HAESE & HARRIS PUBLICATIONS Specialists in mathematics publishingMathematicsfor the international studentMathematics SL John Owen Robert Haese Sandra Haese Mark Bruce International Baccalaureate Diploma Programme
FOREWORDMathematics for the International Student: Mathematics SL has been written to embracethe syllabus for the new two-year Mathematics SL Course, which is one of the courses ofstudy in the International Baccalaureate Diploma Programme. It is not our intention to definethe course. Teachers are encouraged to use other resources. We have developed the book inde-pendently of the International Baccalaureate Organization (IBO) in consultation with manyexperienced teachers of IB Mathematics. The text is not endorsed by the IBO.This package is language rich and technology rich. The combination of textbook and interac-tive Student CD will foster the mathematical development of students in a stimulating way.Frequent use of the interactive features on the CD is certain to nurture a much deeper under-standing and appreciation of mathematical concepts.The book contains many problems from the basic to the advanced, to cater for a wide rangeof student abilities and interests. While some of the exercises are simply designed to buildskills, every effort has been made to contextualise problems, so that students can see every-day uses and practical applications of the mathematics they are studying, and appreciate theuniversality of mathematics.Emphasis is placed on the gradual development of concepts with appropriate worked exam-ples, but we have also provided extension material for those who wish to go beyond thescope of the syllabus. Some proofs have been included for completeness and interest al-though they will not be examined.For students who may not have a good understanding of the necessary background knowl-edge for this course, we have provided printable pages of information, examples, exercisesand answers on the Student CD. To access these pages, simply click on the ‘Backgroundknowledge’ icon when running the CD.It is not our intention that each chapter be worked through in full. Time constraints will notallow for this. Teachers must select exercises carefully, according to the abilities and priorknowledge of their students, to make the most efficient use of time and give as thorough cov-erage of work as possible. Investigations throughout the book will add to the discovery aspectof the course and enhance student understanding and learning. Many Investigations are suit-able for portfolio assignments and have been highlighted in the table of contents. Reviewsets appear at the end of each chapter and a suggested order for teaching the two-year courseis given at the end of this Foreword.The extensive use of graphics calculators and computer packages throughout the book en-ables students to realise the importance, application and appropriate use of technology. Nosingle aspect of technology has been favoured. It is as important that students work with apen and paper as it is that they use their calculator or graphics calculator, or use a spreadsheetor graphing package on computer.The interactive features of the CD allow immediate access to our own specially designedgeometry packages, graphing packages and more. Teachers are provided with a quick andeasy way to demonstrate concepts, and students can discover for themselves and re-visitwhen necessary.
Instructions appropriate to each graphic calculator problem are on the CD and can be printedfor students. These instructions are written for Texas Instruments and Casio calculators.In this changing world of mathematics education, we believe that the contextual approachshown in this book, with the associated use of technology, will enhance the students’ under-standing, knowledge and appreciation of mathematics, and its universal application.We welcome your feedback.Email: email@example.comWeb: www.haeseandharris.com.au JTO RCH SHH MFBThank youThe authors and publishers would like to thank all those teachers who offered advice andencouragement. Many of them read the page proofs and offered constructive comments andsuggestions. These teachers include: Marjut Mäenpää, Cameron Hall, Paul Urban, FranO’Connor, Glenn Smith, Anne Walker, Malcolm Coad, Ian Hilditch, Phil Moore, JulieWilson, David Martin, Kerrie Clements, Margie Karbassioun, Brian Johnson, Carolyn Farr,Rupert de Smidt, Terry Swain, Marie-Therese Filippi, Nigel Wheeler, Sarah Locke, RemaGeorge. TEACHING THE TWO-YEAR COURSE – A SUGGESTED ORDER For the first year, it is suggested that students work progressively from Chapter 1 through to Chapter 16, although some teachers may prefer to leave Chapter 16 ‘Vectors in 3 dimensions’ until the second year. ‘Descriptive statistics’ and ‘Probability’ (Chapters 18 and 19) could possibly be taught the first year. Alternatively, calculus could be introduced (Chapters 20-22), but most teachers will probably prefer to leave calculus until the second year and have students work progressively from Chapter 20 though to Chapter 29. We invite teachers who have their preferred order, to email their suggestions to us. We can put these suggestions on our website to be shared with other teachers.
USING THE INTERACTIVE STUDENT CDThe CD is ideal for independent study. Frequent use will nurture a deeper understanding ofMathematics. Students can revisit concepts taught in class and undertake their own revisionand practice. The CD also has the text of the book, allowing students to leave the textbook atschool and keep the CD at home.The icon denotes an active link on the CD. Simply ‘click’ the icon to access a range ofinteractive features: l spreadsheets and worksheets l video clips CD LINK l graphing and geometry software l graphics calculator instructions l computer demonstrations and simulations l background knowledgeFor those who want to make sure they have the prerequisite levels of understanding for thisnew course, printable pages of background information, examples, exercises and answers areprovided on the CD. Click the ‘Background knowledge’ icon.Graphics calculators: Instructions for using graphics calculators are also given TIon the CD and can be printed. Instructions are given for Texas Instruments andCasio calculators. Click on the relevant symbol (TI or C) to access printable Cinstructions.NOTE ON ACCURACYStudents are reminded that in assessment tasks, including examination papers, unless other-wise stated in the question, all numerical answers must be given exactly or to three significantfigures.ERRATAIf you find an error in this book please notify us by emailing firstname.lastname@example.org.As a help to other teachers and students, we will include the correction on our website andcorrect the book at the first reprint opportunity.
6 TABLE OF CONTENTSTABLE OF CONTENTS E Algebraic expansion 73 F Exponential equations 74 * denotes ideas for possible Portfolio Assignments G Graphs of exponential functions 75 * Investigation: Exponential graphs 76 Abbreviations used in this book 10 H Growth 79 I Decay 81 BACKGROUND KNOWLEDGE Review set 3A 84 – to access, ‘click’ active icon on CD 11 Review set 3B 84 A Operations with surds (radicals) CD Review set 3C 85 B Standard form (scientific notation) CD Review set 3D 86 C Number systems and set notation CD 4 LOGARITHMS 87 D Algebraic simplification CD E Linear equations and inequalities CD A Introduction 88 F Absolute value (modulus) CD B Logarithms in base 10 90 G Product expansion CD Investigation: Discovering the laws of logarithms 92 H Factorisation CD C Laws of logarithms 93 Investigation: Another factorisation technique CD D Exponential equations (using logarithms) 96 I Formula rearrangement CD E Growth and decay revisited 97 J Adding and subtracting algebraic fractions CD F Compound interest revisited 99 K Congruence and similarity CD G The change of base rule 100 ANSWERS CD Review set 4A 101 Summary of circle properties 11 Review set 4B 102 Summary of measurement facts 12 5 NATURAL LOGARITHMS 103 1 FUNCTIONS 15 A Introduction 104 A Relations and functions 16 Investigation 1: e occurs naturally 104 B Interval notation, domain and range 19 Investigation 2: Continuous compound interest 105 C Function notation 22 B Natural logarithms 108 Investigation: Fluid filling functions 24 Investigation 3: The laws of natural D Composite functions, 26 f±g logarithms 109 E The reciprocal function 1 27 C Laws of natural logarithms 110 x x D Exponential equations involving e 112 F Inverse functions 28 G The identity function 31 E Growth and decay revisited 112 Review set 1A 32 F Inverse functions revisited 114 Review set 1B 33 Review set 5A 116 Review set 5B 117 2 SEQUENCES AND SERIES 35 A Number patterns 36 6 GRAPHING AND TRANS- B Sequences of numbers 38 FORMING FUNCTIONS 119 C Arithmetic sequences 41 A Families of functions 120 D Geometric sequences 44 * Investigation: Function families 120 E Series 51 B Key features of functions 122 F Sigma notation 57 C Transformations of graphs 123 * Investigation: Von Koch’s Snowflake curve 58 D Functional transformations 126 Review set 2A 59 Review set 6 127 Review set 2B 59 7 COORDINATE GEOMETRY 129 Review set 2C 60 A Assumed knowledge 131 3 EXPONENTS 61 * Investigation: Finding where lines A Index notation 62 meet using technology 135 B Negative bases 63 B Equations of lines 138 C Index laws 65 C Distance between two points 141 D Rational indices 71 D Midpoints and perpendicular bisectors 144 Review set 7A 146
TABLE OF CONTENTS 7 Review set 7B 147 Review set 11 233 Review set 7C 148 12 NON RIGHT ANGLED TRIANGLE8 QUADRATIC EQUATIONS AND TRIGONOMETRY 235 FUNCTIONS 149 A Areas of triangles 236A Function notation f : ` ax2 + bx + c ! 153 B Sectors and segments 239B Graphs of quadratic functions 154 C The cosine rule 241* Investigation 1: Graphing D The sine rule 244 y = a (x ¡ ®) (x ¡ ¯) 155 Investigation: The ambiguous case 245* Investigation 2: Graphing E Using the sine and cosine rules 248 2 y = a (x ¡ h) + k 155 Review set 12A 251C Completing the square 160 Review set 12B 252D Quadratic equations 162E The quadratic formula 168 13 PERIODIC PHENOMENA 255F Solving quadratic equations with A Observing periodic behaviour 257 technology 170 B Radian measure and periodic propertiesG Problem solving with quadratics 171 of circles 260H Quadratic graphs (review) 174 C The unit circle (revisited) 265I The discriminant, D 178 D The sine function 270J Determining the quadratic from a graph 182 * Investigation 1: The family y = A sin x 271K Where functions meet 185 * Investigation 2: The familyL Quadratic modelling 186 y = sin B x; B > 0 271 Review set 8A 190 * Investigation 3: The families Review set 8B 191 y = sin (x ¡ C) and y = sin x + D 273 Review set 8C 191 E Modelling using sine functions 275 Review set 8D 192 F Equations involving sine 278 Review set 8E 193 * Investigation 4: The area under an arch of y = sin µ 2849 THE BINOMIAL THEOREM 195 G The cosine function 285A Binomial expansions 196 H Solving cosine equations 287 Investigation 1: The binomial expansions I Trigonometric relationships 289 of (a + b)n ; n > 4 196 J Double angle formulae 292 n Investigation 5: Double angle formulae 292 Investigation 2: n C r or Cr values 199B The general binomial expansion 199 K The tangent function 295 Review set 9 202 L Tangent equations 298 M Other equations involving tan x 30110 PRACTICAL TRIGONOMETRY Review set 13A 302 WITH RIGHT ANGLED Review set 13B 302 TRIANGLES 203 Review set 13C 303 Review set 13D 304A Pythagoras’ rule (review) 205 Review set 13E 305B Pythagoras’ rule in 3-D problems 207 Investigation: Shortest distance 208 14 MATRICES 307C Right angled triangle trigonometry 209 A Introduction 308D Finding sides and angles 211 B Addition and subtraction of matrices 311E Problem solving using trigonometry 217 C Multiples of matrices 314F The slope of a straight line 221 D Matrix algebra for addition 316 Review set 10A 222 E Matrix multiplication 317 Review set 10B 223 F Using technology 321 Review set 10C 223 G Some properties of matrix multiplication 32511 THE UNIT CIRCLE 225 H The inverse of a 2 £ 2 matrix 328 I Solving a pair of linear equations 331A The unit quarter circle 226 J The 3 £ 3 determinant 334B Obtuse angles 228 K The inverse of a 3 £ 3 matrix 337C The unit circle 231 Investigation: Parametric equations 232 L 3 £ 3 systems with unique solutions 337
8 TABLE OF CONTENTS * Investigation: Using matrices in C Presenting and interpreting data CD cryptography 341 ANSWERS CD Review set 14A 342 A Continuous numerical data and histograms 421 Review set 14B 343 B Measuring the centre of data 425 Review set 14C 344 Investigation: Merits of the mean Review set 14D 345 and median 427 Review set 14E 345 C Cumulative data 442 D Measuring the spread of data 44515 VECTORS IN 2-DIMENSIONS 347 E Statistics using technology 453 A Vectors 348 F Variance and standard deviation 456 B Operations with vectors 352 G The significance of standard deviation 461 C Vectors in component form 360 Review set 18A 463 D Vector equations 365 Review set 18B 465 E Vectors in coordinate geometry 366 F Parallelism 368 19 PROBABILITY 467 G Unit vectors 369 A Experimental probability 470 H Angles and scalar product 371 Investigation 1: Tossing drawing pins 470 Review set 15A 376 Investigation 2: Coin tossing experiments 471 Review set 15B 376 Investigation 3: Dice rolling experiments 472 Review set 15C 377 B Sample space 474 Review set 15D 378 C Theoretical probability 475 D Using grids to find probabilities 47916 VECTORS IN 3-DIMENSIONS 379 E Compound events 480 A 3-dimensional coordinates 380 Investigation 4: Probabilities of B 3-dimensional vectors 383 compound events 481 C Algebraic operations with 3-D vectors 386 Investigation 5: Revisiting drawing pins 481 D Parallelism 389 F Using tree diagrams 485 E Unit vectors 390 G Sampling with and without replacement 487 F Collinear points and ratio of division – * Investigation 6: Sampling simulation 490 extension 391 Investigation 7: How many should I plant? 491 G The scalar product of 3-D vectors 392 H Pascal’s triangle revisited 492 Review set 16A 395 I Sets and Venn diagrams 494 Review set 16B 396 J Laws of probability 49917 LINES IN THE PLANE AND K Independent events revisited 503 IN SPACE 397 Review set 19A 505 Review set 19B 505 A Vector and parametric form of a line in 2-dimensional geometry 399 20 INTRODUCTION TO CALCULUS 507 B The velocity vector of a moving object 401 Investigation 1: The speed of falling C Constant velocity problems 403 objects 508 Investigation: The two yachts problem 405 A Rate of change 509 D The closest distance 406 B Instantaneous rates of change 513 E Geometric applications of r = a + tb 409 Investigation 2: Instantaneous speed 513 F Lines in space 411 Review set 20 519 G Line classification 414 Review set 17A 416 21 DIFFERENTIAL CALCULUS 521 Review set 17B 417 A The idea of a limit 522 Investigation 1: The slope of a tangent 52218 DESCRIPTIVE STATISTICS 419 B Derivatives at a given x-value 525 BACKGROUND KNOWLEDGE IN C The derivative function 530 STATISTICS – to access, ‘click’ Investigation 2: Finding slopes of functions with technology 531 active icon on CD 420 D Simple rules of differentiation 533 A Statistical enquiries CD Investigation 3: Simple rules of Investigation: Statistics from the internet CD differentiation 534 B Populations and samples CD E The chain rule 537
TABLE OF CONTENTS 9 Investigation 4: Differentiating composites 538 26 INTEGRATION 649F Product and quotient rules 541 A Reviewing the definite integral 650G Tangents and normals 545 B The area function 655H The second derivative 550 Investigation 1: The area function 656 Review set 21A 552 C Antidifferentiation 656 Review set 21B 553 D The fundamental theorem of calculus 658 Review set 21C 554 E Integration 662 n F Integrating eax +b and (ax + b) 66822 APPLICATIONS OF 0 G Integrating f (u) u (x) by substitution 670 DIFFERENTIAL CALCULUS 555 H Distance from velocity 673A Functions of time 556 I Definite integrals Rb 675B Time rate of change 558 Investigation 2: a f (x) dx and areas 677C General rates of change 560 J Finding areas 677D Motion in a straight line 564 K Problem solving by integration 682 Investigation: Displacement, velocity Review set 26A 685 and acceleration graphs 568 Review set 26B 686E Curve properties 571 Review set 26C 687F Rational functions 579 27 TRIGONOMETRICG Inflections and shape type 583 INTEGRATION 689H Optimisation 587 A Basic trigonometric integrals 690I Economic models 597 B Integrals of trigonometric functions Review set 22A 600 of the form f (ax + b) 692 Review set 22B 602 C Definite integrals 695 D Area determination 69823 DERIVATIVES OF Review set 27A 700 EXPONENTIAL AND Review set 27B 700 LOGARITHMIC FUNCTIONS 605A Derivatives of exponential functions 606 28 VOLUMES OF REVOLUTION 701 Investigation 1: The derivative of y = ax 606 A Solids of revolution 702 Investigation 2: Finding a when y = ax B Volumes for two defining functions 705 dy Review set 28 708 and = ax 607 dx 29 STATISTICAL DISTRIBUTIONS 709B Using natural logarithms 611 A Discrete random variables 710C Derivatives of logarithmic functions 615 B Discrete probability distributions 712 Investigation 3: The derivative of ln¡x 615 C Expectation 714D Applications 618 Investigation 1: Concealed number tickets 717 Review set 23A 622 D The mean and standard deviation of a Review set 23B 623 discrete random variable 717 E The binomial distribution 72124 DERIVATIVES OF F Mean and standard deviation of a TRIGONOMETRIC FUNCTIONS 625 binomial random variable 724A The derivative of sin¡x, cos¡x, tan¡x 627 Investigation 2: The mean and standardB Maxima/minima with trigonometry 632 deviation of a binomial random variable 726 Review set 24 634 G Normal distributions 727 Investigation 3: Standard deviation25 AREAS WITHIN CURVED significance 730 BOUNDARIES 637 Investigation 4: Mean and standardA Areas where boundaries are curved 638 deviation of z = x ¡ xs 733 Investigation 1: Finding areas using H The standard normal distribution rectangles 640 (z-distribution) 734B Definite integrals 642 R1¡ ¢ I Applications of the normal distribution 739 Investigation 2: 0 1 ¡ x2 dx, Review set 29A 741 R1¡ 2 ¢ 0 x ¡ 1 dx 646 Review set 29B 742 Review set 29C 743 Review set 25 648 ANSWERS AND INDEX 745
SYMBOLS AND NOTATION USED IN THIS BOOKThis notation is based on that indicated by the International Organisation of Standardisation. dyN the set of all natural numbers the derivative of y with respect to x dx f0, 1, 2, 3, 4, 5, ..... g f 0 (x) the derivative of f(x) withZ the set of all integers respect to x f0, §1, §2, §3, §4, §5, .....g ZZ+ the set of all positive integers y dx the indefinite integral of y with f1, 2, 3, 4, 5, .....g respect to x Z bQ the set of all rational numbers y dxthe definite integral of y withQ+ the set of all positive rational a respect to x from x = a to numbers x=bR the set of all real numbers ex the exponential functionR+ the set of all positive real numbers loga x the logarithm of x, in base an(S) the number of elements in set S ln x the natural logarithm of x, loge x2 is an element of sin x, cos x the circular functions2= is not an element of and tan x? the empty set, or null set P(x, y) point P with coordinates x and yU the universal set ]A the angle at A[ union ]PQR the angle between QP and QR intersection ¢PQR the triangle with vertices P, Q and Rjxj the modulus of x, v vector v or the absolute value of x ¡! AB the vector from point A to point B j x j = x if x > 0 or ¡x if x 6 0 ¡! a the position vector of A, OAun the nth term of a sequence i, j, k unit vectors in the direction of thed the common difference of an x-, y - and z-axis respectively arithmetic sequence jaj the magnitude of ar the common ratio of a geometric a²b the scalar product of a and b sequence ¡1 A the inverse of matrix ASn u1 + u2 + u3 + ..... +un the sum of the first n terms of a det A the determinant of matrix A sequence I the identity matrix under £S1 the sum to infinity of a P(A) the probability of event A sequence P(A0 ) the probability of event ‘not A’nX P(A/B) the probability of A occurring ui u1 + u2 + u3 + ..... +un given that B has occurredi=1µ ¶ N(¹, ¾ 2 ) the normal distribution with n the binomial coefficient of the mean ¹ and variance ¾ 2 r (r + 1)th term in the expansion ¹ population mean of (a + b)n ¾2 population variancef : x 7! y f is the function where x goes to y ¾ population standard deviationf (x) the image of x operated on by f x sample meanf ¡1 (x) the inverse function of f(x) sn 2 sample variancef ±g the composite function of f and g sn sample standard deviationlim f (x) the limit of f(x) as x tends to ax!a
BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS 11 BACKGROUND KNOWLEDGE Before starting this course you can make sure that you BACKGROUND have a good understanding of the necessary background KNOWLEDGE knowledge. Click on the icon alongside to obtain a printable set of exercises and answers on this background knowledge.SUMMARY OF CIRCLE PROPERTIES ² A circle is a set of points which are equidistant from circle a fixed point, which is called its centre. centre ² The circumference is the distance around the entire circle boundary. ² An arc of a circle is any continuous part of the circle. chord arc ² A chord of a circle is a line segment joining any two points of a circle. ² A semi-circle is a half of a circle. diameter ² A diameter of a circle is any chord passing through its centre. radius ² A radius of a circle is any line segment joining its centre to any point on the circle. tangent ² A tangent to a circle is any line which touches the circle in exactly one point. point of contactBelow is a summary of well known results called theorems. Click on the appropriate icon torevisit them. Name of theorem Statement Diagram B Angle in a The angle in a semi- If then ]ABC = 90o. semi-circle circle is a right angle. GEOMETRY PACKAGE A O C
12 BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS Name of theorem Statement Diagram Chords of a The perpendicular If then AM = BM. circle from the centre of O GEOMETRY A a circle to a chord PACKAGE bisects the chord. M B Radius-tangent The tangent to a cir- If then ]OAT = 90o . cle is perpendicular O to the radius at the GEOMETRY PACKAGE point of contact. A T A Tangents from Tangents from an ex- If then AP = BP. an external ternal point are equal point in length. O P GEOMETRY PACKAGE B C Angle at the The angle at the centre If then ]AOB = 2]ACB. centre of a circle is twice the angle on the circle sub- O GEOMETRY PACKAGE tended by the same arc. A B C then ]ADB = ]ACB. If D Angles Angles subtended by an subtended arc on the circle are by the equal in size. GEOMETRY same arc PACKAGE A B Angle between The angle between a tan- If C then ]BAS = ]BCA. a tangent and gent and a chord at the a chord point of contact is equal B GEOMETRY PACKAGE to the angle subtended by the chord in the al- ternate segment. T A SSUMMARY OF MEASUREMENT FACTSPERIMETER FORMULAE The distance around a closed figure is its perimeter.
BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS 13For some shapes we can derive a formula for perimeter. The formulaefor the most common shapes are given below: a b r d r l w q° l c square rectangle triangle circle arcP = 4l P = 2(l + w) P = a+ b+ c C =2¼r l = ( 360)2¼r µ or C = ¼d The length of an arc is a fraction of the circumferenceAREA FORMULAE of a circle. Shape Figure Formula Rectangle Area = length £ width width length Triangle Area = 1 base £ height 2 height base base Parallelogram Area = base £ height height base a Trapezium or h Area = (a + b) £h 2 Trapezoid b Circle Area = ¼r2 r Sector Area = (360) £¼r µ 2 q rSURFACE AREA FORMULAERECTANGULAR PRISM c A = 2(ab + bc + ac) b a
14 BACKGROUND KNOWLEDGE AND GEOMETRIC FACTSCYLINDER CONE Object Outer surface area Object Outer surface area Hollow cylinder A = 2¼rh hollow Open cone A = ¼rs (no ends) r (no base) h s r hollow Open can A = 2¼rh + ¼r2 hollow (one end) Solid cone A = ¼rs + ¼r2 r (solid) h r solid s 2 Solid cylinder A = 2¼rh + 2¼r solid (two ends) h SPHERE r Area, r solid A = 4¼r2VOLUME FORMULAE Object Figure Volume Solids of uniform height Volume of uniform solid cross-section = area of end £ length end height end height height Pyramids h Volume of a pyramid and or cone cones = 1 (area of base £ height) 3 base base r Volume of a sphere Spheres = 4 ¼r3 3
Functions Chapter 1 Contents: A Relations and functions B Interval notation, domain and range C Function notation Investigation: Fluid filling functions D Composite functions, f¡±¡g 1 E ! The reciprocal function x ` x F Inverse functions G The identity function Review set 1A Review set 1B
16 FUNCTIONS (Chapter 1) A RELATIONS AND FUNCTIONSThe charges for parking a car in a short-term car park at an Car park chargesAirport are given in the table shown alongside. Period (h) ChargeThere is an obvious relationship between time spent and the 0 - 1 hours $5:00cost. The cost is dependent on the length of time the car is 1 - 2 hours $9:00parked. 2 - 3 hours $11:00Looking at this table we might ask: How much would be 3 - 6 hours $13:00charged for exactly one hour? Would it be $5 or $9? 6 - 9 hours $18:00 9 - 12 hours $22:00To make the situation clear, and to avoid confusion, we 12 - 24 hours $28:00could adjust the table and draw a graph. We need to indicatethat 2-3 hours really means for time over 2 hours up to andincluding 3 hours i.e., 2 < t 6 3.So, we Car park chargesnow have Period Charge 0 < t 6 1 hours $5:00 1 < t 6 2 hours $9:00 2 < t 6 3 hours $11:00 3 < t 6 6 hours $13:00 6 < t 6 9 hours $18:00 9 < t 6 12 hours $22:00 12 < t 6 24 hours $28:00 30 charge ($)In mathematical terms, because we have arelationship between two variables, time andcost, the schedule of charges is an example 20of a relation. exclusionA relation may consist of a finite number of inclusionordered pairs, such as f(1, 5), (¡2, 3), 10(4, 3), (1, 6)g or an infinite number of or-dered pairs. time (t) 3 6 9 12 15 18 21 24The parking charges example is clearly the latter as any real value of time ( t hours) in theinterval 0 < t 6 24 is represented.The set of possible values of the variable on the horizontal axis is called the domain of therelation.For example: ² ft : 0 < t 6 24g is the domain for the car park relation ² f¡2, 1, 4g is the domain of f(1, 5), (¡2, 3), (4, 3), (1, 6)g.
FUNCTIONS (Chapter 1) 17The set which describes the possible y-values is called the range of the relation.For example: ² the range of the car park relation is f5, 9, 11, 13, 18, 22, 28g ² the range of f(1, 5), (¡2, 3), (4, 3), (1, 6)g is f3, 5, 6g.We will now look at relations and functions more formally.RELATIONS A relation is any set of points on the Cartesian plane.A relation is often expressed in the form of an equation connecting the variables x and y.For example y = x + 3 and x = y2 are the equations of two relations.These equations generate sets of ordered pairs.Their graphs are: y y y=x+3 2 3 x 4 -3 x x = y2However, a relation may not be able to be defined by an equation. Below are two exampleswhich show this: (1) y All points in the (2) y first quadrant These 13 points are a relation. form a relation. x > 0, y > 0 x xFUNCTIONS A function is a relation in which no two different ordered pairs have the same x-coordinate (first member).We can see from the above definition that a function is a special type of relation.TESTING FOR FUNCTIONSAlgebraic Test: If a relation is given as an equation, and the substitution of any value for x results in one and only one value of y, we have a function.
18 FUNCTIONS (Chapter 1)For example: ² y = 3x ¡ 1 is a function, as for any value of x there is only one value of y ² x = y 2 is not a function since if x = 4, say, then y = §2.Geometric Test (“Vertical Line Test”): If we draw all possible vertical lines on the graph of a relation, the relation: DEMO ² is a function if each line cuts the graph no more than once ² is not a function if one line cuts the graph more than once. Example 1 Which of the following relations are functions? a y b y c y x x x a y b y c y x x x a function a function not a functionGRAPHICAL NOTE ² If a graph contains a small open circle end point such as , the end point is not included. ² If a graph contains a small filled-in circle end point such as , the end point is included. ² If a graph contains an arrow head at an end such as then the graph continues indefinitely in that general direction, or the shape may repeat as it has done previously.EXERCISE 1A 1 Which of the following sets of ordered pairs are functions? Give reasons. a (1, 3), (2, 4), (3, 5), (4, 6) b (1, 3), (3, 2), (1, 7), (¡1, 4) c (2, ¡1), (2, 0), (2, 3), (2, 11) d (7, 6), (5, 6), (3, 6), (¡4, 6) e (0, 0), (1, 0), (3, 0), (5, 0) f (0, 0), (0, ¡2), (0, 2), (0, 4)
FUNCTIONS (Chapter 1) 19 2 Use the vertical line test to determine which of the following relations are functions: a b c y y y x x x d y e y f y x x x g h i y y y x x x 3 Will the graph of a straight line always be a function? Give evidence. 4 Give algebraic evidence to show that the relation x2 + y2 = 9 is not a function. B INTERVAL NOTATION, DOMAIN AND RANGEDOMAIN AND RANGE The domain of a relation is the set of permissible values that x may have. The range of a relation is the set of permissible values that y may have.For example: y All values of x > ¡1 are permissible. (1) x So, the domain is fx: x > ¡1g. (-1,-3) All values of y > ¡3 are permissible. So, the range is fy: y > ¡3g. (2) y x can take any value. (2, 1) So, the domain is fx: x is in Rg. x y cannot be > 1 ) range is fy: y 6 1g.
20 FUNCTIONS (Chapter 1) y (3) x can take all values except x = 2: So, the domain is fx: x 6= 2g. Likewise, the range is fy: y 6= 1g. y=1 x x=2The domain and range of a relation are best described where appropriate using intervalnotation. yFor example: The domain consists of all real range x such that x > 3 and we write this as 2 (3, 2) fx : x > 3g x 3 domain the set of all such thatLikewise the range would be fy: y > 2g. profit ($) For this profit function: 100 ² the domain is fx: x > 0g range ² the range is fy: y 6 100g. items made (x) 10 domainIntervals have corresponding graphs.For example:fx: x > 3g or [3, 1[ is read “the set of all x such that x is greater than or equal to 3” and has number line graph x 3fx: x < 2g or ]¡1, 2[ has number line graph x 2fx: ¡2 < x 6 1g or ]¡2, 1] has number line graph x -2 1fx: x 6 0 or x > 4gi.e., ]¡1, 0] or ]4, 1[ has number line graph 0 4 xNote: a b for numbers between a and b we write a < x < b or ]a, b[. for numbers ‘outside’ a and b we write x < a or x > b a b i.e., ]¡1, a[ or ]b, 1[.
FUNCTIONS (Chapter 1) 21 Example 2 For each of the following graphs state the domain and range: a y b y (4, 3) x x (8,-2) (2,-1) a Domain is fx: x 6 8g. b Domain is fx: x is in Rg. Range is fy: y > ¡2g. Range is fy: y > ¡1g.EXERCISE 1B1 For each of the following graphs find the domain and range: a y b y c y (-1, 3) (5, 3) (-1, 1) x x x y = -1 x=2 d y e y f y (Qw_ , 6 Qr_) (0, 2) x x x (1, -1) g y h y i y (-1, 2) x x -1 x (2, -2) y =-2 x =-2 x=2 (-4,-3)2 Use a graphics calculator to help sketch carefully the graphs of the following functions and find the domain and range of each: p 1 p a f(x) = x b f(x) = 2 c f(x) = 4 ¡ x x 1 d y = x2 ¡ 7x + 10 e y = 5x ¡ 3x2 f y =x+ x
22 FUNCTIONS (Chapter 1) x+4 3x ¡ 9 g y= h y = x3 ¡ 3x2 ¡ 9x + 10 i y= x¡2 x2 ¡x¡2 1 j y = x2 + x¡2 k y = x3 + l y = x4 + 4x3 ¡ 16x + 3 x3 C FUNCTION NOTATIONFunction machines are sometimes used to illustrate how functions behave.For example: x So, if 4 is fed into the machine, 2(4) + 3 = 11 comes out. I double the input and then add 3 2x + 3The above ‘machine’ has been programmed to perform a particular function.If f is used to represent that particular function we can write: f is the function that will convert x into 2x + 3.So, f would convert 2 into 2(2) + 3 = 7 and ¡4 into 2(¡4) + 3 = ¡5.This function can be written as: f : x` ! 2x + 3 function f such that x is converted into 2x + 3Two other equivalent forms we use are: f(x) = 2x + 3 or y = 2x + 3So, f(x) is the value of y for a given value of x, i.e., y = f(x).Notice that for f (x) = 2x + 3, f (2) = 2(2) + 3 = 7 and f(¡4) = 2(¡4) + 3 = ¡5:Consequently, f (2) = 7 indicates that the point y (2, 7) lies on the graph of the function. (2, 7)Likewise f (¡4) = ¡5 indicates that the ƒ(x) = 2x + 3 point (¡4, ¡5) also lies on the graph. 3 3 x (-4,-5)
FUNCTIONS (Chapter 1) 23Note: ² f (x) is read as “f of x” and is the value of the function at any value of x. ² If (x, y) is any point on the graph then y = f (x). ² ! f is the function which converts x into f (x), i.e., f : x ` f (x). ² f (x) is sometimes called the image of x. Example 3 If f : x ` 2x2 ¡ 3x, find the value of: ! a f(5) b f (¡4) f(x) = 2x2 ¡ 3x a f(5) = 2(5)2 ¡ 3(5) freplacing x by (5)g = 2 £ 25 ¡ 15 = 35 b f(¡4) = 2(¡4)2 ¡ 3(¡4) freplacing x by (¡4)g = 2(16) + 12 = 44EXERCISE 1C ! 1 If f : x ` 3x + 2, find the value of: a f(0) b f (2) c f (¡1) d f(¡5) e f (¡ 1 ) 3 4 2 If g : x ` x ¡ ! , find the value of: x a g(1) b g(4) c g(¡1) d g(¡4) e g(¡ 1 ) 2 3 If f : x ` 3x ¡ x2 + 2, find the value of: ! a f(0) b f (3) c f (¡3) d f(¡7) e f(3) 2 Example 4 If f (x) = 5 ¡ x ¡ x2 , find in simplest form: a f (¡x) b f(x + 2) a f(¡x) = 5 ¡ (¡x) ¡ (¡x)2 freplacing x by (¡x)g = 5 + x ¡ x2 b f (x + 2) = 5 ¡ (x + 2) ¡ (x + 2)2 freplacing x by (x + 2)g = 5 ¡ x ¡ 2 ¡ [x2 + 4x + 4] = 3 ¡ x ¡ x2 ¡ 4x ¡ 4 = ¡x2 ¡ 5x ¡ 1 4 If f(x) = 7 ¡ 3x, find in simplest form: a f(a) b f (¡a) c f (a + 3) d f(b ¡ 1) e f (x + 2)
24 FUNCTIONS (Chapter 1) 5 If F (x) = 2x2 + 3x ¡ 1, find in simplest form: a F (x + 4) b F (2 ¡ x) c F (¡x) d F (x2 ) e F (x2 ¡ 1) 2x + 3 6 If G(x) = : x¡4 a evaluate i G(2) ii G(0) iii G(¡ 1 ) 2 b find a value of x where G(x) does not exist c find G(x + 2) in simplest form d find x if G(x) = ¡3: 7 f represents a function. What is the difference in meaning between f and f(x)? 8 If f (x) = 2x , show that f (a)f (b) = f(a + b). 9 Given f(x) = x2 find in simplest form: f(x) ¡ f(3) f(2 + h) ¡ f (2) a b x¡3 h10 If the value of a photocopier t years after purchase is given by V (t) = 9650 ¡ 860t dollars: a find V (4) and state what V (4) means b find t when V (t) = 5780 and explain what this represents c find the original purchase price of the photocopier.11 On the same set of axes draw the graphs of three different functions f (x) such that f(2) = 1 and f (5) = 3:12 Find f (x) = ax + b, a linear function, in which f (2) = 1 and f(¡3) = 11. b13 Find constants a and b where f (x) = ax + and f (1) = 1, f (2) = 5. x14 Given T (x) = ax2 + bx + c, find a, b and c if T (0) = ¡4, T (1) = ¡2 and T (2) = 6: INVESTIGATION FLUID FILLING FUNCTIONS When water is added at a constant rate to a cylindrical container the depth of water in the container is a function of time. DEMO This is because the volume of water water depth added is directly proportional to the time taken to add it. If water was not added at a constant rate the direct proportionality depth would not exist. The depth-time graph for the case of a time cylinder would be as shown alongside.
FUNCTIONS (Chapter 1) 25The question arises: ‘What changes in appearance of the graph occur for different shapedcontainers?’ Consider a vase of conical shape. depth DEMO timeWhat to do:1 For each of the following containers, draw a ‘depth v time’ graph as water is added: a b c d e f g h2 Use the water filling demonstration to check your answers to question 1.3 Write a brief report on the connection between the shape of a vessel and the corre- sponding shape of its depth-time graph. You may wish to discuss this in parts. For example, first examine cylindrical containers, then conical, then other shapes. Slopes of curves must be included in your report.4 Draw possible containers as in question 1 which have the following ‘depth v time’ graphs: a depth b depth c depth time time time d e f depth depth depth time time time
26 FUNCTIONS (Chapter 1) D COMPOSITE FUNCTIONS, f¡±¡g ! ! Given f : x ` f (x) and g : x ` g(x), then the composite function of f and g will convert x into f(g(x)). f ± g is used to represent the composite function of f and g. f ± g means f following g and (f ± g)(x) = f(g(x)), i.e., f ± g : x ` f (g(x)). !Consider f : x ` x4 ! ! and g : x ` 2x + 3. DEMOf ± g means that g converts x to 2x + 3 and then f converts (2x + 3) to (2x + 3)4 .This is illustrated by the two function machines below. x g-function machine I double and then add 3 2x + 3 f-function machine 2x + 3 I raise a number to the power 4 (2!+3)VAlgebraically, if f(x) = x4 and g(x) = 2x + 3, then (f ± g)(x) = f (g(x)) = f (2x + 3) fg operates on x firstg = (2x + 3)4 ff operates on g(x) nextgLikewise, (g ± f )(x) = g(f (x)) = g(x4 ) ff operates on x firstg = 2(x4 ) + 3 fg operates on f (x) nextg = 2x4 + 3So, in general, f (g(x)) 6= g(f(x)).The ability to break down functions into composite functions is useful in differential calculus. Example 5 Given f : x ` 2x + 1 and g : x ` 3 ¡ 4x find in simplest form: ! ! a (f ± g)(x) b (g ± f )(x)
FUNCTIONS (Chapter 1) 27 f(x) = 2x + 1 and g(x) = 3 ¡ 4x a ) (f ± g)(x) = f (g(x)) b (g ± f)(x) = g(f(x)) = f (3 ¡ 4x) = g(2x + 1) = 2(3 ¡ 4x) + 1 = 3 ¡ 4(2x + 1) = 6 ¡ 8x + 1 = 3 ¡ 8x ¡ 4 = 7 ¡ 8x = ¡8x ¡ 1Note: If f(x) = 2x + 1 then f(¢) = 2(¢) + 1 f(¤) = 2(¤) + 1 and f(3 ¡ 4x) = 2(3 ¡ 4x) + 1EXERCISE 1D 1 Given f : x ` 2x + 3 and g : x ` 1 ¡ x, find in simplest form: ! ! a (f ± g)(x) b (g ± f)(x) c (f ± g)(¡3) 2 Given f : x ` x2 ! and g : x ` 2 ¡ x find (f ± g)(x) and (g ± f )(x). ! 3 Given f : x ` x2 + 1 and g : x ` 3 ¡ x, find in simplest form: ! ! a (f ± g)(x) b (g ± f)(x) c x if (g ± f )(x) = f (x) 4 a If ax + b = cx + d for all values of x, show that a = c and b = d. (Hint: If it is true for all x, it is true for x = 0 and x = 1.) b Given f (x) = 2x + 3 and g(x) = ax + b and that (f ± g)(x) = x for all values of x, deduce that a = 1 and b = ¡ 3 . 2 2 c Is the result in b true if (g ± f)(x) = x for all x? E THE RECIPROCAL FUNCTION x 1 x 1 1x `! , i.e., f (x) = is defined as the reciprocal function. x xIt has graph: Notice that: y 1 y=-x y=x ² f(x) = is meaningless when x = 0 x 1 (1, 1) ² The graph of f (x) = exists in the first x x and third quadrants only. (-1,-1) 1 ² f(x) = is symmetric about y = x and x y = ¡x GRAPHING PACKAGE 1 The two branches of y = x
28 FUNCTIONS (Chapter 1) 1 ² f (x) = is asymptotic (approaches) ² as x ! 1, f(x) ! 0 (above) x to the x-axis and to the y-axis. as x ! ¡1, f(x) ! 0 (below) as y ! 1, x ! 0 (right) as y ! ¡1, x ! 0 (left) ! reads approaches or tends toEXERCISE 1E 1 2 4 1 Sketch the graph of f (x) = , g(x) = , h(x) = on the same set of axes. x x x Comment on any similarities and differences. 1 2 4 2 Sketch the graphs of f(x) = ¡ , g(x) = ¡ , h(x) = ¡ on the same set of axes. x x x Comment on any similarities and differences. F INVERSE FUNCTIONSA function y = f (x) may or may not have an inverse function. If y = f(x) has an inverse function, this new function ² must indeed be a function, i.e., satisfy the vertical line test and it ² must be the reflection of y = f (x) in the line y = x. The inverse function of y = f(x) is denoted by y = f ¡1 (x). If (x, y) lies on f , then (y, x) lies on f ¡1 . So reflecting the function in y = x has the algebraic effect of interchanging x and y, e.g., f : y = 5x + 2 becomes f ¡1 : x = 5y + 2.For example, y=ƒ(x) y = f ¡1 (x) is the inverse of y y = f (x) as 1 y=ƒ -1(x) ² it is also a function ² it is the reflection of y = f (x) 1 x in the oblique line y = x. y=x This is the reflection of y = f (x) in y = x, but it is not the inverse y function of y = f (x) as it fails the y=ƒ(x) vertical line test. x We say that the function y = f (x) y=ƒ(x), x¡>¡0 does not have an inverse. Note: y = f (x) subject to x > 0 y=x does have an inverse function. y=ƒ -1(x), y¡>¡0 Also, y = f(x) subject to x 6 0 does have an inverse function. y=x
FUNCTIONS (Chapter 1) 29 Example 6 ! Consider f : x ` 2x + 3. a On the same axes, graph f and its inverse function f ¡1 . b Find f ¡1 (x) using i coordinate geometry and the slope of f ¡1 (x) from a ii variable interchange. a f(x) = 2x + 3 passes through (0, 3) and (2, 7). ) f ¡1 (x) passes through (3, 0) and (7, 2). y y=ƒ(x) 2¡0 1 y=x b i This line has slope = . (2, 7) 7¡3 2 So, its equation is (0, 3) y=ƒ -1(x) y¡0 1 = (7, 2) x¡3 2 (3, 0) x x¡3 i.e., y = 2 ¡1 x¡3 i.e., f (x) = 2 ii f is y = 2x + 3, so f ¡1 is x = 2y + 3 ) x ¡ 3 = 2y x¡3 x¡3 ) =y i.e., f ¡1 (x) = 2 2Note: If f includes point (a, b) then f ¡1 includes point (b, a), i.e., the point obtained by interchanging the coordinates.EXERCISE 1F ! 1 Consider f : x ` 3x + 1. a On the same axes graph y = x, f and f ¡1 . b Find f ¡1 (x) using coordinate geometry and a. c Find f ¡1 (x) using variable interchange. x+2 2 Consider f : x `! . 4 a On the same set of axes graph y = x, f and f ¡1 . b Find f ¡1 (x) using coordinate geometry and a. c Find f ¡1 (x) using variable interchange. 3 For each of the following functions f i find f ¡1 (x) ii sketch y = f (x), y = f ¡1 (x) and y = x on the same axes: 3 ¡ 2x a ! f : x ` 2x + 5 b f: x`! c ! f : x ` x+3 4
30 FUNCTIONS (Chapter 1) 4 Copy the graphs of the following functions and in each case include the graphs of y = x and y = f ¡1 (x). y y y a b c 5 5 x x 4 -2 x (3,-6) d y e y f y 2 1 1 -3 x x x 2 5 a Sketch the graph of f : x ` x2 ¡ 4 and reflect it in the line y = x. ! b Does f have an inverse function? c Does f where x > 0 have an inverse function? 6 Sketch the graph of f : x ` x3 ! and its inverse function f ¡1 (x). 7 The ‘horizontal line test’ says that: for a function to have an inverse function, no horizontal line can cut it more than once. a Explain why this is a valid test for the existence of an inverse function. b Which of the following functions have an inverse function? i ii iii y y y 1 -1 x x x 2 Example 7 Consider f : x ` x2 where x > 0. ! ¡1 a Find f (x). b Sketch y = f (x), y = x and y = f ¡1 (x) on the same set of axes. a f is defined by y = x2 , x > 0 b @=!X !>0 ¡1 2 y ) f is defined by x = y , y > 0 p ) y = § x, y > 0 @=~`! p i.e., y = x x p fas ¡ x is 6 0g p So, f ¡1 (x) = x y=x
FUNCTIONS (Chapter 1) 31 8 Consider f : x ` x2 where x 6 0. ! ¡1 a Find f (x). b Sketch y = f(x), y = x and y = f ¡1 (x) on the same set of axes. 9 a Explain why f : x ` x2 ¡ 4x + 3 is a function but does not have an inverse ! function. b Explain why f for x > 2 has an inverse function. p c Show that the inverse function of the function in b is f ¡1 (x) = 2 + 1 + x. d If the domain of f is restricted to x > 2, state the domain and range of i f ii f ¡1 .10 Consider f (x) = 1 x ¡ 1. 2 a Find f ¡1 (x). b Find i (f ± f ¡1 )(x) ii (f ¡1 ± f )(x).11 Given f : x ` (x + 1)2 + 3 where x > ¡1, ! a find the defining equation of f ¡1 b sketch, using technology, the graphs of y = f (x), y = x and y = f ¡1 (x) c state the domain and range of i f ii f ¡1 . 8¡x !12 Consider the functions f : x ` 2x + 5 and g : x `! . 2 a Find g ¡1 (¡1). b Solve for x the equation (f ± g ¡1 )(x) = 9. p13 Given f : x ` 5x ! and g : x ` ! x, a find i f(2) ii g ¡1 (4) b solve the equation (g ¡1 ± f )(x) = 25.14 Given f : x `! 2x and g : x ` 4x ¡ 3 show that ! ¡1 ¡1 ¡1 (f ± g )(x) = (g ± f) (x).15 Which of these functions are their own inverses, that is f ¡1 (x) = f(x)? 1 6 a f(x) = 2x b f (x) = x c f(x) = ¡x d f(x) = e f (x) = ¡ x x G THE IDENTITY FUNCTIONIn question 10 of the previous exercise we considered f (x) = 1 x ¡ 1. 2We found that f ¡1 (x) = 2x + 2 and that (f ± f ¡1 )(x) = x and (f ¡1 ± f )(x) = x. e(x) = x is called the identity function of function y = f (x) ¡1 ¡1 It is the unique solution of (f ± f )(x) = (f ± f )(x) = e(x).
32 FUNCTIONS (Chapter 1)EXERCISE 1G 1 For f (x) = 3x + 1, find f ¡1 (x) and show that (f ± f ¡1 )(x) = (f ¡1 ± f )(x) = x. x+3 2 For f (x) = , find f ¡1 (x) and show that (f ± f ¡1 )(x) = (f ¡1 ± f )(x) = x. 4 p 3 For f (x) = x, find f ¡1 (x) and show that (f ± f ¡1 )(x) = (f ¡1 ± f )(x) = x. 4 y a B is the image of A under a reflection in the B line y = x. y = f -1( x) If A is (x, f(x)), what are the coordinates of A B under the reflection? x b Substitute your result from a into y = f ¡1 (x). What result do you obtain? y = f (x) c Explain how to establish that f (f ¡1 (x)) = x also. REVIEW SET 1A 1 Draw a graph to show what happens in the following jar-water-golf ball situation: Water is added to an empty jar at a constant rate for two minutes and then one golf ball is added. After one minute another golf ball is added. Two minutes later both golf balls are removed. Half the water is then removed at a constant rate over a two minute period. 2 If f(x) = 2x ¡ x2 find: a f (2) b f(¡3) c f(¡ 1 ) 2 3 For the following graphs determine: i the range and domain ii the x and y-intercepts iii whether it is a function. a y b y 1 x -1 x -1 5 -Wl_T_ -3 (2,-5) 4 For each of the following graphs find the domain and range: a y b y (3, 2) 3 1 (3, 1) (-2, 1) x x -1 5 If h(x) = 7 ¡ 3x: a find in simplest form h(2x ¡ 1) b find x if h(2x ¡ 1) = ¡2 6 If f (x) = ax + b where a and b are constants, find a and b for f(1) = 7 and f(3) = ¡5: