Unit 5: Relations & Functions - Miss Zukowski's Class...1)Relations & Functions: Introduction . 1. Using the following graph, answer the questions below. The graph shows the distance

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Unit 5: Relations&Functions

& Textbook Pages: __________________________________

Date Assignment/Worksheet Due Date Completed?

Quizzes & Tests:

What? When? Completed?

Quiz 1

Quiz 2

Unit/ Chapter test

1) Relations & Functions: Introduction

1. Using the following graph, answer the questions below. Thegraph shows the distance a rock climber is from the base of thecliff as time passes.

a) Place each line segment in the appropriate section of thetable. OA, AB, BC, CD, DE, EF, FG.

b) Describe one property a line segment has if the climber is climbing.

c) Describe one property a line segment has if the climber is resting.

d) Describe one property a line segment has if the climber is descending.

e) How would the graph of the line segment be different if he increased his speed for the first time heclimbed?

f) What would you add to the graph to show the climbers return to the bottom of the cliff?

Climbing Resting Descending

2. Match each graph below with a situation from the list given. Then, draw each graph carefully labeling eachaxis to show the quantities being compared.

3. Create a speed-time graph for the following scenario. Label each section of your graph with capital letters, andwrite a description of what is happening at each line segment.

Graph:

Hint: make sure to graph the independent variable on the horizontal axis, and the dependent variable onthe vertical axis.

Explanation:

Summary Ideas:

A graph represents the relationship between two quantities.

Straight lines are used to indicate a constant rate of change.

Horizontal lines are used if one quantity is NOT changing relative to the change in the other quantity.

A steeper line indicates a _______________ rate of change. This line could represent either an increase or

a decrease.

A curve shows that the rate of change is ______________________.

Constant Rate of Change No Rate of Change Rate of Change is not Constant

ASSIGNMENT # 1 pages 3-5, 39-40 Questions #1-6 & #171-176

FMPC 10 Updated June 2018

P a g e 3 |Relations Copyright Mathbeacon.com. Use with permission. Do not use after June 2019

Key Terms Term Definition Example

Relation

Function

Ordered pair

Coordinate Plane

x-axis

y-axis

Domain

Range

Element

Permissible values

Dependent Variable

Independent Variable



Introduction to Relations Relationshipsexisteverywherewelook…

• Thereisarelationshipbetweenthelengthsoflineupsatthefairandhowexcitingtheridesare.

• Thereisarelationshipbetweentheheightofaballandhowlongagoitwaskicked.• Thereisarelationshipbetweentrafficandthetimeofday.• Thereisarelationshipbetweendistancetravelledandthespeedofthecar.

Somerelationshipsdon’tevenseemtohaveamathematicalrelationshipbutareconnectedinsomeotherway.Forexample:Thestudentsinyourclassallhaveabirthmonthandheight.Wecouldwritealistmatchingeachstudent’sbirthmonthandheight.

Asorderedpairs…(3,155),(5,138),(11,162),(12,135),(7,142),…

Somenoteshere…

ChallengeQuestion:1. Giveexamplesofthreeotherrelationshipsyouseeonaneverydaybasis:

2. Writeasetof3orderedpairsforoneofyourrelationshipsabove.Explainwhattheorderedpairmeans.

(March, 155 cm tall)



Usethefollowinginformationtoanswerquestionsbelow.

Considerthedatagiveninfollowingtable:

Student Height(cm) ArmSpan(cm)Lulu 135 137Bones 144 151PhatCharlie 150 148Lucky 150 156DizzyDee 165 165Crash 155 152Anjohkinu 160 164Sam 200 210Talloola 125 127

Writtenasorderedpairs.

(135,137)(144,151)(150,148)(150,156)(165,165)(155,152)(160,164)(200,210)(125,127)

3. Whydoyouthinkthenumbersinbracketsarecalled“orderedpairs”?

4. Thedataaboverepresentsarelationbetweenwhattwoquantities?

5. Graphthedatainthetableabove(Troublegraphing?Seenextpage).

6. Describetherelationshipyouseeonthegraphabove.(Whatdoesitlooklike?Whatshapeisit?)

Graphs help to show if there is a pattern in the data.

If there is a pattern, a graph will show us what type of pattern.

2) Relations & Functions: Graphing Relations

Warm-Up #1: You are walking to school at a rate of 100 ft/min.

a) Copy and complete the table of values for this scenario.

b) Graph your data on the grid below.

Warm-Up #2: You are walking to school at a rate of 150 ft/min.

a) Copy and complete the table of values for this scenario.

b) On the same grid as warm-up #1, graph the new walking data in a different colour.

Warm-Up #3: You live 5 km from school. You sleep in for the first 10 min of class.

a) Complete the table of values for this scenario.

b) Graph your data on the grid below.

Time, t (s) Distance Walked,

d (ft)

30

60

90

120

Time, t (s) Distance Walked,

d (ft)

30

60

90

120

Time (min), t Distance From School, d (km)

0

2

4

6

What is a Relation??

Relation

Relations can be Represented in Many Ways:

1. In Words: _____________________________________________________________________________

2. A Table of Values:

3. A Set of Ordered Pairs: __________________________________________________________________

4. An Equation: _____________________________

5. A Graph:

Example 1: The value of a car depreciates with each year. Using the information from the table of values, present the relation in each of the other ways.

Ordered Pairs:

Equation: _________________________

Words: _____________________________________________________________________________________

Time (years) Value (thousands of dollars)

0 15 2 13 4 11 6 9 8 7

Example 2: Using a table of values, plot the relation described by 𝑦𝑦 = 𝑥𝑥.

Example 3: Using a table of values, plot the relation described by 𝑦𝑦 = 3𝑥𝑥 − 4.

Example 4: Using a table of values, plot the relation described by 𝑦𝑦 = −3

2𝑥𝑥.

x y

x y

x y

ASSIGNMENT # 2pages 6-12 Questions #8-35



Whatisarelation?Definition1:Iftwogroupsofitemsarerelated,thesetofallpossiblepairingsiscalledarelation.

Forexample:Aperson’sheightandtheirarmspan.Distancetravelledanddrivingtime.Examscoreandstudytime.

Definition2:Arelationisthesetoforderedpairsthatconnectstwosets.

Definition3:7. Writeyourown…

Domain:Thesetoffirstitemsinarelation.

Range:Thesetofseconditemsinarelation.

Somenoteshere(possibly)…



GraphingRelationsonaCoordinatePlane

BelowaretwoexamplesoftheCoordinatePlane

8. Theverticallinewithnumbersonitiscalledthe_______________9. Thehorizontallinewithnumbersonitiscalledthe_______________

10. Whatisthedifferencebetweeneachofthegraphsshownabove?

11. Describeascenariowhereitismoreappropriatetousethegraphontheright.

12. Describeascenariowhereitismoreappropriatetousethegraphontheleft.

13. Howcouldyoudescribetoanotherstudentwheretoplotapointontheplane?Forexample:(2,5)

14. Plotandlabelthefollowingorderedpairsoneachofthegridsabove(wheneverpossible):A(1,2),B(-3,5),C(10,4),D(-3,-7),E(8,-2)



CoordinateGeometry(CartesianCoordinateGeometry)

Basedonthecoordinateplane.

Thecoordinateplanehastwoaxes.

Theverticaly-axis.

Thehorizontalx-axis.

Thepointwheretheaxesmeetiscalledtheorigin.

Everypointontheplanecanbelocatedusingtwonumberscalledcoordinatesoranorderedpair.

Theorderedpairisalwaysgivenas(x,y).Thecoordinatesoftheoriginare(0,0).

Example: PointAhascoordinates(4,2).4isanelementofthedomain.2isanelementoftherange.

15. ChallengeQuestion:Usingthegraphbelow,plottherelationdescribedbytheequation𝑦 = 2𝑥.



GraphingrelationsusingaTableofValues.

Usingthegraphbelow,plottherelationdescribedbytheequation𝑦 = 2𝑥.𝑦 = 2𝑥x y-2 -4-1 -20 01 -22 -4

Usingthegraphbelow,plottherelationdescribedbytheequation𝑦 = √𝑥 + 2.

𝑦 = √𝑥 + 2x y-2 0-1 12 27 3

17. Arethereanyinputvaluesthatwouldnotmakesense?Arethereanythatare“notpermitted”?

18. Arethereanyinputvaluesthatwouldnotmakesense?Arethereanythatare“notpermitted”?

19. Consideryouranswerstotheprevioustwoquestions.Whateffectdo“notpermitted”inputvalueshaveonthegraphoftherelation?

Whencreatingatableofvalues,youshouldconsider:

• PositiveInputs(domain)• NegativeInputs(domain)• Zeros• Non-permittedinputvalues

A Table of Values: Choose a few reasonable input values (x), then calculate output values (y).

This produces some ordered pairs to plot our relation.

Can you see why I chose these particular input (x) values?



GraphingRelationscontinued…

20. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =3𝑥.

𝑦 = 3𝑥.x y

21. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =𝑥 + 2.

𝑦 = 𝑥 + 2.x y

24. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =3 − 𝑥.

𝑦 = 3 − 𝑥x y

25. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =2𝑥 + 1.

𝑦 = 2𝑥 + 1x y




26. Usingthetableandgraphbelow,plotthe relationdescribedbytheequation𝑦 = 𝑥+ .

𝑦 = 𝑥+x y

30. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =−2𝑥 − 1.

𝑦 = −2𝑥 − 1

x y

31. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =-+𝑥.

𝑦 =32𝑥

x y




32. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =1 − 2𝑥.

𝑦 = 1 − 2𝑥x y

33. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑦 =𝑥.

𝑦 = 𝑥x y

34. Usingthetableandgraphbelow,plottherelationdescribedbytheequation𝑥 =2𝑦.

𝑥 = 2𝑦.x y

35. **CHALLENGE**Usingthetableandgraphbelow,plottherelationdescribed

bytheequation𝑦 = |𝑥|

(where x is an absolute value, meaning the magnitude of a real number without regard to its sign)

𝑦 = |𝑥|.x y

3) Relations & Functions: Domain and Range

Warm-Up #1: List all the values of x in each relation.

a) (1, 3), (2, 5), (9, 4)

b)

c) y = 2x – 3

Warm- Up #2: List all the values of y in each relation.

a) (1, 3), (2, 5), (9, 4)

b)

c) y = 2x – 3

Warm-Up #3: For each relation, write an inequality statement.

a)

b)

c)

d)

What is different about part d in the warm up?

x y

–3 4

–1 7

0 1

x y

–3 4

–1 7

0 1

2

−2

−4 3

−1 0 1 2 3

Warm-Up #4: Fill in the table below, with the correct symbol for each number set. Then, identify the number set represented in the examples from warm-up #1.

Number Set Symbol Real

Rational

Irrational

Integer

Whole

Natural

When interpreting information to solve a problem, it is important to make sense of the possible values of each quantity being compared.

Example #1: Determine the possible values for each quantity in the given relation.

When comparing two quantities, the words DOMAIN and RANGE are used to describe the values that are appropriate.

The __________________ is the set of all possible values for the independent variable in a relation.

The ___________________ is the set of all possible values for the dependent variable in a relation.

Review your answer in the warm up. Identify the appropriate number set for each inequality.

a) ______________________________

b) ______________________________

c) ______________________________

d) ______________________________

There are a variety of ways to express the domain and range of a relation.

1. Words – A description of the value that are allowed.

Example: the range is the set of all whole numbers less than twenty

2. Number Line - A picture of the values that are allowed.

Example:

3. A List – used for discrete data

Example: For the relation {(3, 1), (2,−3), (7, 0.4)}

the domain is _______________________

the range is ________________________

4. Set Notation – a formal way to give the values of the domain and range.

Example: {𝑥𝑥|𝑥𝑥 ≥ −1, 𝑥𝑥 ∈ 𝑍𝑍}

5. Interval Notation – uses different brackets to indicate an interval

Example: [0, 10] means all numbers between zero to ten inclusive.

Example: (0, 10) means all numbers between zero and ten (not including 0 or 10)

Example: (10,∞) means all numbers greater than 10

What does it all mean???

{ } = type of brackets used for a set | means “such that” ∈ means “is an element of” (or “belongs to“) This statement is read as:

Example #2: Consider the Relation: all real numbers between -5 and 2 including -5 but not including 2.

Number Line:

Set Notation:

Interval Notation:

Could I use a list? Explain.

Example #3: Complete the table.

Words: your age from grade 1 until now Number Line:

List:

Set Notation:

Interval Notation:

Let’s see how this applies to graphs!

Example #4: Write the domain and range for each relation, as specified.

a) b)

c)

Domain as a list:

Range in set notation:

Domain in words:

Range in interval notation:

Domain in set notation:


Domain in interval notation:

Range in interval notation:

d)

e) {(4,−1), (−1, 4), (−1, 3), (4,−3), (−3, 0)}

Domain in set notation:





Domain&Range(continued)

Recall,(2,5)and(-3,7)arecalledorderedpairsbecausetheorderofthetwoelementsisimportant.

• Thefirstsetofelementsintheorderedpairiscalledthedomainoftherelation.• Thesecondsetofelementsintheorderedpairiscalledtherangeoftherelation.

36. ChallengeQuestion:Listthedomainandrangefortherelation(1,1),(2,4),(3,9),(4,16)

37. Whichofthefollowingis/aretrue?

a. Thedomainisthesetofpermissiblevaluesofx.b. Thedomainisthesetofpermissiblevaluesofy.c. Therangeisthesetofpermissiblevaluesofx.d. Therangeisthesetofpermissiblevaluesofy.

Yournoteshere…

Answer: Domain: {1,2,3,4} Range: {1,4,9,16}



Domain&RangeofDiscreteData(points): [Definitiononpage25]

Remember,domainisall“firstelements”andrangeisall“secondelements”.

Domainisoftendescribedasallpermissiblevaluesofx.

Rangeisoftendescribedasallpermissiblevaluesofy.

Findthedomainandrange:Example:Findthedomainandrangeoftherelation:(2,3),(3,4),(4,5),(5,6)

Solution:Simplylistthefirstelements,thensecond:

domain:{2,3,4,5} range:{3,4,5,6}

Example:Findthedomainandrangeofthefollowingrelation.

Solution:First,findthecoordinatesoftheBIGpoints:(2,4),(7,9),(10,-3)

Domain:{2,7,10}

Range:{4,9,-3}

**It’sOKthatthereisnoapparentpattern…thisisstillarelation.

Findeachofthefollowing.

38. Findthedomainforthefollowingrelation.

(−2,4), (3,5), (5,7), (8,11)

39. Findtherangefortherelationbelow.

(2,3), (4,3), (6,3), (8,3)

40. Findthedomainforthegraphedrelation.

All the y-values

Since we are often working with graphs that have an x-axis and a y-axis.



41. ChallengeQuestion:Findthedomainofthefollowinggraph.

42. Howmanyitemsarethereinthedomainoftherelationabove?

43. Whatisthesmallestiteminthedomain?

44. Whatisthebiggestvalueinthedomain?

45. Howmanyitemsarethereintherange?

46. Whatisthesmallestitemintherange?

47. Whatisthebiggestitemintherange?



Domain&RangeofContinuousData(LinesandCurves): [Definitiononpage25]

Whenthegraphofarelationisalineorcurve,thedomainandrangecannotbeexpressedasalistofnumbersasintheearlierquestions.Whyisthisso?

ConsiderExampleAandB.

ExampleA ExampleB

UseInequalities UseIntervalNotation Useanumberline

ExampleA

Domain:−4 ≤ 𝑥 < 3

Range:2 < 𝑦 ≤ 4

ExampleA

Domain:[−4,3)

Range:(2,4]

ExampleA

Domain:

Range:

ExampleB

Domain:𝑥 ≥ −4

Range:𝑦 ≤ 3

ExampleB

Domain:[−4,∞)

Range:(∞,3]

ExampleB

Domain:

Range:

Theinequalitysymbols:<,>,≤,≥,≠

SetNotation:𝒙𝝐𝑹:Thedomainisthesetofrealnumbers.

{𝒚|𝒚 ≤ 𝟎, 𝒚𝝐𝑹}:Therangeisthesetofrealnumberslessthanorequaltozero.

Bracketsareusedtoshowtheinterval.

[ifthenumberisincluded(ifthenumberisnotincluded

∞isusedifthesetdoesnotend.

(−∞,∞):Noupperorlowerlimit,or,“allrealnumbers”.

(3,∞):Allrealnumbersgreaterthan3.

Solidcirclesindicatethenumberisincluded.

Hollowcirclesindicatethenumberisnotincluded.



48. Ifarelationcontinuesinbothdirections:

UseIntervalNotation:Domain:Range:

49. Therelationhasastartingpointbutnoendingpoint:

UseInequalities:Domain:Range:

50. Therelationhasanon-permissiblevalue:

UseNumberLines:Domain:Range:

51. Therelationhasastartingpointandafinishingpoint:

Usewords:Domain:Range:

52. Therelationhasastartingpointandafinishingpoint:

UseInequalities:Domain:Range:

53. Therelationhasnostartingpointorfinishingpoint:

UseIntervalNotation:Domain:Range:

Writeasetofinstructionsforfindingthedomainofafunctionin:

54. IntervalNotation:

55. UsingaNumberLine:

56. UsingInequalities:



57. Trytomatcheachofthefollowinggraphswithdomainandrangebelow.(Therearethreeoneachgraph)

𝑨.𝑥𝜖𝑅, 𝑦𝜖𝑅

𝑩.[1,9]𝑎𝑛𝑑[−7,1]

𝑪.{𝑥|𝑥𝜖𝑅},{𝑦|𝑦 ≤ 0,𝑦𝜖𝑅}

𝑫.𝑑𝑜𝑚𝑎𝑖𝑛[4,9], 𝑟𝑎𝑛𝑔𝑒[4,7]

𝑬.{𝑥|𝑥 ≥ −7, 𝑥𝜖𝑅}, {𝑦|𝑦 ≥ 1,𝑦𝜖𝑅}

𝑭.Domainisallrealnumbersfrom-5to8.Rangeisallrealnumbersfrom3to8.

Findthedomainandrangeforeachofthefollowinggraphs.

58. Usesetnotation:

domain__________

range_____________

59. Useintervalnotation:

domain__________

range_____________

60. Usenumberlines:

domain__________

range_____________



Findthedomainandrangeforeachofthefollowinggraphs.

61. Usesetnotation:

domain__________

range_____________


domain__________

range_____________

63. Usenumberlines:

domain__________

range_____________

64. Usealist(discrete):

domain__________

range_____________

65. Usealist(discrete):

domain__________

range_____________


domain__________

range_____________

4) Relations & Functions: Graphing in a Domain

In Lesson 2, when we graphed functions, we graphed in the domain of all real numbers.

Now, when you graph a relation, the domain may be:

1. Given as R (all real numbers)2. Given as a list, for example: {-2, -1, 0, 1, 2}3. Given as an inequality, for example: {𝑥𝑥 ≥ 0}

Example #1: Graph the relation 𝑦𝑦 = 2𝑥𝑥 + 1 in the following domains.

a) {𝑥𝑥| 𝑥𝑥 ∈ 𝑅𝑅}

b) {𝑥𝑥|𝑥𝑥 = −4,−3, 0, 4 }

x y

x y

The domain that you are given just tells you what values of x you are allowed to plug into your table of values

c) {𝑥𝑥|𝑥𝑥 < −1, 𝑥𝑥 ∈ 𝑅𝑅 }

Example #2: Find the domain and range of the following relations.

a) 𝑦𝑦 = 2𝑥𝑥 − 8

Domain =

Range =

b) 𝑦𝑦 = √𝑥𝑥 + 3

Domain =

Range =

c) 𝑦𝑦 = 𝑥𝑥2 − 1

Domain =

Range =

x y

Hint: Think about this in one of two ways:

1. Visualize the graph2. Consider if there are any x-values

that you can’t plug into the equation(ones that will give you an ERRORin your calculator)




67. Drawagraphusingthefollowingdomainandrange.

{𝑥|𝑥 ≥ 4, 𝑥𝜖𝑅},{𝑦|𝑦 ≤ −3, 𝑦𝜖𝑅}


{𝑥|−3 ≤ 𝑥 ≤ 2, 𝑥𝜖𝑅}{𝑦|5 ≤ 𝑦 ≤ 8, 𝑦𝜖𝑅}


{𝑥|𝑥 = 5},{𝑦|𝑦 ≥ −4, 𝑦𝜖𝑅}

70. ChallengeQuestion:Graphtherelationrepresentedbytheequation𝑦 = 3𝑥.

71. Whatisthedomainof𝑦 = 3𝑥?

72. ChallengeQuestion:

Graphthelinerepresentedbytheequation𝑦 = 3𝑥ifthedomainis𝑥 ≥ −2.



GraphingRelationsandDomain:

Whengraphingarelation,thedomainmaybe:

• GivenasR(allrealnumbers)• Givenasalistsuchas{-2,-1,0,1,2}• Givenasaninequalitysuchas𝑥 ≥ 0.

Wewillconsidertheimpacteachofthesehavewhengraphingtherelation𝑦 = 2𝑥

73. Graphtherelation𝑦 =2𝑥.

Domain:{𝑥|𝑥𝜖𝑅}

Seenextpageforsolution


Thedomainis{-2,-1,0,1,2}



Domain:(0,∞)


Somenoteshere…

When graphing equations earlier, we always assumed the domain was all real numbers. ..Not anymore.



Graphtherelation𝑦 = 2𝑥.Thedomainis{𝑥|𝑥𝜖𝑅}.

Anyvaluesofxwouldbepermissible.Thisresultsinacontinuouslineinbothdirections.

Twoarrowheads!

Graphtherelation𝑦 = 2𝑥.Thedomainis{-2,-1,0,1,2}

Onlythespecifiedvaluesofxwouldbepermissible{-2,-1,0,1,2}.Thisresultsinfivediscretepointsonthegraph.

Findtheyvaluesthatgowiththesefivexvalues.

Graphtherelation𝑦 = 2𝑥.Domain:(0,∞).

Anyvaluesofxgreaterthan0wouldbepermissible.Thisresultsinacontinuouslinestartingatx=0andmovinginthepositivedirection.

Onearrowhead!

Grapheachofthefollowingforthegivendomain.76. 𝑦 = 3𝑥 + 2

Domain:{𝑥|𝑥𝜖𝑅}

77. 𝑦 = −2𝑥 − 4

Domainis{−3, −1, 1, 3}.

78. 𝑦 = Z+𝑥+

Domainis:(∞, 0)

79. 𝑦 = 𝑥

Domainis{𝑥| − 𝟐 ≤ 𝒙 ≤ 𝟐, 𝑥𝜖𝑅}

80. 𝑦 = 𝑥+ − 2,

Domainis{−2, −1,2,1}.

81. 𝑦 − 5 = 𝑥 + 2,

Domainis{𝑥|𝑥𝜖𝑅}



82. Inyourownwords,describethedifferentwaysarelationmaylookduetorestrictionsonthedomain.

Findingthedomainandrangeofanequation.

Becomingmorefamiliarwiththeequationofparticularrelations(assumingthereisone)allowsyoutoquicklydeterminethedomainorrange.

PossibleStrategies:

• Visualizethegraphfrommemory(oractuallyplotit).

• Considerpossiblerestrictionsbasedontheequation.Forexample,𝑦 = √𝑥hasadomain𝑥 ≥ 0becauseallnegativevaluesofxproducea“notreal”output.

83. Findthedomainoftherelation:

𝑦 = 3𝑥

84. Findthedomainoftherelation:𝑦 = √𝑥 − 2

85. Findthedomainoftherelation:

𝑦 = 𝑥+

86. Findtherangeoftherelation:

𝑦 = 3𝑥

87. Findtherangeoftherelation:𝑦 = √𝑥 − 2

88. Findtherangeoftherelation:

𝑦 = 𝑥+



89. ChallengeQuestion:Considerthevariouswaysgraphslookbecauseoftherestrictionsontheirdomainbeforeyouanswerthefollowingquestion.

Usetheequation𝐶 = 10𝑛tographthecost,C,ofafamilywith‘n’peopletogotothemovies.

90. ChallengeQuestion:Findareasonabledomainforthefunctionabove.

Findareasonablerangeforthefunctionabove.

Somenoteshere…

5) Relations & Functions:

Continuous/Discrete & Vertical Line Test

Warm-Up: Students at Reynolds are selling t-shirts during lunch for $10 each.

a) Complete the following table of values:

b) Graph the relation:

c) Can the dots be connected? Explain.

d) Fill in the table below, and add in your own example in.

Type of Data Continuous Discrete

Characteristics

• Graph will appear as ______________.• Occurs when quantities don’t

______________ values.• Occurs when having _____________

of quantities makes sense.

• Graph will appear as _____________________________________________.

• Occurs when quantities can only be_______________________________.

• Occurs when part numbers __________________________________________.

Example

•

•

•

•

•

•

Number of t-shirts sold

Total amount of money made, in

dollars 0

1

2

3

4

5

Introduction to Functions

A function is

When a relation is presented as a graph, a quick method to determine whether or not it is a function is known as the VERTICAL LINE TEST.

Example #1: Do these graphs represent functions?

Example #2: Do these relations represent functions? Justify your choice.

If a vertical line intersects the graph at more than one point, the relation IS NOT a function.




DiscreteandContinuousData

Frompreviouspage…

Usetheequation𝐶 = 10𝑛tographthecost,C,ofafamilywith‘n’peopletogotothemovies.

91. Whyisthegraphaboveaseriesofdots,notacontinuousgraph?

ContinuousData:• Graphwillappearasaline.• Occurswhenquantitiesdon’t“skip”

values…continuousthingsliketimeandtemperature.

DiscreteData:• Graphswillappearasaseriesofdots.• Quantitiessuchaswholeitems(people,

cars,hamburgers,etc.)• Whenpartnumbersdon’t“makesense.”

Note:Thefirsttwopointsaremathematicallycorrectbutdonotmakesenseintermsoftheproblem.



Answerthefollowingquestions.

92. Acupofcoffeesitsonthecounterforseveralhours.Describewhatwouldhappentothetemperatureofthecoffeeinthecup.

93. Acupofcoffeesitsonthecounterforseveralhours.Temperatureisafunctionofthetimethecoffeeisonthecounter.

a) Whatwouldbeareasonabledomainforthisfunction?

b) DiscreteorContinuous(Circleone)

94. Ahighschoolstudentissurveyingthevolumeoftrafficintheschoolparkinglot.Thenumberofcarsintheparkinglotisafunctionofthetimeofday.

a) Whatwouldbeareasonablerangeforthisfunction?

b) DiscreteorContinuous(Circleone)

95. Dalleepisplottinghisheightasafunctionoftime.

a) Whatquantitywouldrepresentthedomain?HeightorTime(Circleone)

b) Height:DiscreteorContinuous(Circleone)

c) Time:DiscreteorContinuous(Circleone)

d) Whatwouldbeareasonablerange?

96. TheCostofEnergyBars:

Costisafunctionof_________________.

DiscreteorContinuous(Circleone)

97. TotalEarnings(withhourlywage):

Earningsareafunctionof_________________.

DiscreteorContinuous(Circleone)



Functions:

Aspecialclassofrelationinwhichthereisonlyoneoutput(y)foreveryvalidinput(x).

Considerthefollowingtables:

Domain(x):{-1,0,2,4} Domain(x):{-2,-1,0,1} Domain(x):{4,3,2,4}

Range(y):{3,0,-6,-12} Range(y):{4,1,0,1} Range(y):{2,7,12,17}

NoticeinTable3,whentheinput(x)is4,therearetwopossibleoutputs,2or17.ThisisNOTafunction.

Tables1and2arebothfunctions.Eachelementinthedomainproducesonlyoneelementintherange.

Somenotesherepossibly…

Table 1 x y -1 3 0 0 2 -64 -12

Table 2 x y -2 4 -1 1 0 0 1 1

Table 3 x y 4 2 3 7 2 12 4 17

I find it easiest to determine if a relation is a function from a graph!



Whichofthefollowingrelationsarefunctions?Indicatewhyorwhynot.

98. (2,1),(3,5),(7,8),(9,12)

YESorNOWhy?__________________________________

99. (1,3),(5,3),(7,3),(12,3)

YESorNOWhy?__________________________________

100. (3,1),(3,5),(2,6),(1,4)

YESorNOWhy?__________________________________

101. 𝑦 = ±√𝑥

YESorNO

Why?__________________________________

102. x y

-3 72 35 51 7

YESorNO

Why?__________________________________

103.

YESorNO

Why?__________________________________

x y-3 112 5-3 -68 7

TheVerticalLineTest:Youcantestwhetherarelationisafunctionbyusingtheverticallinetest.

Ifyoumoveaverticallinethroughtherelationfromlefttoright,theverticallinewillonlyevercontactafunctiononce.Iftheverticallinecontactsthegraphmorethanonceatagiventime,itisnotafunction.

Eg.1.

Yes,itisafunction.

Averticallinewillcontactthefunctiononlyonce.

Eg.2.

No,notafunction.

Averticallinemaycontactthefunctiontwice.



Determineifeachofthefollowingrelationsisafunctionornot.

104. 105. 106.

107. Istherelation𝑦 = 2𝑥 − 3afunction?

YESorNO

Howdoyouknow?

108. Istherelation𝑦 = 𝑥+afunction?

YESorNO

Howdoyouknow?

109. Istherelation𝑥 = 𝑦+afunction?

YESorNO

Howdoyouknow?


6) Relations & Functions: Function Notation

Warm-Up #1: Complete the table of values for the equation 𝑦𝑦 = 3𝑥𝑥 + 2

Does this equation represent a function? __________

Warm-Up #2: Evaluate y = 2x2 – 3x + 5 for each of the given values.

a) x = –3 b) x = 3

Warm- Up #3: Evaluate y = 3x – 5 for each of the given values.

a) y = 10 b) y = –26

Investigation: If 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 + 2 determine the following.

a) 𝑓𝑓(−3)

b) 𝑓𝑓(−1)

c) Predict the value of 𝑓𝑓(0), 𝑓𝑓(3).

x y −3

−1

0

1

3

Functions can be written using function notation. For example, 𝒚𝒚 = 𝟑𝟑𝟑𝟑 + 𝟐𝟐 can be written as 𝒇𝒇(𝟑𝟑) = 𝟑𝟑𝟑𝟑 + 𝟐𝟐

𝑓𝑓(𝑥𝑥) is read as “𝑓𝑓 of 𝑥𝑥”

-10 -5 5 10

-10

-5

5

10

Example #1: If 𝑝𝑝(𝑎𝑎) = −2𝑎𝑎 + 5, determine the following:

a) 𝑝𝑝(−2) b) 𝑝𝑝(1)

c) 𝑎𝑎, if 𝑝𝑝(𝑎𝑎) = 1 d) 𝑎𝑎, if 𝑝𝑝(𝑎𝑎) = −5

e) Use your results to create a table ofvalues.

f) Graph the function.

g) Is this discrete or continuous data? Explain.

Example #2: If 𝑔𝑔(𝑥𝑥) = 5𝑥𝑥 − 1, determine a simplified expression for each of the following:

a) 𝑔𝑔(2𝑥𝑥) b) 𝑔𝑔(𝑥𝑥 − 5)

x y

Example #3: Use the graph below to determine the values.

a) 𝑓𝑓 (−1)

b) 𝑓𝑓 (0)

c) 𝑥𝑥 when 𝑓𝑓 (𝑥𝑥 ) = 1

d) 𝑥𝑥 when 𝑓𝑓 (𝑥𝑥 ) = 4

ASSIGNMENT # 6pages 30-35 & 38 Questions #110-160 +169-170

pg's 36, 37 & 41 (optional extra practice)



FunctionNotation:Thereisaspecialwaytowritefunctions.Thisiscalledfunctionnotation.

Considerthefollowingcomparisons:

Equation FunctionNotation

𝑦 = 𝑥 + 2 𝑓(𝑥) = 𝑥 + 2𝑦 = 3𝑥 − 5 𝑓(𝑥) = 3𝑥 − 5𝐶 = 20𝑡 + 1200 𝐶(𝑡) = 20𝑡 + 1200𝐺 = 3ℎ+ − 2 𝑔(ℎ) = 3ℎ+ − 2

• Functionnotationallowsustouselettersappropriatetoourfunctionanddifferentiatebetweenseveralfunctions(givethemuniquenames).

• Alsothenotationtellsuswhichvariableisdependentontheother.

Eg.𝑔(ℎ) = 3ℎ+ − 2 tellsusthatfunctiongiswrittenintermsofh.Thatis,gdependsonh.

Functionnotationcanalsobeusedtotellustoperformanoperation.

Evaluate𝑓(2), 𝑓(−3), 𝑓(𝑥 + 2)forthefunction𝑓(𝑥) = 3𝑥 + 7

𝑓(2) = 3(2) + 7

𝑓(2) = 13

𝑓(−3) = 3(−3) + 7

𝑓(−3) = −2

𝑓(𝑥 + 2) = 3(𝑥 + 2) + 7

𝑓(𝑥 + 2) = 3𝑥 + 6 + 7

𝑓(𝑥 + 2) = 3𝑥 + 13

If𝑓(𝑥) = 5𝑥 − 6,find110. 𝑓(4) 111. 𝑓(−1) 112. 𝑓(−3 + 𝑥)

If𝑔(𝑥) = 2𝑥 − 4,find113. 𝑔(4) 114. 𝑔(−1) 115. 𝑔(𝑥 − 1)

Notice the f(x) part simply replaces the y. There is no new operation; it is only a new way of writing the equation. It immediately tells you “this is a function.” Notice the letter in brackets is always the same as the one on the right.



Ifℎ(𝑥) = 5𝑥+ − 6,find

116. ℎ(4) 117. ℎ(−1) 118. ℎ(𝑥 + 1)h(x + 1) = 5(x + 1)+ − 6= 5(x + 1)(x + 1) − 6 distributiveproperty= 5(x+ + 1x + 1x + 1) − 6 collectlike-terms= 5(x+ + 2x + 1) − 6 distributiveproperty= 5x+ + 10x + 5 − 6 collectlike-terms= 5x+ + 10x − 1

If𝑞(𝑥) = 𝑥+ + 2𝑥 + 3,find

119. 𝑞(4) 120. 𝑞(𝑥 + 1) **CHALLENGE 121. 𝑞(2𝑥 − 3) **CHALLENGE

If𝑔(𝑥) = −2𝑥 + 1,find122. 𝑔(4)

123. Whichvariableistheindependentvariable?

124. 𝑔(−1) 125. Graph𝑔(𝑥)Domain:𝑥𝜖𝑅

**UseyourgraphinQ125tocheckanswersforQ122andQ124.

If𝑓(𝑥) = 2𝑥+,find126. 𝑓(4)

127. Whichvariableisthedependentvariable?

128. 𝑓(−1) 129. Graph𝑓(𝑥)iftheDomain:𝑥𝜖𝑅



Usethegraphofh(t)toanswerthefollowingquestions.

130. Findℎ(2). 131. Findℎ(−6). 132. ℎ(𝑡) = 4Findbothvaluesoft.

Usethegraphofd(x)toanswerthefollowingquestions.

133. Find𝑑(−3). 134. Find𝑑(5). 135. 𝑑(𝑥) = 2Findbothvaluesofx.

Usethegraphof𝑓(𝑥)toanswerthefollowingquestions.

136. Find𝑓(−3). 137. Find𝑓(5). 138. Findxif𝑓(𝑥) = 0.



Describingthesamerelationinvariousways.

Belowarethreedescriptionsofthesamerelation.

Equation:

𝑓(𝑥) = 𝑥 − 2

Or𝑦 = 𝑥 − 2

Graph: Words:

Eachelementintherangeistwolessthantheelementinthedomain.

Fillinremainingtwocellsbelowforthegivenrelation.Equation:

GIVENTHEFUNCTION…

𝑓(𝑥) = 3𝑥 − 1

139. Graph: 140. Words:

141. Equation: GIVENTHEFUNCTION… 142. Words:

143. Equation: 144. Graph: Words:

GIVENTHEFUNCTION…

Eachelementoftherangeisequivalenttothesquareofanelementinthedomain.

Thefollowingtwoexamplesdemonstratearelationshipbetweentwoquantities.



Acomputerservicetechnicianchargesafeeof$120toassessaproblemandafeeof$60perhourtofixtheproblem.Ifthehighschoolnetworkrequires12hoursofwork,whatwillthetotalcostbe?

Cost:= $120 + 12($60)= $120 + $720= $840

Thisisarelationshipbetweentimeworkedandcost.

Wecouldshowthisas(12, 840).

Theheightofathrownobjectcanbemodeledasafunctionoftime(sinceitwasthrown)bythefollowingequation.

ℎ(𝑡) = −5𝑡+ + 12.5𝑡 + 100

Findtheheightoftheobject2secondsafterithasbeenthrown.

Heightisfoundbysubstituting2intotherightsideoftheequation.

ℎ(2) = −5(2)+ + 12.5(2) + 100ℎ(2) = −20 + 25 + 100ℎ(2) = 105𝑚

Thisisarelationshipbetweenheightandtime.

Wecouldshowthisas(2, 105).

145. Createasetofdatafortherelationabove.

146. Graphthedatayoucreatedinthequestionabove.

147. Createasetofdatafortherelationabove.

148. Graphthedatayoucreatedinthequestionabove.



Solveeachproblemusinganystrategythatworks.

Acomputerservicetechnicianchargesafeeof$120toassessaproblemandafeeof$60perhourtofixtheproblem.

149. Ifthehighschoolnetworkrequires7hoursofwork,whatwillthetotalcostbe?

150. Whatarethetwovariablesinthisproblem?

_________________________________

_________________________________

151. Whichvariablewouldbethe“dependentvariable?”(seepg.30)

_________________________________

152. Doesthedependentvariablecorrespondtothedomainorrange?

_________________________________

_________________________________

153. Doyouthinkthisproblemmodelsdiscreteorcontinuousdata?Explain.

_________________________________

_________________________________

154. Whatissignificantaboutthepoint(0,120)?_________________________________

_________________________________

Theheightofathrownobjectcanbemodeledasafunctionoftimesincethrownbythefollowingequation.

ℎ(𝑡) = −5𝑡+ + 12.5𝑡 + 100

155. Findtheheightoftheobject3secondsafterithasbeenthrown.

156. Canyouthinkofanyvaluesfortime(t)thatdon’tmakesense?

_________________________________

157. Whatdoestimerepresentdomainorrange?

_________________________________

158. Canyouthinkofanyvaluesforheight(h)thatdon’tmakesense?

_________________________________

159. Isheightthedependentorindependentvariable?

_________________________________

160. Usethegraphonthepreviouspagetoestimatethetimeittakestheobjecttoreachmaximumheight.

BONUS:Canyoucalculatethetimeittakestheobjecttoland?



Solveeachproblemusinganystrategythatworks.

161. Abiketechniciancharges$40forabasictune-upand$20/hforanyadditionalwork.

Writeanequationthatrelatescost(C)totime(t)forthescenarioabove.

162. Createatableforthescenarioabove.

Time(hours) Cost($)

163. Graphtherelationabove.

164. Thepopulationofacolonyofbacteriagrowsthroughcelldivision.Thedoublingtimeforthepopulationis30minutes.Completethetablebelowforthegrowthofbacteriastartingwithonebacterium.

Time(minutes)

NumberofBacteria

0

30

60

90

120

210

165. Graphtherelationabove.

166. Whatnumbersareacceptablevaluesforthehorizontalaxis(domain)ofthegraphabove?(Thinkaboutwhatnumberswouldnotmakesense.)



167. Goingtothemovies.Thecostofgoingtothemoviesforagroupofgrade10studentsisrepresentedbytheequation𝐶 = 10.5𝑛.

a) Whatisareasonablerangeforthisfunction?

b) Whatisthedependentvariable?

c) Writetheequationusingfunctionnotation.

168. DrivingDistance.JJleavesNanaimodrivingnorth.Atthetimeheleft,hewas105kmfromhome.Thefollowinggraphrepresentstherelationshipbetweendistancefromhomeandelapseddrivingtime.

Theequationfortherelation is𝑑(𝑡) = 50𝑡 + 100.

a) Explainwhythefunctioniscalled𝑑(𝑡).

b) Suggestareasonabledomainforthefunction𝑑(𝑡).

c) Find𝑑(3).

d) Whyisthegraphalineandnotaseriesofdots?



169. Halloweendance.Student’sCouncilplansonhiringDJ-Jae-Sunforthisyear’sHalloweendance.Jae-Sunappreciateswhatheremembersofmathfunctionsandsendsthecouncilthefollowingpricinginformation.

𝐶(𝑛) = 2000 + 17.50𝑛

a) Explainwhatyouthinktheequationabovemeans.

b) Whatwouldbeareasonabledomainatyourschool?

c) Whatisareasonablerangeforyourschool?

d) Whatdoestherangerepresent?

e) Isthisthedependentorindependentvariable?

170. Weddingbanquet.Lin-Karenisplanningherdreamwedding.Cateringcostsareafunctionofthenumberofpeoplethatattendthewedding.AhighendcatererquotedLin-Karenaset-upcostof1500dollarsplus75dollarsperguest.

a) Writethecostasafunctionofthenumberofguestsusingfunctionnotation.

b) IsthisDiscreteorContinuousdata?

c) Graphtherelationaboveusingareasonabledomain.Usearulertomarkyouraxes.Labelyouraxeswith“NumberofGuests”onthehorizontalaxis.



GeneralrelationsWhenconsideringsomerelationships,itissolelythepatternortrendthatweareinterestedin.

Canyouvisualizeagraphforthefollowingrelationships?

• TheheightabovethegroundofapassengeronaFerrisWheelasafunctionoftime.• Thenumberofcarsinaparkinglotasafunctionofthetimeofday.• Temperatureofacupofcoffeeasafunctionoftimesinceitwaspoured.• Thecostofmailingapackageasafunctionofitsmass.• Theheightofafootballasafunctionoftimesinceitwaskicked.

Matcheachofthefollowingwithanexamplefromabove.Thendescribebelowwhyyoumadethatchoice.

_____________ ____________________ ____________________

_____________ ____________________ ____________________

_____________ ____________________ ____________________


Answerthequestionsassociatedwitheachgraph.



171. Describearelationshipthatcouldberepresentedbythegraphbelow.


173.

Aboveisagraphofthetemperatureinsideanovensetto425℃.Whydoesthegraphfluctuate?

174.

Thegraphaboverepresentstheheightofakickedballasafunctionoftime.Whyisthegraphnotsymmetrical?


176.

Explainwhythisgraphofpostageratesappearsstepped.



177. Ahotcupofcoffeewasleftonthetabletocool.Graphthedatabelow.

178. Tohireaplumbertofixhisdrain,Mr.Jhadtopayaninitial“call-out”feeof$60thenhehadtopaytheplumber$45perhour.GraphtheCostasafunctionofTimeinhoursforthisservice.

Time(min)

0 2 4 6 8 10 12 14

Temp.(℃)

84 60 44 34 26 23 21 21



Answers:

1. Answersmayvary.Possiblerelations:Ageandheight,costandtime,wageandhours,…

2. (27years,180cm),(9years,110cm)(0.6years,55cm)

3. Theirorderisimportant.4. HeightandArmSpan.5.

6. Pointsappeartoresemblealine.7. Checkwithaclassmateorteacher.8. y-axis9. x-axis10. Graphonleftincludesnegative

coordinates.11. Graphsofdatawherenegativesare

notincluded.(eg.Distancevs.Time)12. Graphsofdatawherenegativesare

appropriate.(eg.Altitudes,temperatures)13. Twounitsrightandfiveunitsupof

theorigin(middle).14.

15.

16.

17. SameasQ15.Allinputvalues“makesense”

mathematically.18. SameasQ16.Anyvalueof‘x’lessthan-2will

maketheequation(function)undefinedoverthesetofrealnumbers.

19. Thegraphwillhavebreaksorstoppingpointswherevaluesarenotpermitted.

20.



21.

22. −2, 5.Anyrealvalueisapossibleelementofthedomain.

23. −7, 1.Anyrealvalueisapossibleelementoftherange.

24.

25.

26.

27.

28. Nonegativevaluescanbecontainedintherange.

29. Nonegativevaluescanbecontainedintherange.

30.

31.

32.



33.

34.

35.

36. Domain:{1,2,3,4},Range:{1,4,9,16}or{𝑥|1,2,3,4}, {𝑦|1,4,9,16}

37. aandd38. {𝑥|−2,3,5,8}39. {𝑦|3}40. {𝑥|−1, 0,1,2,3}41. {𝑥 ≥ 0,𝑥 ∈ ℝ}Thedomainisthereal

numbersgreaterthanorequaltozero.42. Infinite43. 044. Thereisnotalargestvalue.Infinity.45. Infinite46. 247. Thereisnotalargestvalue.Infinity.48. (−∞,∞), (−∞,∞)49. {𝑥|𝑥 ≥ −4,𝑥𝜖𝑅}, {𝑦|𝑦 ≥ 1,𝑦𝜖𝑅}50.

x:

y:

51. Thedomainconsistsofallrealnumbersfrom-6to5inclusive.Therangeconsistsofallrealnumbersfrom-1to2inclusive.

52. {𝑥| − 5 < 𝑥 < 7, 𝑥𝜖𝑅}, {𝑦|𝑦 = 4}53. (−∞,∞), (−∞,∞)54. Findtheupperandlowerlimits.Separate

themwithacomma.Useasquarebracketifthelimitisincluded,acurvedbracketifitisnot.

55. Plotthelimit(s)onanumberlinewithasoliddotifincludedorhollowdotifexcluded.Drawanarrow/lineintheappropriatedirection(unlessthedataisonlyapointorpoints).

56. Findthelimit(s).Choosethecorrectsymbol,<,≤,>,≥,≠.Fillouttheinequalityusingoneofthefollowingasaguide:𝑥 ≥__________≤ 𝑥 ≤_____

57. A.4B.5C.3D.6E.1F.258. {𝑥𝜖𝑅}, {𝑦𝜖𝑅}59. [−7,6], [−5,7]60. x:

y:

61. {𝑥| − 5 ≤ 𝑥 ≤ 8, 𝑥𝜖𝑅}, {𝑦| − 3 ≤ 𝑦 ≤ 5, 𝑦𝜖𝑅}62. (−6,∞),(∞, 3)63. x:

y:

64. {𝑥|2},{𝑦|−3,−1, 1, 3}65. {𝑥|−7,−5,−3,6}, {𝑦|−4,5,7}66. (−8,9), [−5,7)67.

68.



69.

70.

71. {𝑥|𝑥𝜖𝑅}72.

73.

74.

75.

76.

77.

78.

79.

80.

81.

82. Consider,points,lines,breaks,…83. {𝑥|𝑥𝜖𝑅}or[−∞,∞]



84. 𝑥 ≥ 2or[2,∞)orAllrealnumbersgreaterthanorequalto2.

85. {𝑥|𝑥𝜖𝑅}or(−∞,∞)orAllrealnumbers.86. {𝑦|𝑦𝜖𝑅}or(−∞,∞)orAllrealnumbers.87. 𝑦 ≥ 0or[0,∞)orAllrealnumbersgreater

thanorequalto0.88. {𝑦|𝑦 ≥ 0}or([0,∞)orRealnumbersgreater

thanzero.89.

90. {𝑥|2,3,4,5,6,7,8,9},{𝑦|20,30,40,50,60,70,80,90}91. Thespacebetweenpointsrepresents

“fractionsofpeople”andthecorrespondingcost.Thedomainislimitedtowholenumbersinthiscase.

92. Thetemp.ofthecoffeewillcooluntilitreachesroomtemperature.

93. a)Several(3or4)hours.b)Continuous

94. a){𝑛|0 ≤ 𝑛 ≤ 75, 𝑛𝜖𝑊}b)discrete95. a)Time

b)Continuousc)Continuousd) Anhuman’sheightwouldrangefromabout45cmtoabout200cm.Thereareexceptionsofcourse!

96. Costisafunctionofnumberofbars.Discretedata.

97. Earningsareafunctionofhoursworked.Discretedata.

98. Yes.Eachx-valuehasonlyonecorrespondingy-value.


100. No.Therearetwopossibleoutputswhenxis3.

101. No.The±indicatesthateachinput,exceptzero,willhavetwooutputs.


103. No.Therearetwopossibleoutputswhenxis−3.

104. No.105. No.106. Yes.107. Yes.Eachinputvalue(x)willproduceonlyone

outputvalue(y).108. Yes.Eachinputvalue(x)willproduceonlyone

outputvalue(y).109. No.Therewillbetwooutputs(onepositive,

onenegative)foreachinput.Exceptwhentheinputiszero.

110. 14111. −11

112. 5𝑥 −21113. 4114. −6115. 2𝑥 −6116. 74117. −1118. 5𝑥+ + 10𝑥 − 1119. 27120. 𝑥+ + 4𝑥 + 6121. 4𝑥+ − 8𝑥 + 6122. −7123. 𝑥isindependent.124. 3125.

126. 32127. 𝑓(𝑥)whichhasreplacedy.128. 2129.

130. 4131. 3132. 2,−8133. 6134. 10135. 𝑥 = −1,1136. 9137. Notpossible.5isnotanelementofthedomain

inthisfunction.138. 𝑥 = 7139.

140. Eachelementintherangeisonelessthantripleanelementinthedomain.



141. 𝑓(𝑥) = 2𝑥142. Eachelementintherangeisdoublean

elementinthedomain.143. 𝑓(𝑥) = 𝑥+144.

145. (1,180), (2,240), (3,300),(4,360), (5,420),…146.

147. (1,107.5), (2,105), (3,92.5),…148.

149. $540150. C:totalcost

h:hoursworked151. Cost(C)152. Range153. Explainyouranswer.154. Thetechnicianhasnotperformedanyhoursof

repairworkbutthereisstillacostof$120.155. 92.5m156. Negativevalues.157. Domain158. Negativevalues(unlesstheobjectthrowncan

fallbelowtheheightthatwehavecalledheightzero.Forexample,itmaylandinaholeorcrevace.

159. Dependant160. Approximately1.5seconds.161. 𝐶 = 40 + 20ℎor

𝐶(ℎ) = 20ℎ + 40.

162. Time(hours) Cost($)

0 401 602 803 1004 1205 140

163.

164. Time

(minutes)NumberofBacteria

0 130 260 490 8120 16210 128

165.

166. Realnumbersgreaterthanorequaltozero.167. a)

{𝐶|10.5,21,31.5, 42,52.5,63,73.5,84,94.5, 105}b)C,cost.c)𝐶(𝑛) = 10.5𝑛

168. a)Thefunctioniscalled𝑑(𝑡)beacure“distance”isafunctionof“time”.b)Realnumbersbetween0and5inclusive.c)250d)Thisdataisalinebecausebothdistanceandtimearecontinuous(nogaps).

169. a)𝐶(𝑛) = 2000+ 17.50𝑛TheDJchargesisa$2000fixedcostinadditionto$17.50perhourorpossiblyperstudent.Itisnotclearwhatnrepresents.Iwillworkassumingnrepresentshours.b)eg.{𝑛|0 ≤ 𝑛 ≤ 4, 𝑛𝜖𝑊}c)eg.{𝐶|2000, 2017.50, 3035, 2052.50, 2070}d)CostoftheDJ.e)Dependent

170. a)𝐶(𝑛) = 1500+ 75𝑛b)Discrete



c)

171. Manyanswers.Eg.Wage($/hr)172. Manyanswers.Eg.Costtohireataxi.173. Theheatingelementmustturnonandoffto

maintainanapproximatelyconstanttemperature.

174. Theballiskickedfromaheightabovezero(ground).

175. Theriseandfallofastock’spriceovertime.176. Eachlineonthegraphrepresentsarangeof

masses.Thereisnotadifferentpriceforeverypossiblemass.Forexample,itcurrentlycosts$0.64tosendanyletterunder30gfromonepartofVictoria,B.C.toanother.

177.

178.

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Unit 5: Relations & Functions - Miss Zukowski's Class...1)Relations & Functions: Introduction . 1. Using the following graph, answer the questions below. The graph shows the distance