AnnouncementsTopics:

-  sections1.3,2.1,1.4,and2.2

*Readthesesectionsandstudysolvedexamplesinyourtextbook!

Homework:

-  reviewlecturenotesthoroughly-  workonpracticeproblemsfromthetextbookandassignmentsfromthecoursepackasassignedonthecoursewebpage(underthe“SCHEDULE+HOMEWORK”link)

WorkingWithFunctions

•  Reviewaddition,subtraction,multiplication,division,andcompositionoffunctionsonyourown…

•  Reviewtransformationsofgraphsandinversefunctions(we’lldoabriefreviewhere)

InverseFunctions

Thefunctionistheinverseofifand.Eachofandundoestheactionoftheother.Somesimpleexamples:

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f

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f −1

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f −1( f (x)) = x

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f ( f −1(x)) = x

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f −1

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f

WhatFunctionsHaveInverses?

Afunctionhasaninverseifandonlyifitisaone-to-onefunction.Afunctionfisone-to-oneifforeveryy-valueintherangeoff,thereisexactlyonex-valueinthedomainoffsuchthaty=f(x).

HorizontalLineTest

Ifeveryhorizontallineintersectsthegraphofafunctioninatmostonepoint,thenthegraphrepresentsaone-to-onefunction.

FindingtheInverseofaFunctionAlgorithm:1.  Writetheequationy=f(x).2.  Solveforxintermsofy.3.  Replacexby(x)andybyx.

Note:ThedomainandrangeareinterchangedExample:.

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f −1

TemperatureConversion

TherelationshipbetweendegreesCelsius(C)anddegreesFahrenheit(F)islinear.Weknowthatcorrespondstoandcorrespondsto.

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32oF

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0oC

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100oC

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212oF

TemperatureConversion

(a) FindthefunctionthatconvertstoNote:inputisandoutputisDataPoints:Function:

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oC

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oF .

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slope =change in outputchange in input

=ΔFΔC

=212 − 32100 − 0

=1.8

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(0oC, 32oF) and (0oC, 32oF)€

oC

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oF

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F(C) =1.8C + 32

TemperatureConversion

(b)FindthefunctionthatconvertstoNote:inputisandoutputisOneapproach:FindtheINVERSEofF(c):

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oC.

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oF

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oF

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oC

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F =1.8C + 32F − 32 =1.8CF − 321.8

= C

∴C(F) =F − 321.8

Note:wedonotinterchangevariablesattheendsinceFandChaveaphysicalmeaning

ExponentialFunctions

Anexponentialfunctionisafunctionoftheformwhereisapositiverealnumbercalledthebaseandisavariablecalledtheexponent.Domain:Range:

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f (x) = ax

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a

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x

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x∈ R

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y > 0*Note:PleasereviewEXPONENTLAWSonyourown!

GraphsofExponentialFunctions

y

x

y

x

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f (x) = 3x

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f (x) = 12( )x

Whena>1,thefunctionisincreasing. Whena<1,thefunctionisdecreasing.

y=0isahorizontalasymptote

TransformationofanExponentialFunction

GraphRecall:isaspecialirrationalnumberbetween2and3thatiscommonlyusedincalculusApproximation:

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e ≈ 2.718

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e

y

x€

f (x) = e−2x + 3.

LogarithmicFunctions

Theinverseofanexponentialfunctionisalogarithmicfunction,i.e.Cancellationequations:Ingeneral: Forexponentials&logarithms:

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aloga x = x

loga ax = x

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f ( f −1(x)) = x

f −1( f (x)) = x

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If f (x) = ax, then f −1(x) = loga x.

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e ln x = x

lnex = x

GraphsofLogarithmicFunctions

Recall:Forinversefunctions,thedomainandrangeareinterchangedandtheirgraphsarereflectionsinthelineExample:Graph

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f (x) = ln x.€

y = x.

GraphsofLogarithmicFunctionsy

x

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f −1(x) = ex

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(−1,e−1)

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(0,1)€

(1,e)€

e ≈ 2.7

GraphsofLogarithmicFunctions

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f −1(x) = ex

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(−1,e−1)

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(0,1)€

(1,e)€

e ≈ 2.7

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y = xy

x

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(e,1)

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(1,0)

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(e−1,−1)€

f (x) = ln x

Memorize!!!

LawsofLogs

Forx,y>0andpanyrealnumber:

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ln(xy) = ln x + ln yln(x / y) = ln x − ln yln(x p ) = pln x

SemilogGraphs

Definition:Asemiloggraphplotsthelogarithmoftheoutputagainsttheinput.Thesemiloggraphofafunctionhasareducedrangemakingthekeyfeaturesofcertainfunctionseasiertodistinguish.

SemilogGraphs

Example:

SemilogGraphs

Example:Sketchthesemiloggraphof

f (x) =10e−4x.

Double-LogGraphs

Definition:Adouble-loggraphplotsthelogarithmoftheoutputagainstthelogarithmoftheinput.

SemilogandDouble-LogGraphs

Example:BloodCirculationTimeinMammalsSketchthesemiloganddouble-loggraphsforthemodel

T (B) =17.73B0.25.

B

ExponentialModels

Whenthechangeinameasurementisproportionaltoitssize,wecandescribethemeasurementasafunctionoftimebytheformulawhereisthevalueofthemeasurementattimeistheinitialvalueofthemeasurement,andisaparameterwhichdescribestherateatwhichthemeasurementchanges

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S(t) = S(0)eα t

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S(t)

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S(0)

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α

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t

DoublingTime.

Example:Abacterialculturestartswith100bacteriaandafter3hoursthepopulationis450bacteria.Assumingthattherateofgrowthofthepopulationisproportionaltoitssize,findthetimeittakesforthepopulationtodouble.

Half-LivesofDrugs Half-life

Tetrahydrocannabinol … Marijuana (infrequent users)

1.3-3 days

Marijuana (frequent users) 1-10 days

Marijuana (if taken orally as pills) 25-36 hours

Marijuana (smoking/inhaling) 1.6-59 hours

LSD (Lysergic acid diethylamide) 3-5 hours

MDMA … ecstasy Methylenedioxymethamphetamine

6-10 hours

Caffeine adults 4-5 hours

Caffeine infants 10-20 hours

Caffeine with oral contraceptives 5-10 hours

Caffeine (if pregnant) 9-11 hours

Caffeine (liver disease) several days

Codeine (Tylenol 3) 3-6 hours

Demerol (pain killer) 3-5 hours

Morphine (pain killer) 2-3 hours

Heroin (IV or inhaled) 3-5 minutes

Cocaine (benzoylmethylecgonine) 1 hour

Psilocin … magic mushrooms, shrooms

2-3 hours

Phencyclidine … rocket fuel, killer weed, angel dust

7-46 hours

Half-LivesofDrugs

Example:ThinkinginHalf-Lives

**Manydrugsarenoteffectivewhenlessthan5%oftheiroriginallevelremainsinthebody.

#ofhalf-lives amountleftinbody %amountleftinbody

0 M(0) 100

1

2

3

4

5