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1.1CHECK Your Understanding1. State the domain and range of each relation. Then determinewhether the relation is a function, and justify your answer.a)y4c)2–12 –8 –40–2x4d) y 5 3x 2 58e)–4–6b)4y2–4 –2 0–25(1, 4), (1, 9), (2, 7),(3, 25), (4, 11)6–40–311223x246f ) y 5 25x 2–4–62. State the domain and range of each relation. Then determine whetherthe relation is a function, and justify your answer.a) y 5 22(x 1 1) 2 2 3b) y 51x13c) y 5 22xe) x 2 1 y 2 5 9d) y 5 cos x 1 1f ) y 5 2 sin xPRACTISING3. Determine whether each relation is a function, and state its domainand range.a)c)1357246b) 5 (2, 3), (1, 3), (5, 6),(0, 21)6NELe)012321410100d) 5 (2, 5), (6, 1), (2, 7),(8, 3)60123f ) 5(1, 2), (2, 1), (3, 4),(4, 3) 6Chapter 111

4. Determine whether each relation is a function, and state its domainKa)8and range.yb)6644220–2x–4 –2 0–2246c) x 2 5 2y 1 1yd) x 5 y 2x481216 203xf ) f (x) 5 3x 1 1e) y 5–45. Determine the equations that describe the following function rules:a) The input is 3 less than the output.b) The output is 5 less than the input multiplied by 2.c) Subtract 2 from the input and then multiply by 3 to find the output.d) The sum of the input and output is 5.6. Martin wants to build an additional closet in a corner of his bedroom.lBecause the closet will be in a corner, only two new walls need to bebuilt. The total length of the two new walls must be 12 m. Martin wantsthe length of the closet to be twice as long as the width, as shown in thediagram.a) Explain why l 5 2w.b) Let the function f (l ) be the sum of the length and the width. Findthe equation for f (l ).c) Graph y 5 f (l ).d) Find the desired length and width.wallwclosetwall7. The following table gives Tina’s height above the ground while riding aAFerris wheel, in relation to the time she was riding it.Time (s)020406080100120140160180200220240Height (m)5105051050510505a) Draw a graph of the relation, using time as the independent variableand height as the dependent variable.What is the domain?What is the range?Is this relation a function? Justify your answer.Another student sketched a graph, but used height as the independentvariable. What does this graph look like?f ) Is the relation in part e) a function? Justify your answer.b)c)d)e)121.1FunctionsNEL

1.18. Consider what happens to a relation when the coordinates of all itsordered pairs are switched.a) Give an example of a function that is still a function when itscoordinates are switched.b) Give an example of a function that is no longer a function whenits coordinates are switched.c) Give an example of a relation that is not a function, but becomes afunction when its coordinates are switched.9. Explain why a relation that fails the vertical line test is not a function.10. Consider the relation between x and y that consists of all points (x, y)such that the distance from (x, y) to the origin is 5.a) Is (4, 3) in the relation? Explain.b) Is (1, 5) in the relation? Explain.c) Is the relation a function? Explain.11. The table below lists all the ordered pairs that belong to thefunction g(x).x012345g(x)347121928a) Determine an equation for g(x).b) Does g(3) 2 g(2) 5 g(3 2 2) ? Explain.12. The factors of 4 are 1, 2, and 4. The sum of the factors isT1 1 2 1 4 5 7. The sum of the factors is called the sigma function.Therefore, f (4) 5 7.a) Find f (6), f (7), and f (8).c) Is f (12) 5 f (3) 3 f (4) ?b) Is f (15) 5 f (3) 3 f (5) ?d) Are there others that will work?13. Make a concept map to show what you have learned about functions.CPut “FUNCTION” in the centre of your concept map, and includethe following words:algebraic modeldependent variabledomainfunction notationgraphical modelindependent variablemapping modelnumerical modelrangevertical line testExtending14. Consider the relations x 2 1 y 2 5 25 and y 5 "25 2 x 2. Drawthe graphs of these relations, and determine whether each relation isa function. State the domain and range of each relation.15. You already know that y is a function of x if and only if the graphCommunicationTipA concept map is a type ofweb diagram used forexploring knowledge andgathering and sharinginformation. A concept mapconsists of cells that contain aconcept, item, or question andlinks. The links are labelled anddenote direction with an arrowsymbol. The labelled linksexplain the relationshipbetween the cells. The arrowdescribes the direction of therelationship and reads like asentence.passes the vertical line test. When is x a function of y? Explain.NELChapter 113

1.2Exploring Absolute ValueYOU WILL NEEDGOAL graph paper graphing calculatorDiscover the properties of the absolute value function.EXPLORE the MathAn average person’s blood pressure is dependent on their age and gender.For example, the average systolic blood pressure, Pn , for a 17-year-old girlis about 127 mm Hg. (The symbol mm Hg stands for millimetres ofmercury, which is a unit of measure for blood pressure.) The averagesystolic blood pressure for a 17-year-old boy is about 134 mm Hg.When doctors measure blood pressure, they compare the blood pressure tothe average blood pressure for people in the same age and gender group.This comparison, Pd , is calculated using the formula Pd 5 0 P 2 Pn 0 , whereP is the blood pressure reading and Pn is the average reading for people inthe same age and gender group.TechSupportTo use the absolute valuecommand on a graphingcalculator, press MATHand scroll right to NUM.Then press ENTER.?A.Jim is a 17-year-old boy whose most recent blood pressure reading was142 mm Hg. Calculate Pd for Jim.B.Joe is a 17-year-old boy whose most recent blood pressure reading was126 mm Hg. Calculate Pd for Joe.C.Blood PressureReading, PHow can the blood pressure readings of a group of people becompared?Compare the values of P 2 Pn and 0 P 2 Pn 0 that were used todetermine Pd for each boy. What do you notice?D.Complete the following table by calculating the values of Pd for thegiven blood pressure readings for 17-year-old boys.95100E.Draw a scatter plot of Pd as a function of blood pressure, ring Absolute ValueNEL

1.2Describe these characteristics of your graph:i) domainii) rangeiii) zerosiv) existence of any asymptotesv) shape of the graphvi) intervals of the domain in which the values of the function Pdare increasing and decreasing.vii) behaviour of the values of the function Pd as P becomes largerand smallerF.ReflectingG.Why might you predict the range of your graph to be greater than orequal to zero?H.What other function with domain greater than Pn could you haveused to plot the right side of your graph? Why does this make sense?I.What other function with domain less than Pn could you have used toplot the left side of your graph? Why does this make sense?J.How will the graph of y 5 0 x 0 compare with the graph ofPd 5 0 P 2 Pn 0 , if Pd is the y-coordinate and P is the x-coordinate?Use the characteristics you listed in part F to make your comparison.In SummaryKey Idea f(x) 5 0 x 0 is the absolute value function. On a number line, this functiondescribes the distance, f(x), of any number x from the origin.Need to Know For the function f(x) 5 0 x 0 , there is one zero located at the origin the graph is comprised of two linear functions and is defined as follows:x, if x 0f(x) 5 e2x, if x , 0 the graph is symmetric about the y-axis as x approaches large positive values, y approaches large positive values as x approaches large negative values, y approaches large positive values the absolute value function has domain 5xPR6 and range 5 yPR 0 y 06 every input in an absolute value returns an output that is non-negativeNEL0 230 5 30 30 5 3–5 –4 –3 –2 –1 0 1f(x) –x, x 062 3 4 5f(x) x, x 0y42–6 –4 –2 0–2x246–4–6Chapter 115

FURTHER Your Understanding1. Arrange these values in order, from least to greatest:0 25 0 , 0 20 0 , 0 2150 , 0 120 , 0 225 02. Evaluate.a) 0 222 0b) 2 0 235 0c) 0 25 2 13 00 28 0242160 222 0f)10 211 00 24 0e)d) 0 4 2 7 0 1 0 210 1 2 03. Express using absolute value notation.a) x , 23 or x . 3c) x # 21 or x 1b) 28 # x # 8d) x 2 65CommunicationTipTo show that a number is notincluded in the solution set,use an open dot at this value.A solid dot shows that thisvalue is included in thesolution set.4. Graph on a number line.a) 0 x 0 , 8b) 0 x 0 16c)0 x 0 # 24d) 0 x 0 . 275. Rewrite using absolute value notation.a)b)c)d)–4 –3 –2 –1 0 1 2 3 4–4 –3 –2 –1 0 1 2 3 4–4 –3 –2 –1 0 1 2 3 4–4 –3 –2 –1 0 1 2 3 46. Graph f (x) 5 0 x 2 8 0 and g (x) 5 0 2x 1 80 .a) What do you notice?b) How could you have predicted this?7. Graph the following functions.a) f (x) 5 0 x 2 2 0b) f (x) 5 0 x0 1 2c) f (x) 5 0 x 1 20d) f (x) 5 0 x 0 2 28. Compare the graphs you drew in question 7. How could you usetransformations to describe the graph of f (x) 5 0 x 1 3 0 2 4?9. Predict what the graph of f (x) 5 0 2x 1 10 will look like. Verify yourprediction using graphing technology.10. Predict what the graph of f (x) 5 3 2 0 2x 2 5 0 will look like. Verifyyour prediction using graphing technology.161.2Exploring Absolute ValueNEL

1.3Properties of Graphsof FunctionsYOU WILL NEEDGOAL graphing calculatorCompare and contrast the properties of various types of functions.INVESTIGATE the MathTwo students created a game that they called “Which function am I?” Inthis game, players turn over cards that are placed face down and match thecharacteristics and properties with the correct functions. The winner is theplayer who has the most pairs at the end of the game.The students have studied the following parent functions:2–4 –2 0–2h(x) 1xg(x) x2f(x) x22–4 –2 0–242–4 –2 0–2x24222–4 –2 0–24p(x) 2xm(x) xk(x) x –4 –2 0–2424q(x) sin x2–4 –2 0–22224 –180 0–2 180interval of increase?Which criteria could the students use to differentiate betweenthese different types of functions?A.Graph each of these parent functions on a graphing calculator, andsketch its graph. State the domain and range of each function, anddetermine its zeros and y-intercepts.B.NELDetermine the intervals of increase and the intervals of decrease foreach of the parent functions.the interval(s) within a function’sdomain, where the y-values ofthe function get larger, movingfrom left to rightinterval of decreasethe interval(s) within a function’sdomain, where the y-values ofthe function get smaller, movingfrom left to rightChapter 117

odd functionany function that hasrotational symmetry aboutthe origin; algebraically,all odd functions havethe propertyf(2x) 5 2f(x)yxC.State whether each parent function is an odd function, aneven function, or neither.D.Do any of the functions have vertical or horizontal asymptotes? If so,what are the equations of these asymptotes?E.Which graphs are continuous ? Which have discontinuities ?F.Complete the following statements to describe the end behaviour ofeach parent function.a) As x increases to large positive values, y . . .b) As x decreases to large negative values, y . . .even functionany function that is symmetricabout the y-axis; algebraically,all even functions have theproperty f(2x) 5 f(x)CommunicationTipIt is often convenient to use the symbol for infinity, , and the following notationto write the end behaviour of a function: For “As x increases to large positive values, y . . . ,” write “As x S , y S . . . ” For “As x decreases to large negative values, y . . . ,” write “As x S 2 , y S . . . ”yxG.Summarize your findings.continuous functionReflectingany function that does notcontain any holes or breaksover its entire domainH.Which of the parent functions can be distinguished by their domain?Which can be distinguished by their range? Which can bedistinguished by their zeros?I.An increasing function is one in which the function’s values increasefrom left to right over its entire domain. A decreasing function is onein which the function’s values decrease from left to right over its entiredomain. Which of the parent functions are increasing functions?Which are decreasing functions?J.Which properties of each function would make the function easy toidentify from a description of it?yxdiscontinuitya break in the graph of afunction is called a point ofdiscontinuityyx181.3Properties of Graphs of FunctionsNEL

1.3APPLY the MathEXAMPLE1Connecting the graph of a function with its characteristicsMatch each parent function card with a characteristic of its graph. Each card may only be used for one parentfunction.f(x) 2xf(x) x2–4 –2 0–2f(x) x 224Range:{yPR y 0}–4 –2 0–2x2Domain:{xPR}4f(x) sin (x)2–4 –2 0–222InfiniteNumber ofZeros4 –180 0–2 180As x 2 ,y 0.Solutionf(x) 2xAs x 2 ,y 0.2–4 –2 0–2244f(x) sin x2 –180 0–2NEL 180InfiniteNumber ofZerosThis property describesthe end behaviour:as x becomes negativelylarge, y approacheszero. The function musthave a horizontalasymptote defined byy 5 0. The functionmust be y 5 2x.The sine function isperiodic and continuesinfinitely, intersectingthe x-axis an infinitenumber of times.Chapter 119

f(x) x 2–4 –2 0–224f(x) xDomain:{xPR}2–4 –2 0–22The range 5 yPR Z y 06 indicates thatall y-values of the function must benon-negative. This is true for both f(x) 5 2xand f (x) 5 0 x 0. However, f(x) 5 2x hasthe x-axis as its horizontal asymptote,so y 2 0. Choose f(x) 5 0 x 0.Range:{yPR y 0}The remaining function, f(x) 5 x, matchesthe property 5xPR6.4If you are given some characteristics of a function, you may be able todetermine the equation of the function.EXAMPLE2Using reasoning to determine the equationof a parent functionState which of the parent functions in this lesson have the followingcharacteristics:a) Domain 5 5xPR6b) Range 5 5 yPR 0 21 # y # 16Solutiona) Domain 5 5xPR6f (x) 5 xg(x) 5 x 21h(x) 5 (Domain 5 5xPR 0 x 2 06)xk(x) 5 ZxZThere are five parentfunctions that matchthis characteristic andtwo that do not.m(x) 5 "x (Domain 5 5xPR 0 x 06)p(x) 5 2xq(x) 5 sin xb) Range 5 5 yPR Z 21 # y # 16f (x)g(x)k(x)p(x)q(x)201.35 x (Range 5 5 yPR6)5 x 2 (Range 5 5yPR 0 y 06)5 Zx Z (Range 5 5 yPR 0 y 06)5 2x (Range 5 5yPR 0 y 06)5 sin xProperties of Graphs of FunctionsOf these five functions,only the sine functionhas the range5 yPR 0 21 # y # 16.NEL

1.3Visualizing what the graph of a function looks like can help you remembersome of the characteristics of the function.EXAMPLE3Connecting the characteristics of afunction with its equationWhich of the following are characteristics of the parent function p(x) 5 2x ?Justify your reasoning.a) The graph is decreasing for all values in the domain of p(x).b) The graph is continuous for all values in the domain of p(x).c) The function p(x) is an even function.d) The function p(x) has no zeros.Solutionp(x) 5 2x10y864y 2x2–8 –6 –4 –2 0–2x24The function p(x) is an exponentialfunction with a base that is greaterthan 1.This type of function is increasingfor all values in its domain.–4a) This function is increasing forall values in the domain of p(x).b) The graph is continuous for all valuesin the domain of p(x).c) The function p(x) is not an evenfunction.d) The function p(x) has no zeros.This function has no breaks.This type of function is notsymmetric about the y-axis.f(2x) 5 22x. This substitutiondoes not result in f(x).As x approaches negative infinity,the graph gets arbitrarily close tothe x-axis but does not intersect it.Only b) and d) are characteristics of p(x).NELChapter 121

EXAMPLECommunicationTipThe interval (2 , 0) isdescribed using intervalnotation and is equivalent tox , 0 in set notation. The useof round brackets in intervalnotation indicates that theendpoint is not included in theinterval. The use of squarebrackets in interval notationindicates that the endpoint isincluded in the interval. Forexample, 323, 5) is equivalentto 23 # x , 5.4Determine a possible transformed parent function that has the followingcharacteristics, and sketch the function: D 5 5xPR6 R 5 5 yPR Z y 226 decreasing on the interval (2 , 0) increasing on the interval (0, )SolutionIntervals ofIncrease(0, )(2 , 0)k(x) 5 Z x Z(0, )(2 , 0)y21–2 –10–1–212y x2 – 2–3Properties of Graphs of Functionsy1x–2 –1State the intervals ofincrease and decreasefor the two remainingfunctions. Check to see ifthese intervals match thegiven conditions. Thereare two possible parentfunctions that have thegiven characteristics.Intervals ofDecreaseg(x) 5 x221.3List the functions thathave domain 5xPR6.Eliminate the functionsthat cannot have therange 5 yPR Z y 226.Each of the remainingfunctions can betranslated down twounits to have this range.f (x) 5 xg(x) 5 x 2k(x) 5 ZxZp(x) 5 2xq(x) 5 sin xFunction22Connecting the characteristics of a functionwith its equation and its graph0–1x12Sketch the graph of eachparent function shifted2 units down.–2 y x – 2–3NEL

1.3In SummaryKey IdeaFunctions can be categorized based on their graphical characteristics: domain and range intervals of increase and decrease x-intercepts and y-intercepts symmetry (even odd) continuity and discontinuity end behaviourNeed to Know Given a set of graphical characteristics, the type of function that has thesecharacteristics can be determined by eliminating those that do not have thesecharacteristics. Some characteristics are more helpful than others when determining the typeof function.CHECK Your Understanding1. Which graphical characteristic is the least helpful for differentiatingamong the parent functions? Why?2. Which graphical characteristic is the most helpful for differentiatingamong the parent functions? Why?3. One of the seven parent functions examined in this lesson istransformed to yield a graph with these characteristics: D 5 5xPR6 R 5 5yPR Z y . 26 As x S 2 , y S 2.What is the equation of the transformed function?PRACTISING4. For each pair of functions, give a characteristic that the two functionsKhave in common and a characteristic that distinguishes between them.1a) f (x) 5 and g(x) 5 xc) f (x) 5 x and g(x) 5 x 2xb) f (x) 5 sin x and g(x) 5 xd) f (x) 5 2x and g(x) 5 Zx Z5. For each function, determine f (2x) and 2f (2x) and compare itwith f (x) . Use this to decide whether each function is even, odd,or neither.a) f (x) 5 x 2 2 4d) f (x) 5 2x 3 1 xb) f (x) 5 sin x 1 xe) f (x) 5 2x 2 2 x1c) f (x) 5 2 xf ) f (x) 5 Z2x 1 3 ZxNELChapter 123 pa

NEL Chapter 1 5 1.1 The length is 60 m. The width is 30 m. The dimensions that are 30 m apart and will produce an area of are 60 m 330 m. 1800 m2 y 5 1800 60 530 x 560 or x 5230 x 260 50 or x 130 50 (x 2