Nelson Functions 11 Textbook PDF

Nelson Series Author and Senior Consultant Marian Small Lead Author Chris Kirkpatrick Authors Barb...

Nelson Series Author and Senior Consultant Marian Small Lead Author Chris Kirkpatrick Authors Barbara Alldred Andrew Dmytriw Shawn Godin Angelo Lillo David Pilmer Susanne Trew Noel Walker Australia Canada Mexico Singapore Spain United Kingdom United States  Functions 11 Series Author and Senior Authors Senior Consultant Consultant Barbara Alldred, Andrew Dmytriw, David Zimmer Marian Small Shawn Godin, Angelo Lillo, David Pilmer, Susanne Trew, Math Consultant Lead Author Noel Walker Kaye Appleby Chris Kirkpatrick Contributing Authors Kathleen Kacuiba, Ralph Montesanto General Manager, Mathematics, Contributing Editors Interior Design Science, and Technology Alasdair Graham, First Folio Peter Papayanakis Lenore Brooks Resource Group, Inc., Tom Shields, David Gargaro, Robert Templeton, Cover Design Publisher, Mathematics First Folio Resource Group, Inc., Eugene Lo Colin Garnham Caroline Winter Cover Image Associate Publisher, Mathematics Editorial Assistant The famous thrust stage of the Sandra McTavish Caroline Winter Stratford Festival of Canada’s Festival Theatre. 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Or you can visit our Internet site at http://www.nelson.com For permission to use material from this text or product, submit a request online at www.thomsonrights.com Reviewers and Advisory Panel Paul Alves Halyna Kopach Kathy Pilon Mathematics Department Head Mathematics Department Head Mathematics Program Leader Stephen Lewis Secondary School Mary Ward Catholic Secondary School St. John Catholic High School Peel District School Board Toronto Catholic District School Board Catholic District School Board of Mississauga, ON Toronto, ON Eastern Ontario Perth, ON Terri Blackwell Richard Long Mathematics Teacher Mathematics Department Head Joshua Plant Burlington Central High School Centennial Secondary School Mathematics Teacher Halton District School Board Hastings and Prince Edward District Huntsville High School Burlington, ON School Board Trillium Lakelands District School Board Belleville, ON Huntsville, ON Karen Bryan Program Resource Teacher, Numeracy & Frank Maggio Margaret Russo Literacy 7–12 Department Head of Mathematics Mathematics Teacher Upper Canada District School Board Holy Trinity High School Madonna Catholic Secondary School Brockville, ON Halton Catholic District School Board Toronto Catholic District School Board Oakville, ON Toronto, ON Angela Conetta Mathematics Teacher Peter Matijosaitis Patricia Steele Chaminade College School Mathematics Department Head Mathematics Resource Teacher Toronto Catholic District School Board Loretto Abbey Catholic Secondary Simcoe County District School Board Toronto, ON School Midhurst, ON Toronto Catholic District School Board Justin De Weerdt Toronto, ON Scott Taylor Mathematics Department Head Head of Mathematics, Business, and Huntsville High School Cheryl McQueen Computer Science Trillium Lakelands District School Board Mathematics Learning Coordinator, Bell High School Huntsville, ON 7–12 Ottawa-Carleton District School Board Thames Valley District School Board Nepean, ON Robert Donato London, ON Secondary Mathematics Resource Salvatore Trabona Teacher Ian McTavish Mathematics Department Head Toronto Catholic District School Board Librarian/Mathematics Teacher Madonna Catholic Secondary School Toronto, ON Huntsville High School Toronto Catholic District School Board Trillium Lakelands District School Board Toronto, ON Richard Gallant Huntsville, ON Secondary Curriculum Consultant Dave Wright Simcoe Muskoka Catholic District Grace Mlodzianowski Mathematics Teacher School Board Mathematics Department Head Woburn Collegiate Institute Barrie, ON Cardinal Newman High School Toronto District School Board Toronto Catholic District School Board Toronto, ON Jacqueline Hill Toronto, ON K–12 Mathematics Facilitator Krista Zupan Durham District School Board Elizabeth Pattison Resource Teacher for Student Success Whitby, ON Head of Mathematics Durham Catholic District School Board Westlane Secondary School Oshawa, ON Punitha Kandasamy District School Board of Niagara Classroom Teacher (Secondary— Niagara Falls, ON Mathematics) Mississauga Secondary School Peel District School Board Mississauga, ON NEL v Table of Contents Chapter 1: Introduction to Functions 1 2.7 Adding and Subtracting Rational Expressions 124 Getting Started 2 Chapter Review 131 1.1 Relations and Functions 4 Chapter Self-Test 134 Curious Math 13 Chapter Task 135 1.2 Function Notation 14 1.3 Exploring Properties of Parent Functions 25 Chapter 3: Quadratic Functions 136 1.4 Determining the Domain and Range of a Function 29 Getting Started 138 Mid-Chapter Review 38 3.1 Properties of Quadratic Functions 140 1.5 The Inverse Function and Its Properties 41 3.2 Determining Maximum and Minimum Values of a Quadratic Function 148 1.6 Exploring Transformations of Parent Functions 50 3.3 The Inverse of a Quadratic Function 155 1.7 Investigating Horizontal Stretches, 3.4 Operations with Radicals 163 Compressions, and Reflections 52 Mid-Chapter Review 169 1.8 Using Transformations to Graph Functions of the Form y ! af [k(x " d )] # c 61 Curious Math 171 3.5 Quadratic Function Models: Solving Chapter Review 74 Quadratic Equations 172 Chapter Self-Test 78 3.6 The Zeros of a Quadratic Function 179 Chapter Task 79 3.7 Families of Quadratic Functions 187 3.8 Linear–Quadratic Systems 194 Chapter 2: Equivalent Algebraic Chapter Review 200 Expressions 80 Chapter Self-Test 204 Getting Started 82 Chapter Task 205 2.1 Adding and Subtracting Polynomials 84 2.2 Multiplying Polynomials 91 Chapters 1–3 Cumulative Review 206 Curious Math 97 Chapter 4: Exponential Functions 210 2.3 Factoring Polynomials 98 Getting Started 212 Mid-Chapter Review 105 4.1 Exploring Growth and Decay 214 2.4 Simplifying Rational Functions 108 4.2 Working with Integer Exponents 217 2.5 Exploring Graphs of Rational Functions 115 4.3 Working with Rational Exponents 224 2.6 Multiplying and Dividing Rational Expressions 117 4.4 Simplifying Algebraic Expressions Involving Exponents 231 NEL Table of Contents vii Mid-Chapter Review 238 6.1 Periodic Functions and Their Properties 346 4.5 Exploring the Properties of Exponential 6.2 Investigating the Properties of Sinusoidal Functions 240 Functions 357 4.6 Transformations of Exponential Functions 244 6.3 Interpreting Sinusoidal Functions 365 4.7 Applications Involving Exponential Mid-Chapter Review 374 Functions 254 6.4 Exploring Transformations of Sinusoidal Curious Math 264 Functions 377 Chapter Review 265 6.5 Using Transformations to Sketch the Graphs of Sinusoidal Functions 380 Chapter Self-Test 270 6.6 Investigating Models of Sinusoidal Chapter Task 271 Functions 386 6.7 Solving Problems Using Sinusoidal Models 394 Chapter 5: Trigonometric Ratios 272 Curious Math 402 Getting Started 274 Chapter Review 403 5.1 Trigonometric Ratios of Acute Angles 276 Chapter Self-Test 406 5.2 Evaluating Trigonometric Ratios for Special Chapter Task 407 Angles 283 Curious Math 288 Chapters 4–6 Cumulative Review 408 5.3 Exploring Trigonometric Ratios for Angles Greater than 90° 289 Chapter 7: Discrete Functions: 5.4 Evaluating Trigonometric Ratios for Any Sequences and Series 412 Angle Between 0° and 360° 293 Getting Started 414 Mid-Chapter Review 302 7.1 Arithmetic Sequences 416 5.5 Trigonometric Identities 305 7.2 Geometric Sequences 426 5.6 The Sine Law 312 7.3 Creating Rules to Define Sequences 433 5.7 The Cosine Law 321 7.4 Exploring Recursive Sequences 441 5.8 Solving Three-Dimensional Problems by Using Trigonometry 328 Curious Math 444 Mid-Chapter Review 445 Chapter Review 336 7.5 Arithmetic Series 448 Chapter Self-Test 340 7.6 Geometric Series 454 Chapter Task 341 7.7 Pascal's Triangle and Binomial Expansions 462 Chapter Review 467 Chapter 6: Sinusoidal Functions 342 Chapter Self-Test 470 Getting Started 344 Chapter Task 471 viii Table of Contents NEL Chapter 8: Discrete Functions: Financial Chapter Review 532 Applications 472 Chapter Self-Test 536 Getting Started 474 Chapter Task 537 8.1 Simple Interest 476 8.2 Compound Interest: Future Value 483 Chapters 7–8 Cumulative Review 538 8.3 Compound Interest: Present Value 493 Appendix A: Review of Essential Skills Curious Math 500 and Knowledge 540 Mid-Chapter Review 501 Appendix B: Review of Technical Skills 576 8.4 Annuities: Future Value 504 Glossary 610 8.5 Annuities: Present Value 513 Answers 617 8.6 Using Technology to Investigate Financial Problems 523 Index 687 Credits 690 NEL Table of Contents ix Help wanted: NEL  Chapter 1 Introduction to Functions GOALS You will be able to Identify a function as a special type of relation Recognize functions in various representations and use function notation Explore the properties of some basic functions and apply transformations to those functions Investigate the inverse of a linear function and its properties ? Anton needs a summer job. How would you help him compare the two offers he has received? NEL 1  1 Getting Started SKILLS AND CONCEPTS You Need 1. Simplify each expression. Study Aid 1 2 3 a) 3(x 1 y) 2 5(x 2 y) c) (x 1 1) 2 (x 2 2 1) For help, see the Essential Skills 2 2 Appendix. b) (4x 2 y) (4x 1 y) d) 4x(x 1 2) 2 2x(x 2 4) Question Appendix 2. Evaluate each expression in question 1 when x 5 3 and y 5 25. 1 A-8 3. Solve each linear equation. 2 A-7 5 3 a) 5x 2 8 5 7 c) y 2 y 5 23 6 4 4 A-5 x22 2x 1 1 5 A-15 b) 22(x 2 3) 5 2(1 2 2x) d) 5 4 3 6 A-12 4. Graph each linear relation. 7 A-14 a) y 5 2x 2 3 b) 3x 1 4y 5 12 8 A-9, A-10 5. Graph each circle. a) x2 1 y2 5 9 b) 3x2 1 3y2 5 12 6. Graph each parabola, labelling the vertex and the axis of symmetry. a) y 5 x2 2 6 c) y 5 23(x 1 4) 2 1 2 b) y 5 (x 2 2) 2 2 1 d) y 5 2x2 1 6x 7. For each quadratic relation, list the transformations you need to apply to y 5 x2 to graph the relation. Then sketch the graph. 1 a) y 5 x2 2 2 c) y 5 (x 2 1) 2 2 4 2 b) y 5 24x2 1 3 d) y 5 22(x 1 3) 2 1 5 8. Solve each quadratic equation. a) x 2 2 5x 1 6 5 0 b) 3x 2 2 5 5 70 9. Compare the properties of linear relations, circles, and quadratic relations. Begin by completing a table like the one shown. Then list similarities and differences among the three types of relations. Property Linear Relations Circles Quadratic Relations Equation(s) Shape of graph Number of quadrants graph enters Descriptive features of graph Types of problems modelled by the relation 2 Chapter 1 NEL  Getting Started APPLYING What You Know Fencing a Cornfield YOU WILL NEED graph paper Rebecca has 600 m of fencing for her cornfield. The creek that goes through her farmland will form one side of the rectangular boundary. Rebecca considers different widths to maximize the area enclosed. ? How are the length and area of the field related to its width? A. What are the minimum and maximum values of the width of the field? B. What equations describe each? i) the relationship between the length and width of the field ii) the relationship between the area and width of the field C. Copy and complete this table of values for widths that go from the least to the greatest possible values in intervals of 50 m. Width (m) Length (m) Area (m2) D. Graph the data you wrote in the first two columns. Use width as the independent variable. Describe the graph. What type of relationship is this? E. Now graph the data you wrote in the first and third columns. Use width as the independent variable again. Describe the graph. What type of relationship is this? F. How could you have used the table of values to determine the types of relationships you reported in parts D and E? G. How could you have used the equations from part B to determine the types of relationships you reported in parts D and E? NEL Introduction to Functions 3  1.1 Relations and Functions YOU WILL NEED GOAL graphing calculator or graph paper Recognize functions in various representations. INVESTIGATE the Math Ang recorded the heights and shoe sizes of students in his class. Shoe Size Height (cm) Shoe Size Height (cm) 10 158 8 156 11.5 175 7.5 161 10 173 12 179 9 164 11 178 9 167 10.5 173 10 170 8.5 177 11 172 8 165 8 160 12 182 8 174 13 177 11 175 13 192 8 166 7.5 157 7.5 153 8.5 163 10 171 12 183 11 181 10 168 11 171 11 180 10 170 ? Can you predict a person’s height from his or her shoe size? A. Plot the data, using shoe size as the independent variable. Describe the relationship shown in the scatter plot. B. Use your plot to predict the height of a person with each shoe size. i) 8 ii) 10 iii) 13 Tech Support C. Use your plot to predict what shoe size corresponds to each height. i) 153 cm ii) 173 cm iii) 177 cm For help drawing a line of best fit on a graphing D. Draw a line of good fit on your plot. Write the equation of your line, and calculator, see Technical use it to determine the heights corresponding to the shoe sizes in part B. How Appendix, B-11. are your results different from those in part B? 4 Chapter 1 NEL  1.1 E. Describe the domain and range of the relationship between shoe size and domain height in Ang’s class. the set of all values of the independent variable of a F. Explain why the relation plotted in part A is not a function. relation G. Is the relation drawn in part D a function? Explain. range H. Which of the relations in parts A and D could be used to predict a single the set of all values of the dependent variable of a relation height for a given shoe size? Explain. relation a set of ordered pairs; values of Reflecting the independent variable are I. How did the numbers in the table of values show that the relation was not paired with values of the dependent variable a function? function J. How did the graph of the linear function you drew in part D differ from the a relation where each value of graph of the relation you plotted in part A? the independent variable K. Explain why it is easier to use the linear function than the scatter plot of the corresponds with only one value of the dependent variable actual data to predict height. APPLY the Math EXAMPLE 1 Representing functions in different ways The ages and soccer practice days of four students are listed. Student Age Soccer Practice Day Communication Tip Use braces to list the values, or Craig 15 Tuesday elements, in a set. Magda 16 Tuesday For example, the set of the Stefani 15 Thursday first five even numbers is Amit 17 Saturday {2, 4, 6, 8, 10}. For each of the given relations, state the domain and range and then determine whether or not the relations are functions. a) students and the day for soccer practice b) ages and the day for soccer practice Jenny’s Solution: Using Set Notation I wrote the relation as a set of ordered pairs, (student’s a) {(Craig, Tuesday), name, day for practice). I wrote the domain by listing the (Magda, Tuesday), students’ names—the independent variable, or first (Stefani, Thursday), elements, in each ordered pair. (Amit, Saturday)} I listed the day for practice—the dependent variable, or Domain 5 {Craig, Magda, Stefanie, Amit} second elements—to write the range. Range 5 {Tuesday, Thursday, Saturday} NEL Introduction to Functions 5  Each element of the domain corresponds with only Each student has only one practice day, so the relation is one element in the range, so the relation between a function. In this case, if I know the student’s name, I students and their soccer practice day is a function. can predict his or her practice day. The first elements appear only once in the list of ordered pairs. No name is repeated. I noticed that one 15-year-old practiced on Tuesday, but b) {(15, Tuesday), (16, Tuesday), another practiced on Thursday, so I can’t predict a (15, Thursday), (17, Saturday)} practice day just by knowing the age. This is not a Domain 5 {15, 16, 17} function. Range 5 {Tuesday, Thursday, Saturday} 15 in the domain corresponds with two different days in the range, so this relation is not a function. Olivier’s Solution: Using a Mapping Diagram a) Student Practice day I drew a diagram of the relation between students and soccer practice days by listing the student names in an Craig Sat oval and the days in another oval. Then I drew arrows to Magda match the students with their practice days. The diagram Thurs Stefani is called a mapping diagram, since it maps the elements Amit Tues of the domain onto the elements of the range. Domain 5 {Craig, Magda, Stefanie, Amit} The elements in the left oval are the values of the Range 5 {Tuesday, Thursday, Saturday} independent variable and make up the domain. The elements in the right oval are the values of the dependent variable and make up the range. I wrote the domain and range by listing what was in each oval. Each element of the domain has only one The relation is a function because each student name has corresponding element in the range, so the relation only one arrow leaving it. is a function. b) Age Practice day I drew another mapping diagram for the age and practice day relation. I matched the ages to the 15 Sat practice days. 16 Thurs 17 Tues Domain 5 {15, 16, 17} Range 5 {Tuesday, Thursday, Saturday} Two arrows go from 15 to two different days. This cannot The value 15 of the independent variable, age, maps be a function. An element of the domain can’t map to to two different values of the dependent variable, days. two elements in the range. This relation is not a function. 6 Chapter 1 NEL  1.1 EXAMPLE 2 Selecting a strategy to recognize functions in graphs Determine which of the following graphs are functions. a) y b) y 5 4 2 x x !5 0 5 !4 !2 0 2 4 6 !2 !4 !5 Ken’s Solution a) y I used the vertical-line test to see how vertical-line test 5 many points on the graph there were if any vertical line intersects the for each value of x. graph of a relation more than once, then the relation is not a x An easy way to do this was to use a function 0 ruler to represent a vertical line and !5 5 move it across the graph. The ruler crossed the graph in two !5 places everywhere except at the leftmost and rightmost ends of the circle. This showed that there are At least one vertical line drawn on x-values in the domain of this relation the graph intersects the graph at that correspond to two y-values in the two points. This is not the graph of range. a function. b) y 4 I used the vertical-line test again. Wherever I placed my ruler, the vertical 2 x line intersected the graph in only one place. This showed that each x-value in !4 !2 0 2 4 6 !2 the domain corresponds with only one y-value in the range. !4 Any vertical line drawn on the graph intersects the graph at only one point. This is the graph of a function. NEL Introduction to Functions 7  EXAMPLE 3 Using reasoning to recognize a function from an equation Determine which equations represent functions. a) y 5 2x 2 5 b) x 2 1 y 2 5 9 c) y 5 2x 2 2 3x 1 1 Keith’s Solution: Using the Graph Defined by its Equation a) This equation defines the graph of a linear function I used my graphing calculator and entered with a positive slope. Its graph is a straight line that y 5 2x 2 5. I graphed the function and checked it increases from left to right. with the vertical-line test. This graph passes the vertical-line test, showing that for each x-value in the domain there is only one y-value in the range. This is the graph of a function. y 5 2x 2 5 is a function. b) This equation defines the graph of a circle centred at I used my graphing calculator and entered the upper (0, 0) with a radius of 3. half of the circle in Y1 and the lower half in Y2. Then I applied the vertical-line test to check. This graph fails the vertical-line test, showing that there are x-values in the domain of this relation that correspond to two y-values in the range. This is not the graph of a function. x 2 1 y 2 5 9 is not a function. c) This equation defines the graph of a parabola I used my graphing calculator to enter that opens upward. y 5 2x 2 2 3x 1 1 and applied the vertical-line test to check. This graph passes the vertical-line test, showing that for each x-value in the domain there is only one y-value in the range. This is the graph of a function. y 5 2x2 2 3x 1 1 is a function. 8 Chapter 1 NEL  1.1 Mayda’s Solution: Substituting Values a) For any value of x, the equation y 5 2x 2 5 I substituted numbers for x in the equation. produces only one value of y. For example, No matter what number I substituted for x, I got only one y 5 2(1) 2 5 5 23 answer for y when I doubled the number for x and then subtracted 5. This equation defines a function. b) Substitute 0 for x in the equation I substituted 0 for x in the equation and solved for y. x 2 1 y 2 5 9. I used 0 because it’s an easy value to calculate with. (0) 2 1 y 2 5 9 I got two values for y with x " 0. y 5 3 or 23 There are two values for y when x 5 0, so the equation defines a relation, but not a function. No matter what number I choose for x, I get only one c) Every value of x gives only one value of y in the number for y that satisfies the equation. equation y 5 2x 2 2 3x 1 1. This equation represents a function. In Summary Key Ideas y A function is a relation in which each value of the independent variable corresponds with only one value of the dependent variable. Functions can be represented in various ways: in words, a table of values, x a set of ordered pairs, a mapping diagram, a graph, or an equation. Need To Know The domain of a relation or function is the set of all values of the independent variable. This is usually represented by the x-values on a coordinate grid. The range of a relation or function is the set of all values of the dependent A relation that is a function variable. This is usually represented by the y-values on a coordinate grid. You can use the vertical-line test to check whether a graph represents a y function. A graph represents a function if every vertical line intersects the graph in at most one point. This shows that there is only one element in the range for each element of the domain. x You can recognize whether a relation is a function from its equation. If you can find even one value of x that gives more than one value of y when you substitute x into the equation, the relation is not a function. Linear relations, which have the general forms y 5 mx 1 b or Ax 1 By 5 C and whose graphs are straight lines, are all functions. Vertical lines are not functions but horizontal lines are. Quadratic relations, which have the general forms y 5 ax2 1 bx 1 c or y 5 a(x 2 h) 2 1 k and whose A relation that is not a function graphs are parabolas, are also functions. NEL Introduction to Functions 9  CHECK Your Understanding 1. State which relations are functions. Explain. a) 5 (25, 1), (23, 2), (21, 3), (1, 2)6 b) !1 !3 1 !1 3 0 5 2 c) 5 (0, 4), (3, 5), (5, 22), (0, 1)6 d) !4 1 !2 2 6 5 2. Use a ruler and the vertical-line test to determine which graphs are functions. a) y d) y 4 4 2 2 x x !4 !2 0 2 4 !4 !2 0 2 4 !2 !2 !4 !4 b) y e) y 4 4 2 2 x x !4 !2 0 2 4 !4 !2 0 2 4 !2 !2 !4 !4 c) y f) y 4 4 2 2 x x !4 !2 0 2 4 !4 !2 0 2 4 !2 !2 !4 !4 3. Substitute 26 for x in each equation and solve for y. Use your results to explain why y 5 x 2 2 5x is a function but x 5 y 2 2 5y is not. 10 Chapter 1 NEL  1.1 PRACTISING 4. The grades and numbers of credits for students are listed. K Student Grade Number of Credits Barbara 10 8 Pierre 12 25 Kateri 11 15 Mandeep 11 18 Elly 10 16 a) Write a list of ordered pairs and create a mapping diagram for the relation between students and grades grades and numbers of credits students and numbers of credits b) State the domain and range of each relation in part (a). c) Which relations in part (a) are functions? Explain. 5. Graph the relations in question 4. Then use the vertical-line test to confirm your answers to part (c). 6. Describe the graphs of the relations y 5 3 and x 5 3. Are these relations functions? Explain. 7. Identify each type of relation and predict whether it is a function. Then graph each function and use the vertical-line test to determine whether your prediction was correct. 3 a) y 5 5 2 2x c) y 5 2 (x 1 3) 2 1 1 4 b) y 5 2x 2 2 3 d) x 2 1 y 2 5 25 8. a) Substitute x 5 0 into each equation and solve for y. Repeat for x 5 22. i) 3x 1 4y 5 5 iii) x 2 1 y 5 2 ii) x 2 1 y 2 5 4 iv) x 1 y 2 5 0 b) Which relations in part (a) appear to be functions? c) How could you verify your answer to part (b)? 9. Determine which relations are functions. a) y 5 !x 1 2 c) 3x 2 2 4y 2 5 12 b) y 5 2 2 x d) y 5 23(x 1 2) 2 2 4 10. Use numeric and graphical representations to investigate whether the relation x 2 y 2 5 2 is a function. Explain your reasoning. 11. Determine which of the following relations are functions. A a) The relation between earnings and sales if Olwen earns $400 per week plus 5% commission on sales b) The relation between distance and time if Bran walks at 5 km/h c) The relation between students’ ages and the number of credits earned NEL Introduction to Functions 11  12. The cost of renting a car depends on the daily rental charge and the number of kilometres driven. A graph of cost versus the distance driven over a one-day period is shown. Cost versus Distance Driven y 120 100 80 Cost ($) 60 40 20 x 0 100 200 300 400 500 Distance (km) a) What are the domain and range of this relation? b) Explain why the domain and range have a lower limit. c) Is the relation a function? Explain. 13. a) Sketch a graph of a function that has the set of integers as its domain and T all integers less than 5 as its range. b) Sketch a graph of a relation that is not a function and that has the set of real numbers less than or equal to 10 as its domain and all real numbers greater than 25 as its range. 14. Use a chart like the following to summarize what you have learned about C functions. Definition: Characteristics: Function Examples: Non-examples: Extending 15. A freight delivery company charges $4/kg for any order less than 100 kg and $3.50/kg for any order of at least 100 kg. a) Why must this relation be a function? b) What is the domain of this function? What is its range? c) Graph the function. d) What suggestions can you offer to the company for a better pricing structure? Support your answer. 12 Chapter 1 NEL  1.1 Curious Math Curves of Pursuit A fox sees a rabbit sitting in the middle of a field and begins to run toward the rabbit. The rabbit sees the fox and runs in a straight line to its burrow. The fox continuously adjusts its direction so that it is always running directly toward the rabbit. Burrow Rabbit’s Path F E Fox B A Rabbit If the fox and rabbit are running at the same speed, the fox reaches point E when the rabbit reaches point A. The fox then changes direction to run along line EA. When the fox reaches point F, the rabbit is at B, so the fox begins to run along FB, and so on. The resulting curve is called a curve of pursuit. 1. If the original position of the rabbit represents the origin and the rabbit’s path is along the positive y-axis, is the fox’s path the graph of a function? Explain. 2. a) Investigate what happens by drawing a curve of pursuit if i) the burrow is farther away from the rabbit than it is in the first example ii) the burrow is closer to the rabbit than it is in the first example b) Where does the fox finish in each case? How does the location of the burrow relative to the rabbit affect the fox’s path? c) Will the path of the fox always be a function, regardless of where the rabbit is relative to its burrow? Explain. 3. a) Draw a curve of pursuit in which i) the rabbit runs faster than the fox ii) the fox runs faster than the rabbit b) Are these relations also functions? How do they differ from the one in question 1? NEL Introduction to Functions 13  1.2 Function Notation YOU WILL NEED GOAL graphing calculator Use function notation to represent linear and quadratic functions. LEARN ABOUT the Math The deepest mine in the world, East Rand mine in South Africa, reaches 3585 m into Earth’s crust. Another South African mine, Western Deep, is being deepened to 4100 m. Suppose the temperature at the top of the mine shaft is 11 °C and that it increases at a rate of 0.015 °C>m as you descend. ? What is the temperature at the bottom of each mine? EXAMPLE 1 Representing a situation with a function and using it to solve a problem a) Represent the temperature in a mine shaft with a function. Explain why function notation your representation is a function, and write it in function notation. notation, such as f(x), used to b) Use your function to determine the temperature at the bottom of East Rand represent the value of the and Western Deep mines. dependent variable—the output—for a given value of the Lucy’s Solution: Using an Equation independent variable, x—the input a) An equation for temperature is I wrote a linear equation for the T 5 11 1 0.015d, where T problem. Communication Tip represents the temperature in I used the fact that T starts at 11 °C The notations y and f(x) are degrees Celsius at a depth of d metres. and increases at a steady rate of interchangeable in the 0.015 ° C>m. equation or graph of a function, so y is equal to f(x). The equation represents a function. Since this equation represents a linear The notation f(x) is read “f at Temperature is a function of relationship between temperature and x” or “f of x.” The symbols depth. depth, it is a function. f(x), g(x), and h(x) are often used to name the outputs of I wrote the equation again. T(d) makes In function notation, functions, but other letters are it clearer that T is a function of d. T(d ) 5 11 1 0.015d. also used, such as v(t) for velocity as a function of time. 14 Chapter 1 NEL  1.2 I found the temperature at the b) T(3585) 5 11 1 0.015(3585) bottom of East Rand mine by 5 11 1 53.775 calculating the temperature at a 5 64.775 depth of 3585 m. I substituted 3585 for d in the equation. T(4100) 5 11 1 0.015(4100) For the new mine, I wanted the 5 11 1 61.5 temperature when d 5 4100, so I calculated T(4100). 5 72.5 The temperatures at the bottom of East Rand mine and Western Deep mine are about 65 °C and 73 °C, respectively. Stuart’s Solution: Using a Graph a) T(d ) 5 11 1 0.015d I wrote an equation to show how the temperature changes This is a function because it is a as you go down the mine. linear relationship. I knew that the relationship was linear because the temperature increases at a steady rate. I used d for depth and called the function T(d) for temperature. b) d (m) T(d)(8C) I made a table of values for the 0 T(0) 5 11 1 0.015(0) 5 11 function. 1000 26 I substituted the d-values into the function equation to get 2000 41 the T(d)-values. 3000 56 4000 71 NEL Introduction to Functions 15  Temperature of a Mine I plotted the points (0, 11), y (1000, 26), (2000, 41), 90 (3000, 56), and (4000, 71). 80 Then I joined them with a Approx. 72.5 straight line. Temperature, T(d), (°C) 70 Approx. 64.5 East Rand mine is 3585 m 60 deep. The temperature at the 50 bottom is T(3585). 15d 40 I interpolated to read T(3585) 0.0 from the graph. It was 11 + 30 approximately 65. )= 20 T(d The other mine is 4100 m 10 3585 4100 deep. x 0 1000 By extrapolating, I found that 3000 5000 T(4100) was about 73. Depth, d, (m) The temperature at the bottom of East Rand mine is about 65 °C. The temperature at the bottom of Western Deep mine is about 73 °C. Tech Support For help using a graphing Eli’s Solution: Using a Graphing Calculator calculator to graph and evaluate functions, see Technical a) Let T(d ) represent the I used function notation to Appendix, B-2 and B-3. temperature in degrees Celsius write the equation. at a depth of d metres. T(d ) 5 11 1 0.015d Temperature increases at a steady rate, so it is a function of depth. b) I graphed the function by entering Y1 5 11 1 0.015X into the equation editor. I used WINDOW settings of 0 # X # 5000, Xscl 200, and 0 # Y # 100, Yscl 10. 16 Chapter 1 NEL  1.2 I used the value operation to find the temperature at the bottom of East Rand mine. This told me that T(3585) 5 64.775. Then I used the value operation again to find the temperature at the bottom of the other mine. I found that T(4100) 5 72.5. As a check, I called up the function on my calculator home screen, using VARS and function notation to display both answers. The temperature at the bottom of the East Rand mine is about 65 °C. The temperature at the bottom of Western Deep mine is about 73 °C. Reflecting A. How did Lucy, Stuart, and Eli know that the relationship between temperature and depth is a function? B. How did Lucy use the function equation to determine the two temperatures? C. What does T(3585) mean? How did Stuart use the graph to determine the value of T(3585)? NEL Introduction to Functions 17  APPLY the Math EXAMPLE 2 Representing a situation with a function model A family played a game to decide who got to eat the last piece of pizza. Each person had to think of a number, double it, and subtract the result from 12. Finally, they each multiplied the resulting difference by the number they first thought of. The person with the highest final number won the pizza slice. a) Use function notation to express the final answer in terms of Tim 5 the original number. b) The original numbers chosen by the family members are Rhea 22 shown. Who won the pizza slice? Sara 7 c) What would be the best number to choose? Why? Andy 10 Barbara’s Solution a) x input Double the number 2x I used a flow chart to show what happens to the original number in the Subtract from 12 game. (12 ! 2x) I chose x to be the original number, or input. Multiply by the original number x(12 ! 2x) output f (x) 5 x(12 2 2x) The expression for the final answer is quadratic, so the final result must be 5 12x 2 2x 2 a function of the original number. I chose f(x) as the name for the final answer, or output. 18 Chapter 1 NEL  1.2 b) Tim: f (5) 5 12(5) 2 2(5) 2 I found the values of f(5), f(22), f(7), and f(10) to see 5 60 2 2(25) who had the highest answer. 5 60 2 50 Tim’s answer was 10. 5 10 Rhea: f (22) 5 12(22) 2 2(22) 2 Rhea’s answer was 232. 5 224 2 2(4) 5 224 2 8 5 232 Sara: f (7) 5 12(7) 2 2(7) 2 Sara’s answer was 214. 5 84 2 2(49) 5 84 2 98 5 214 Andy: f (10) 5 12(10) 2 2(10) 2 Andy’s answer was 280. 5 120 2 2(100) 5 120 2 200 5 280 Tim won the pizza slice. Tim’s answer was the highest. c) f (x) 5 12x 2 2x 2 I recognized that the equation was quadratic and that its graph would be a parabola that opened down, since the coefficient of x2 was negative. This meant that this quadratic function had a maximum value at its vertex. I put the equation back in f (x) 5 22x(x 2 6) factored form by dividing out the The x-intercepts are x 5 0 and x 5 6. common factor, 22x. Vertex: x 5 (0 1 6) 4 2 I remembered that the x-coordinate of the vertex is x53 halfway between the two The best number to choose is 3. x-intercepts. I checked my answer by graphing. NEL Introduction to Functions 19  EXAMPLE 3 Connecting function notation to a graph For the function shown in the graph, determine y each value. 2 a) g(3) y = g(x) 1 b) g(21) x c) x if g(x) 5 1 !1 0 1 2 3 4 d) the domain and range of g(x) !1 !2 Ernesto’s Solution a) y 2 I looked at the graph to find the 1 y = g(x) y-coordinate when x 5 3. x I drew a line up to the graph from the !1 0 1 2 3 4 x-axis at x 5 3 and then a line across !1 from that point of intersection to the !2 y-axis. The y-value was 2, so, in function When x 5 3, y 5 2. notation, g(3) 5 2. g(3) 5 2 I saw that y 5 0 when x 5 21, so b) When x 5 21, y 5 0. 21 is the x-intercept and g(21) 5 0. g(21) 5 0 c) g(x) 5 1 when x 5 0 I saw that the graph crosses the y-axis at y 5 1. The x-value is 0 at this point. d) The graph begins at the point (21, 0) and continues upward. I saw that there was no graph to the The graph exists only for x $ 21 left of the point (21, 0) or below and y $ 0. that point. The domain is all real numbers So the only possible x-values are greater than or equal to 21. x $ 21, and the only possible y-values are y $ 0. The range is all real numbers greater than or equal to 0. 20 Chapter 1 NEL  1.2 EXAMPLE 4 Using algebraic expressions in functions Consider the functions f (x) 5 x 2 2 3x and g(x) 5 1 2 2x. a) Show that f (2). g (2), and explain what that means about their graphs. b) Determine g(3b). c) Determine f (c 1 2) 2 g(c 1 2). Jamilla’s Solution a) f (x) 5 x 2 2 3x I substituted 2 for x in both 2 functions. f (2) 5 (2) 2 3(2) 5426 5 22 g(x) 5 1 2 2x g(2) 5 1 2 2(2) 5124 5 23 22. 23, so f (2). g(2) That means that the point on the graph of f (x) is above the point on the graph of g(x) when x 5 2. I substituted 3b for x. b) g(3b) 5 1 2 2(3b) I simplified the equation. 5 1 2 6b c) f (c 1 2) 2 g(c 1 2) 5 3 (c 1 2) 2 2 3(c 1 2) 4 2 31 2 2(c 1 2)4 5 3 (c 2 1 4c 1 4 2 3c 2 6)4 2 31 2 2c 2 44 I substituted c 1 2 for x in both functions. 5 3c 2 1 c 2 24 2 323 2 2c4 I used square brackets to keep the 5 c 2 1 c 2 2 1 3 1 2c functions separate until I had 5 c 2 1 3c 1 1 simplified each one. NEL Introduction to Functions 21  In Summary Key Idea Symbols such as f(x) are called function notation, which is used to represent the value of the dependent variable y for a given value of the independent variable x. For this reason, y and f(x) are interchangeable in the equation of a function, so y 5 f(x). Need to Know f(x) is read “f at x” or “f of x.” input f(a) represents the value or output of a=2 the function when the input is x 5 a. The output depends on the equation of in the function. To evaluate f(a), substitute out a for x in the equation for f(x). f(a) is the y-coordinate of the point on f(a) = f(2) the graph of f with x-coordinate a. For f(x) output example, if f(x) takes the value 3 at x 5 2, then f(2) 5 3 and the point (2, 3) lies on the graph of f. CHECK Your Understanding 1. Evaluate, where f (x) 5 2 2 3x. a) f (2) c) f (24) e) f (a) 1 b) f (0) d) f a b f ) f (3b) 2 2. The graphs of y 5 f (x) and y 5 g(x) are shown. y y 4 4 2 2 x x !4 !2 0 2 4 !4 !2 0 2 4 !2 !2 y = f (x) y = g(x) !4 !4 Using the graphs, evaluate a) f (1) c) f (4) 2 g(22) b) g(22) d) x when f (x) 5 23 22 Chapter 1 NEL  1.2 3. Milk is leaking from a carton at a rate of 3 mL/min. There is 1.2 L of milk in the carton at 11:00 a.m. a) Use function notation to write an equation for this situation. b) How much will be left in the carton at 1:00 p.m.? c) At what time will 450 mL of milk be left in the carton? PRACTISING 4. Evaluate f (21), f (3), and f (1.5) for a) f (x) 5 (x 2 2) 2 2 1 b) f (x) 5 2 1 3x 2 4x 2 1 5. For f (x) 5 , determine 2x 1 3 a) f (23) b) f (0) c) f (1) 2 f (3) d) f a b 1 f a b 4 4 y 6. The graph of y 5 f (x) is shown at the right. y = f(x) a) State the domain and range of f. 6 b) Evaluate. 4 i) f (3) iii) f (5 2 3) 2 ii) f (5) iv) f (5) 2 f (3) x 7. For h(x) 5 2x 2 5, determine !4 !2 0 2 4 6 a) h(a) c) h(3c 2 1) !2 b) h(b 1 1) d) h(2 2 5x) 8. Consider the function g(t) 5 3t 1 5. a) Create a table of values and graph the function. b) Determine each value. i) g(0) iv) g(2) 2 g(1) ii) g(3) v) g(1001) 2 g(1000) iii) g(1) 2 g(0) vi) g(a 1 1) 2 g(a) 9. Consider the function f (s) 5 s 2 2 6s 1 9. a) Create a table of values for the function. b) Determine each value. i) f (0) iv) f (3) ii) f (1) v) 3 f (2) 2 f (1)4 2 3 f (1) 2 f (0)4 iii) f (2) vi) 3 f (3) 2 f (2)4 2 3 f (2) 2 f (1) 4 c) In part (b), what do you notice about the answers to parts (v) and (vi)? y Explain why this happens. 8 10. The graph at the right shows f (x) 5 2(x 2 3) 2 1. 2 y = 2(x–3)2–1 4 K a) Evaluate f (22). x b) What does f (22) represent on the graph of f ? !8 !4 0 4 8 c) State the domain and range of the relation. !4 d) How do you know that f is a function from its graph? !8 11. For g(x) 5 4 2 5x, determine the input for x when the output of g(x) is 3 a) 26 b) 2 c) 0 d) 5 NEL Introduction to Functions 23  12. A company rents cars for $50 per day plus $0.15/km. a) Express the daily rental cost as a function of the number of kilometres travelled. b) Determine the rental cost if you drive 472 km in one day. c) Determine how far you can drive in a day for $80. 13. As a mental arithmetic exercise, a teacher asked her students to think of a A number, triple it, and subtract the resulting number from 24. Finally, they were asked to multiply the resulting difference by the number they first thought of. a) Use function notation to express the final answer in terms of the original number. b) Determine the result of choosing numbers 3, 25, and 10. c) Determine the maximum result possible. 14. The second span of the Bluewater Bridge in Sarnia, Ontario, is supported by T two parabolic arches. Each arch is set in concrete foundations that are on opposite sides of the St. Clair River. The arches are 281 m apart. The top of each arch rises 71 m above the river. Write a function to model the arch. 15. a) Graph the function f (x) 5 3(x 2 1) 2 2 4. b) What does f (21) represent on the graph? Indicate on the graph how you would find f (21). c) Use the equation to determine i) f (2) 2 f (1) ii) 2f (3) 2 7 iii) f (1 2 x) 16. Let f (x) 5 x 2 1 2x 2 15. Determine the values of x for which a) f (x) 5 0 b) f (x) 5 212 c) f (x) 5 216 17. Let f (x) 5 3x 1 1 and g(x) 5 2 2 x. Determine values for a such that a) f (a) 5 g(a) b) f (a 2 ) 5 g(2a) 18. Explain, with examples, what function notation is and how it relates to the C graph of a function. Include a discussion of the advantages of using function notation. Extending 19. The highest and lowest marks awarded on an examination were 285 and 75. All the marks must be reduced so that the highest and lowest marks become 200 and 60. a) Determine a linear function that will convert 285 to 200 and 75 to 60. b) Use the function to determine the new marks that correspond to original marks of 95, 175, 215, and 255. 20. A function f (x) has these properties: The domain of f is the set of natural numbers. f (1) 5 1 f (x 1 1) 5 f (x) 1 3x(x 1 1) 1 1 a) Determine f (2), f (3), f (4), f (5), and f (6). b) Describe the function. 24 Chapter 1 NEL  1.3 Exploring Properties of Parent Functions GOAL YOU WILL NEED graphing calculator or Explore and compare the graphs and equations of five basic functions. graphing technology graph paper EXPLORE the Math Communication Tip !x always means the positive square root of x. As a child, you learned how to recognize different animal families. In mathematics, every function can be classified as a member of a family. Each family member of a family of functions is related to the simplest, or most basic, function a collection of functions (or lines sharing the same characteristics. This function is called the parent function. Here are or curves) sharing common some members of the linear and quadratic families. The parent functions are in green. characteristics parent function Linear Functions Quadratic Functions the simplest, or base, function in a family y y 4 4 y 2 2 4 x x 2 !4 !2 0 2 4 !4 !2 0 2 4 x !2 !2 !4 !2 0 2 4 !4 !4 !2 !4 Parent function: f(x) 5 x Parent function: f(x) 5 x 2 Family members: f(x) 5 mx 1 b Family members: f(x) 5 a(x 2 h) 2 1 k Examples: f(x) 5 3x 1 2, Examples: f(x) 5 5(x 2 3) 2 2 1, f(x) 5 2 12 x 2 2 f(x) 5 2x 2 1 3 Introduction to Functions 25  Three more parent functions are the square root function f (x) 5 !x , the absolute value 1 reciprocal function f (x) 5 x , and the absolute value function f (x) 5 |x|. written as |x|; describes the distance of x from 0; equals x What are the characteristics of these parent functions that ? when x $ 0 or 2x when x , 0; distinguish them from each other? for example, |3| 5 3 and |23| 5 2 (23) 5 3 A. Make a table like the one shown. Equation of Name of Special Features/ Function Function Sketch of Graph Symmetry Domain Range f(x) 5 x linear function y straight line that goes through the origin slope is 1 x divides the plane exactly in half diagonally graph only in quadrants 1 and 3 f(x) 5 x2 quadratic function y parabola that opens up vertex at the origin x y has a minimum value y-axis is axis of symmetry graph only in quadrants 1 and 2 f(x) 5 "x square root function 1 f(x) 5 reciprocal function x f(x) 5 |x| absolute value function Tech Support B. Use your graphing calculator to check the sketches shown for f (x) 5 x and Use the following WINDOW f (x) 5 x 2 and add anything you think is missing from the descriptions. settings to graph the functions: Explain how you know that these equations are both functions. C. In your table, record the domain and range of each of f (x) 5 x and f (x) 5 x 2. D. Clear all equations from the equation editor. Graph the square root function, f (x) 5 !x. In your table, sketch the graph and describe its shape. Is it a function? Explain. How is it different from the graphs of linear and quadratic functions? You can change to these settings by pressing ZOOM 4. 26 Chapter 1 NEL  1.3 E. Go to the table of values and scroll up and down the table. Does ERR: appear in the Y column? Explain why this happens. Tech Support For help with the TABLE F. Using the table of values and the graph, determine and record the domain function of the graphing and range of the function. calculator, see Technical 1 Appendix, B-6. G. Repeat parts D through F for the reciprocal function f (x) 5. Use the table To graph f(x) 5 |x|, press x of values to see what happens to y when x is close to 0 and when x is far from MATH 1 0. Explain why the graph is in two parts with a break in the middle. X,T,U,n ). H. Where are the asymptotes of this graph? asymptote I. Repeat parts D through F for the absolute value function f (x) 5 |x|. Which a line that the graph of a of the other functions is the resulting graph most like? Explain. When you relation or function gets closer have finished, make sure that your table contains enough information for you and closer to, but never meets, to recognize each of the five parent functions. on some portion of its domain y Reflecting J. Explain how each of the following helped you determine the domain and range. a) the table of values x b) the graph asymptotes c) the function’s equation K. Which graphs lie in the listed quadrants? a) the first and second quadrants Quadrants b) the first and third quadrants L. Which graph has asymptotes? Why? II I M. You have used the slope and y-intercept to sketch lines, vertices, and directions of opening to sketch parabolas. What characteristics of the new III IV 1 parent functions f (x) 5 !x, f (x) 5 , and f (x) 5 |x| could you use to x sketch their graphs? In Summary Key Idea Certain basic functions, called parent functions, form the building blocks for families of more complicated functions. Parent functions include, but are not 1 limited to, f(x) 5 x, f(x) 5 x2, f(x) 5 "x, f(x) 5 , and f(x) 5 |x|. x (continued) NEL Introduction to Functions 27  Need to Know 1 Each of the parent functions f(x) 5 x, f(x) 5 x2, f(x) 5 ! x, f(x) 5 , and x f(x) 5 |x| has unique characteristics that define the shape of its graph. Equation of Function Name of Function Sketch of Graph f(x) 5 x linear fun