom United States

PrefaceThis manual contains solutions/answers to all exercises in the text Precalculus:Functions and Graphs, Thirteenth Edition, by Earl W. Swokowski and Jeffery A. Cole. AStudent's Solutions Manual is also available; it contains solutions for the odd-numbered exercisesin each section and for the Discussion Exercises, as well as solutions for all the exercises in theReview Sections and for the Chapter Tests.For most problems, a reasonably detailed solution is included. It is my hope that bymerely browsing through the solutions, professors will save time in determining appropriateassignments for their particular class.I appreciate feedback concerning errors, solution correctness or style, and manual style—comments from professors using previous editions have greatly strengthened the ancillary packageas well as the text. Any comments may be sent directly to me at [email protected] would like to thank: Marv Riedesel and Mary Johnson for accuracy checking of thenew exercises; Andrew Bulman-Fleming, for manuscript preparation; Brian Morris and the lateGeorge Morris, of Scientific Illustrators, for creating the mathematically precise art package; andLaura Gallus, of Cengage Learning, for checking the manuscript. I dedicate this manual to Carly,Eli, and Mason, my grandchildren.Jeffery A. Cole 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Table of Contents1Topics from Algebra. 11.1Real Numbers . 11.2Exponents and Radicals .51.3Algebraic Expressions . 101.4Equations . 191.5Complex Numbers .301.6Inequalities . 34Chapter 1 Review Exercises . 42Chapter 1 Discussion Exercises . 48Chapter 1 Test . 512Functions and Graphs.552.1Rectangular Coordinate Systems . 552.2Graphs of Equations . 602.3Lines . 732.4Definition of Function . 822.5Graphs of Functions . 932.6Quadratic Functions .1092.7Operations on Functions . 118Chapter 2 Review Exercises . 130Chapter 2 Discussion Exercises . 143Chapter 2 Test . 1463Polynomial and Rational Functions.1493.1Polynomial Functions of Degree Greater Than 2 . 149?3.2Properties of Division . 1623.3Zeros of Polynomials . 1683.4Complex and Rational Zeros of Polynomials . 1753.5Rational Functions . 1833.6Variation . 200Chapter 3 Review Exercises . 204Chapter 3 Discussion Exercises . 210Chapter 3 Test . 2134Inverse, Exponential, and Logarithmic Functions. 2154.1Inverse Functions .2154.2Exponential Functions . 2224.3The Natural Exponential Function . 2364.4Logarithmic Functions . 2444.5Properties of Logarithms . 2534.6Exponential and Logarithmic Equations . 261Chapter 4 Review Exercises . 271Chapter 4 Discussion Exercises . 279Chapter 4 Test . 282 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

v5The Trigonometric Functions.2855.1Angles .2855.2Trigonometric Functions of Angles . 2895.3Trigonometric Functions of Real Numbers . 2975.4Values of the Trigonometric Functions . 3075.5Trigonometric Graphs .3125.6Additional Trigonometric Graphs . 3295.7Applied Problems . 342Chapter 5 Review Exercises . 348Chapter 5 Discussion Exercises . 358Chapter 5 Test . 3606Analytic Trigonometry.3636.1Verifying Trigonometric Identities . 3636.2Trigonometric Equations . 3696.3The Addition and Subtraction Formulas .3786.4Multiple-Angle Formulas . 3866.5Product-to-Sum and Sum-to-Product Formulas . 3946.6The Inverse Trigonometric Functions . 397Chapter 6 Review Exercises . 409Chapter 6 Discussion Exercises . 416Chapter 6 Test . 4187Applications of Trigonometry. 4217.1The Law of Sines . 4217.2The Law of Cosines . 4257.3Vectors .4317.4The Dot Product . 4397.5Trigonometric Form for Complex Numbers . 4437.6De Moivre's Theorem and 8th Roots of Complex Numbers . 448Chapter 7 Review Exercises . 451Chapter 7 Discussion Exercises . 457Chapter 7 Test . 4598Systems of Equations and Inequalities. 4618.1Systems of Equations .4618.2Systems of Linear Equations in Two Variables .4718.3Systems of Inequalities . 4778.4Linear Programming . 4878.5Systems of Linear Equations in More Than Two Variables . 4948.6The Algebra of Matrices . 5048.7The Inverse of a Matrix . 5098.8Determinants .5148.9Properties of Determinants . 5198.10Partial Fractions .524Chapter 8 Review Exercises . 530Chapter 8 Discussion Exercises . 539Chapter 8 Test . 541 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

vi9Sequences, Series, and Probability.5479.1Infinite Sequences and Summation Notation .5479.2Arithmetic Sequences . 5569.3Geometric Sequences . 5629.4Mathematical Induction . 5689.5The Binomial Theorem . 5779.6Permutations . 5829.7Distinguishable Permutations and Combinations . 5869.8Probability . 590Chapter 9 Review Exercises . 598Chapter 9 Discussion Exercises . 605Chapter 9 Test . 60910Topics from Analytic Geometry.61310.1Parabolas .61310.2Ellipses . 62010.3Hyperbolas .63010.4Plane Curves and Parametric Equations . 64110.5Polar Coordinates . 65510.6Polar Equations of Conics . 673Chapter 10 Review Exercises . 679Chapter 10 Discussion Exercises . 689Chapter 10 Test . 69111Limits of Functions.69511.1Introduction to Limits . 69511.2Definition of Limit .70311.3Techniques for Finding Limits . 71111.4Limits Involving Infinity . 718Chapter 11 Review Exercises . 726Chapter 11 Discussion Exercises . 730Chapter 11 Test . 732 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

To the InstructorIn the chapter review sections, the solutions are abbreviated since more detailed solutionswere given in chapter sections. In easier groups of exercises, representative solutions are shown.When appropriate, only the answer is listed.All figures have been plotted using computer software, offering a high degree ofprecision. The calculator graphs are from various TI screens. When possible, we tried to makeeach piece of art with the same scale to show a realistic and consistent graph.This manual was prepared using ê: The Scientific Word Processor.The following notations are used in the manual.Note: Notes to the instructor/student pertaining to hints on instruction or conventions to follow.{}{ comments to the reader are in braces }LS{ Left Side of an equation }RS{ Right Side of an equation }Ê{ implies, next equation, logically follows }Í{ if and only if, is equivalent to }ì{ bullet, used to separate problem statement from solution or explanation }æ{ used to identify the answer to the problem }§{ section references }a{ Example: aB means “for all B” }‘ { }.{ The set of all real numbers except . }QI–QIV{ quadrants I, II, III, IV }{ therefore } 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.vii

Chapter 1: Fundamental Concepts of Algebra1.1 Exercises1(a) Since B and C have opposite signs, the product BC is negative.(b) Since B# ! and C !, the product B# C is positive.Bis negative.CBThus, B is the sum of two negatives, which is negative.C(c) Since B ! {B is negative} and C ! {C is positive}, the quotient(d) Since C ! and B !, C B !.2(a) Since B and C have opposite signs, the quotientBis negative.C(b) Since B ! and C# !, the product BC# is negative.B C(c) Since B C ! and BC !, !.(d) Since C ! and C B !, C C B !.BC3(a) Since ( is to the left of % on a coordinate line, ( %.(b) Using a calculator, we see that1# "Þ&(. Hence,(c) ##& œ "& Note: ##& Á „"&4 "Þ&.(a) Since is to the right of ' on a coordinate line, '.(b) Using a calculator, we see that1% !Þ(*. Hence,(c) #)* œ "( Note: #)* Á „"(51#(a) Since"""œ !Þ!* œ !Þ!*!*á ,(c) Since##(œ Þ"%#)&( and 1 Þ"%"&* ,(a) Since"("("""1% !Þ!*.##( !Þ).(b) Since# œ !Þ' œ !Þ''''á ,# (b) Since&'œ !Þ) œ !Þ) á , !Þ'''. 1.&'6(c) Since # "Þ%"%, # "Þ%.7(a) “B is negative” is equivalent to B !. We symbolize this by writing “B is negative Í B !.”œ !."%#)&(,(b) C is nonnegative Í C !Þ"% .!(d) . is between % and # Í # . % !Þ) .(c) ; is less than or equal to 1 Í ; Ÿ 1(e) is not less than & Í &(f) The negative of D is not greater than Í D Ÿ :"(g) The quotient of : and ; is at most ( ÍŸ((h) The reciprocal of A is at least * Í;A*(i) The absolute value of B is greater than ( Í B (Note: An informal definition of absolute value that may be helpful is something œ itself itself if itself is positive or zeroif itself is negative 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.1

281.1 EXERCISES(a) , is positive Í , !(b) is nonpositive Í Ÿ !(c) A is greater than or equal to % Í A(d) - is between"&and" Í"& - %" (e) : is not greater than # Í : Ÿ #(f) The negative of 7 is not less than # Í 7(g) The quotient of and is at least"&Í "&(i) The absolute value of B is less than % Í B %9(a) % œ ( œ ( {since ( !} œ ( #(h) The reciprocal of 0 is at most "% Í"0Ÿ "%(b) & # œ & # œ & # œ (c) ( % œ ( % œ ( % œ ""(b) ' œ ' œ ' œ (c) ) * œ ) * œ ) * œ "((b) ' Î # œ ' Î # œ 'Î # œ (c) ( % œ ( % œ ( % œ ""(b) &Î # œ &Î # œ &Î#(c) " * œ " * œ " * œ "!10 (a) "" " œ "! œ "! { since "! ! } œ "!11 (a) & ' œ & œ & œ & œ "&12 (a) % ' ( œ % " œ % " œ % " œ %13 (a) Since % 1 is positive, % 1 œ % 1.(b) Since 1 % is negative, 1 % œ 1 % œ % 1.(c) Since # "Þ& is negative, # "Þ& œ # "Þ& œ "Þ& #.14 (a) Since "Þ( is positive, "Þ( œ "Þ(.(b) Since "Þ( is negative, "Þ( œ "Þ( œ "Þ(. (c) "& " œ "& & "&# # œ "&œ "&œ15 (a) . Eß F œ ( œ % œ %(c) . Gß F œ . Fß G œ "#16 (a) . Eß F œ # ' œ % œ %(c) . Gß F œ . Fß G œ '17 (a) . Eß F œ " * œ "! œ "!(c) . Gß F œ . Fß G œ *18 (a) . Eß F œ % ) œ "# œ "#(c) . Gß F œ . Fß G œ #"&(b) . Fß G œ & ( œ "# œ "#(d) . Eß G œ & œ ) œ )(b) . Fß G œ % # œ ' œ '(d) . Eß G œ % ' œ "! œ "!(b) . Fß G œ "! " œ * œ *(d) . Eß G œ "! * œ "* œ "*(b) . Fß G œ " % œ œ (d) . Eß G œ " ) œ * œ *Note: Because œ , the answers to Exercises 19–24 could have a different form. For example, B equivalent to B ).19 E œ B and F œ (, so . Eß F œ ( B . Thus, “. Eß F is less than #” can be written as ( B #. 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.) is

1.1 EXERCISES20 . Eß F œ # B 21 . Eß F œ B 23 . Eß F œ B % ÊÊÊ # B " B B % Ÿ )22 . Eß F œ % B 3Ê % B Ÿ &24 . Eß F œ B # œ B # Ê B # %Note: In Exercises 25–32, you may want to substitute a permissible value for the variable to first test if the expressioninside the absolute value symbol is positive or negative.25 Pick an arbitrary value for B that is less than , say &.Since & œ # is negative, we conclude that if B , then B is negative.26 If B &, then & B !, and & B œ & B œ B &.Hence, B œ B œ B .27 If B #, then # B !, and # B œ # B.28 If B (, then ( B!, and ( B œ ( B.29 If , , then , !, and , œ , œ , .30 If , , then , !, and , œ , .31 Since B# % ! for every B, B# % œ B# %.32 Since B# " ! for every B, B# " œ B# " œ B# ". , , œ œ , - Á RS (which is , - ). , , LS œœ œ , - œ RS. , ,LS œœ œ RS. LS œœ Á RS which is ., ., . , .,. " , LS œ ƒ , ƒ - œ † œ . RS œ ƒ , ƒ - œ ƒ œ † œ. LS Á RS, ,,,33 LS œ3435363738 LS œ , - œ , - . RS œ , - œ , - . LS Á RS39 LS œ , , œœ " œ RS., , 40 LS œ , œ , Á RS (which is , ).41 (a) On the TI-83/4 Plus, the absolute value function is choice 1 under MATH, NUM.Enter abs(3.2 2- (%.#()). Þ## %.#( )Þ"( '(b) "&Þ' "Þ& # %Þ &Þ% # "%Þ"%#)42 (a) Þ%# "Þ#* !Þ#&"&&Þ) #Þ'%(b) 1 "Þ!!' 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

41.1 EXERCISES43 (a)"Þ# ‚ "! !Þ'&&( œ 'Þ&&( ‚ "! " Þ" ‚ "!# "Þ&# ‚ "! Note: For the TI-83/4 Plus, use 1.2E3Î(3.1E2 1.52E3), where E is obtained by pressing #ndEE .(b) "Þ# ‚ "! % %Þ& ‚ "! '(Þ!) œ 'Þ(!) ‚ "!"44 (a) Þ%& "Þ# ‚ "!% "!& %Þ( œ Þ %( ‚ "!#(b) "Þ(* ‚ "!# ‚ *Þ)% ‚ "! œ ",('", '! "Þ('" ‚ "!'45 Construct a right triangle with sides of lengths # and ". The hypotenuse will have length # "# œ .#Next construct a right triangle with sides of lengths and #. # œ &.#The hypotenuse will have length#46 Use G œ #1 with œ ", #, and "! to obtain #1, %1, and #!1 units from the origin.47 The large rectangle has area œ width ‚ length œ , - . The sum of the areas of the two small rectangles is , - . Since the areas are the same, we have , - œ , - .48 B" œ and 8 œ ##ÊB# œ"8" #" %" "("(. B" œ œ œ œ#B"# ## # # '"##B œ"#" "(#" "( #%" &((&(( B# œ "( œ œ œ#B## "##"#"(##!%%!)"#49 (a) Since the decimal point is & places to the right of the first nonzero digit, %#(,!!! œ %Þ#( ‚ "!& .(b) Since the decimal point is ) places to the left of the first nonzero digit, !Þ!!! !!! !* œ *Þ ‚ "! ) .(c) Since the decimal point is ) places to the right of the first nonzero digit, )"!,!!!,!!! œ )Þ" ‚ "!) .50 (a) )&,#!! œ )Þ&# ‚ "!%(b) !Þ!!! !!& % œ &Þ% ‚ "! '(c) #%,*!!,!!! œ #Þ%* ‚ "!(51 (a) Moving the decimal point & places to the right, we have )Þ ‚ "!& œ ) !,!!!.(b) Moving the decimal point "# places to the left, we have #Þ* ‚ "! "# œ !Þ!!! !!! !!! !!# *.(c) Moving the decimal point ) places to the right, we have &Þ'% ‚ "!) œ &'%,!!!,!!!.52 (a) #Þ ‚ "!( œ # ,!!!,!!!(b) (Þ!" ‚ "! * œ !Þ!!! !!! !!( !"(c) "Þ#& ‚ "!"! œ "#,&!!,!!!,!!!53 Since the decimal point is #% places to the left of the first nonzero digit,!Þ!!! !!! !!! !!! !!! !!! !!! !!" ( œ "Þ( ‚ "! #% .54 *Þ" ‚ "! " œ !Þ!!! !!! !!! !!! !!! !!! !!! !!! !!! !!! *"55 It is helpful to write the units of any fraction, and then “cancel” those units to determine the units of the final")',!!! miles '! seconds '! minutes #% hours '& daysanswer.††††† " year &Þ)( ‚ "!"# misecond" minute" hour" day" year 2019 Cengage Learning. All Rights Reserved.May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.