Download - Section 1: Introduction to Functions and Graphs · 1.05 Graphs of Functions • Precalculus (2)(D) • Precalculus (2)(F) • Precalculus (2)(G) • Precalculus (2)(I) Note: Unless

Section1:IntroductiontoFunctionsandGraphsThefollowingmapsthevideosinthissectiontotheTexasEssentialKnowledgeandSkillsforMathematicsTAC§111.42(c).1.01Lines

• Precalculus(1)(A)• Precalculus(1)(B)• Precalculus(1)(C)

1.02Functions

• Precalculus(1)(A)• Precalculus(1)(B)

1.03AlgebraofFunctions

• Precalculus(2)(A)• Precalculus(2)(B)• Precalculus(2)(C)

1.04InverseFunctions

• Precalculus(2)(E)1.05GraphsofFunctions

• Precalculus(2)(D)• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)

Note:Unlessstatedotherwise,anysampledataisfictitiousandusedsolelyforthepurposeofinstruction.

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

1.01 Lines

PropertiesofLines

Slope–

Intercepts–

• Tofindthex-intercepts,set_______andsolvefor_______.• Tofindthey-intercepts,set_______andsolvefor_______.

ParallelLines–Havethesameslope(𝑚" = 𝑚$)

PerpendicularLines–Havethenegativereciprocal(𝑚" = − "&'

)

EquationsofLines

• Slope-InterceptForm: 𝑦 = 𝑚𝑥 + 𝑏o slope=𝑚o y-intercept=𝑏

• Point-SlopeForm: 𝑦 − 𝑦" = 𝑚(𝑥 − 𝑥")o slope=𝑚o point=(𝑥", 𝑦")

• GeneralForm: 𝐴𝑥 + 𝐵𝑦 = 𝐶o slope = −2

3

• VerticalLine: 𝑥 = 𝑎o slope:undefinedo hasanx-intercept

• HorizontalLine: 𝑦 = 𝑏o slope=0o hasay-intercept

1. Findtheequationofalinethatpassesthroughthepoint(4,1)andisperpendiculartothelinethatcontainsthepoints(3, −7)and(−1,9).


2. Findtheequationofalinethatisperpendiculartotheline𝑥=4andpassesthroughthepoint(3,11).

3. Thefreezingpointofwateris0°Cand32°F,whiletheboilingpointis100°Cand212°F.

i. ExpresstheFahrenheittemperatureFintermsoftheCelsiustemperatureC.

ii. DeterminethetemperatureindegreesFahrenheitthatcorrespondsto42°C.


1.02 Functions

Function–arulethatassignstoeveryinputvalueonlyoneoutputvalue

• Notwoorderedpairswillcontainthesamefirstcoordinate.

• Graphically,everyfunctionmustpassthe ______________ .

• 𝑓 𝑥 = 𝑦o Thismeansthatif𝑓 2 = 3,thecoordinateis___________.

o Thecoordinate 𝑥, 𝑓 𝑥 isequivalentto 𝑥, 𝑦 .

o 𝑥isknownastheindependentvariable,and isthesetofallpossible𝑥-values.

o 𝑦isknownasthedependentvariable,and isthesetofallpossible𝑦-values.

1.Whichofthefollowingequations(relations)arefunctions?

i.𝑦 = 3𝑥 + 4 ii.𝑦 = |𝑥| iii. 𝑦 = 𝑥

iv.𝑦 = 3 v.𝑥 = 5 vi.𝑦$ = 𝑥 + 3

vii.𝑥 = 3𝑦 + 1 viii.{(2,3), (4,5), (6,6), (7,6)} ix.{(1,1), (2,2), (2,1), (1,2)}

• In general, an equation is a function if the output variable has an odd

power or root. • An equation cannot be a function if the output variable has an even

power or an absolute value.


2. Giventhegraphof𝑓 𝑥 below,answerthefollowingquestions:

i. Whatis𝑓 0 ?

ii. Forwhatvalue(s)of𝑥does𝑓 𝑥 = 0?

iii. Whatarethedomainandrangeof𝑓 𝑥 ?


3.Given𝑓 𝑥 = 𝑥$ + 2𝑥 − 7

i. Find𝑓 −2 .

ii. Findthe𝑥-valuewhere𝑓 𝑥 = 1.

iii. EvaluateD EFG HD EG

.

When asked to evaluate the expression D EFG HD E

G, the goal is to reduce

the ℎ in the denominator while simplifying. Common techniques include factoring, common denominators, and conjugates.


Domain–theinputvaluesofagivenequation,alsoknownasthesetofallx-values

Therearethreemaindomainrestrictionsuntilyoureachtrigonometry.

Rule1:Afunctioncannothaveanegativenumberunderanevenroot.(Youcanhaveazero.)

• Setwhatisinsidetheradical 0³ andsolve.Theanswerwillbethedomain.• Findthedomainofthefollowingfunction: ( ) 7f x x= -

Rule2:Afunctioncannothaveazerointhedenominator.

• Setthedenominator=0andsolve.Thosearethenumbersthatarenotinthedomain.

• Findthedomainofthefollowingfunction: 1( )( 3)( 1)

xf xx x

+=

+ -

Rule3:Afunctioncannothaveanegativeorzeroinsidealog/ln.

• Settheinsideortheargument>0andsolve.Theanswerwillbethedomain.• Findthedomainofthefollowingfunction: ( )2( ) ln 9f x x= -


4. Findthedomainofthefollowingfunctions:

𝑓 𝑥 = "JHK

𝑓 𝑥 = 𝑥$ + 2𝑥 − 7

𝑓 𝑥 = 1 − L

K'

5. SupposeyouaresellinghotdogsatahotdogstandindowntownAustin.Youstartsellinghotdogsat$4eachandendupsellinganaverageof200hotdogsperday.Whenyouincreasethepriceby$1,salesdecreaseby20hotdogsperday.Assumetherelationshipbetweenhotdogs(x)andprice(p)islinear.

i. Determinetwopointsintheform(𝑥, 𝑝).

ii. Calculatetheslopeofthelinearfunctionbetweenhotdogsandprice.

iii. Findtheequationofthelinepasafunctionofx.


1.03 Algebra of Functions

Addition: 𝑓 + 𝑔 𝑥 = 𝑓 𝑥 + 𝑔 𝑥 Subtraction: 𝑓 − 𝑔 𝑥 = 𝑓 𝑥 − 𝑔 𝑥

Multiplication: 𝑓𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥 Division: DO

𝑥 = D KO K

,𝑔 𝑥 ≠ 0

Tofinddomainoftheabovefunctions,takeintoaccounttheindividualfunctionsandthecombinedfunction.

Composition: 𝑓 ∘ 𝑔 𝑥 = 𝑓 𝑔 𝑥

Tofindthedomainofacompositefunction:

1) Determinethedomainoftheinnerfunction.

2) Then,determinethedomainofthecompositefunction.

3) Lastly,combinesteps1and2tofindtheoveralldomainofthecompositefunction.

1. Let𝑓 𝑥 = 𝑥and𝑔 𝑥 = "KHR

i. Evaluatetheexpression 𝑓 + 𝑔 9 .

ii. Evaluatetheexpression 𝑔 ∘ 𝑓 9 .

iii. Findthefunction 𝑓𝑔 𝑥 anditsdomain.


iv. Findthefunction 𝑓 ∘ 𝑔 𝑥 anditsdomain.

v. Findthefunction 𝑔 ∘ 𝑓 𝑥 anditsdomain.

2. Represent𝑦 = 𝑥 − 2 J + 1asacompositionofthetwofunctions𝑓 𝑥 and𝑔 𝑥 .

3. Thecost,𝐶,ofbuyingagivenproductcanbedeterminedusingthefunction𝐶 𝑥 = 2𝑥 + 5,where𝑥istheamountoftheproductavailable.Theavailabilityoftheproductcanbedeterminedusing𝑥 𝑡 = 80 − 2𝑡where0 < 𝑡 < 32.Determinetheformulaforcost,basedonthedayofthemonth.


1.04 Inverse Functions

PropertiesofInverseFunctions

• Theinverseof𝑓 𝑥 isdenoted𝑓H" 𝑥 .• Fortheinverserelationtobeafunction,itmustpassthehorizontallinetest(and

alreadypasstheverticallinetestbecausetheyarefunctions).• Functionsandtheirinverserelationsaresymmetricovertheline𝑦 = 𝑥,meaning

if𝑓 2 = 3,then____________.• Functionscanbeone-to-one,meaningeach𝑥-valueand𝑦-valueisusedonlyonce.• Ifafunctionisone-to-one,thentheinverserelationisalsoafunction.• Tofindtheinversefunction,switch𝑥and𝑦fortheentireequationandthensolvefor𝑦.• Ifafunctionhasaninverse,then𝑓 𝑓H" 𝑥 = 𝑥and𝑓H" 𝑓 𝑥 = 𝑥.• Sometimesifaninverserelationisnotafunction,thenitsdomaincanberestrictedto

makeitafunction.

1. Let𝑓 𝑥 = 𝑥 − 2 J + 1

i. Find𝑓H" 𝑥 .

ii. Showthat𝑓 𝑓H" 𝑥 = 𝑥and𝑓H" 𝑓 𝑥 = 𝑥.


2. Let𝑓 𝑥 = $KH"KFV

i. Find𝑓H" 𝑥 .

ii. Find𝑓H" 3 .

iii. Findthedomainandrangeof𝑓 𝑥 and𝑓H" 𝑥 .

• The domainof𝑓H" 𝑥 =therangeof𝑓 𝑥 • The rangeof𝑓H" 𝑥 =thedomainof𝑓 𝑥


3. Whichofthefunctionsbelowareone-to-one?i. 𝑓 𝑥 = |𝑥|

ii. 𝑓 𝑥 = 𝑥J

iii. 𝑓 𝑥 = "K'

iv. 𝑓 𝑥 = "K

v. 𝑓 𝑥 = 𝑥W

vi. 𝑓 𝑥 = 𝑥X

vii. 𝑓 𝑥 = 𝑥$ − 4, 𝑥 ≥ 0

• Ifafunctionhasanevenpowerorabsolutevalueforanyvariable,

andithasnodomainrestriction,itwillnotbeone-to-one.• Thatdoesnotmeanthatoddpowerswillnecessarilybeone-to-one.

Forexample:𝑓 𝑥 = 𝑥J − 𝑥.


1.05 Graphs of Functions, Symmetry, Reflections, and Transformations

𝑓 𝑥 = 𝑥$ 𝑓 𝑥 = |𝑥| 𝑓 𝑥 = 𝑥

𝑓 𝑥 = 𝑥J 𝑓 𝑥 = "K 𝑓 𝑥 = 𝑥W

Givenafunction𝑓(𝑥),andaconstant!!c >0 :𝑓 𝑥 + 𝑐 Shifts𝑓(𝑥)up𝑐units

𝑓 𝑥 − 𝑐 Shifts𝑓(𝑥)down𝑐units

𝑓(𝑥 + 𝑐) Shifts𝑓(𝑥)left𝑐units

𝑓(𝑥 − 𝑐) Shifts𝑓(𝑥)right𝑐units

𝑓(−𝑥) Reflects𝑓(𝑥)overthe𝑦-axis

−𝑓(𝑥) Reflects𝑓(𝑥)overthe𝑥-axis


Tographtheequationofafunction,firstdetermineallthereflections,thendeterminethetranslations,andthengraph.

1. Graphthefollowingfunctions.Thendeterminethedomainandrangeforeachfunctionandprovidetheminintervalform.

𝑓 𝑥 = 𝑥 + 1 $ − 3 𝑓 𝑥 = 2 − 𝑥 − 1

Domain: Range: Domain: Range:

Stretches/Compressions

VerticalStretchexample: VerticalCompressionexample:

𝑓 𝑥 = 𝑥$ 𝑓 𝑥 = 2𝑥$ 𝑓 𝑥 = "$𝑥$

Givenafunction,𝑓 𝑥 ,andaconstant,𝑐:

• If 𝑐𝑓 𝑥 ,then𝑓 𝑥 stretchesverticallyif 𝑐 > 1andcompressesverticallyif 𝑐 < 1.• If 𝑓 𝑐𝑥 ,then𝑓 𝑥 compresseshorizontallyif 𝑐 > 1andstretcheshorizontallyif

𝑐 < 1.


2. Provideaverbaldescriptionofthetransformationsappliedto𝑓 𝑥 toobtain𝑔 𝑥 .

𝑓 𝑥 = 𝑥$𝑔 𝑥 = 3 − 6 𝑥 − 2 $

3. Giventhegraph𝑓 𝑥 , answer the following questions: i. Whatistheparentfunction?

ii. Isthegraphshiftedleftorright?Ifso,byhowmanyunits?

iii. Isthegraphshiftedupordown?Ifso,byhowmanyunits?

iv. Isthegraphreflectedonthex-axis,y-axis,ornotatall?

v. Isthegraphstretchedorcompressedverticallyornotatall?

vi. Whatistheequationfor𝑓 𝑥 ?


A chart to help remember symmetry: 𝑥-axis 𝑦-axis origin

4. Giventhat𝑓 𝑥 isanoddfunctionand𝑓 5 = −3,whatisanotherpointon𝑓 𝑥 ?

5. Whichofthefollowingareoddfunctions?i. 𝑓 𝑥 = JKHR

K

ii. 𝑦 = 𝑥J + 4𝑥iii. 𝑥$ + 𝑦$ = 25

EvenFunctions

• Symmetricaroundthe𝑦-axis

• 𝑓 𝑥 =

𝑓 −𝑥

• Replace𝑥with– 𝑥togetthesameequation

OddFunctions

• Symmetricaroundtheorigin

• 𝑓 𝑥 =

−𝑓 −𝑥

• Replace𝑥with– 𝑥and𝑦with– 𝑦togetthesameequation


6. Whichofthefollowingfunctionsareeven?i. 𝑦 = 𝑥$ + 3𝑥 + 1ii. 𝑦 = "

K'FRW iii. 𝑓 𝑥 = |𝑥 + 3|iv. 𝑓 𝑥 = 𝑥 + 3

7. Giventhat𝑓 𝑥 isevenwith𝑓 3 = 4,𝑔 𝑥 isone-to-onewith𝑔 4 = −5,andℎ 𝑥 = 1 − 3𝑥,find:i. ℎ ∘ 𝑓 −3

ii. 𝑔H" ∘ ℎ 2
