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Sections1.1–1.2Functions

V63.0121, CalculusI

September10, 2009

Announcements

I SyllabusisonthecommonBlackboardI OfficeHoursTBA

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OutlineWhatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicity

ClassesofFunctionsLinearfunctionsOtherPolynomialfunctionsOtherpowerfunctionsRationalfunctionsTrigonometricFunctionsExponentialandLogarithmicfunctions

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DefinitionA function f isarelationwhichassignstotoeveryelement x inaset D asingleelement f(x) inaset E.

I Theset D iscalledthe domain of f.I Theset E iscalledthe target of f.I Theset { f(x) | x ∈ D } iscalledthe range of f.

. . . . . .

OutlineWhatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicity

ClassesofFunctionsLinearfunctionsOtherPolynomialfunctionsOtherpowerfunctionsRationalfunctionsTrigonometricFunctionsExponentialandLogarithmicfunctions

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TheModelingProcess

...Real-worldProblems

..Mathematical

Model

..MathematicalConclusions

..Real-worldPredictions

.model.solve

.interpret

.test

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Plato’sCave

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TheModelingProcess

...Real-worldProblems

..Mathematical

Model

..MathematicalConclusions

..Real-worldPredictions

.model.solve

.interpret

.test

.Shadows .Forms

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OutlineWhatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicity

ClassesofFunctionsLinearfunctionsOtherPolynomialfunctionsOtherpowerfunctionsRationalfunctionsTrigonometricFunctionsExponentialandLogarithmicfunctions

. . . . . .

Functionsexpressedbyformulas

Anyexpressioninasinglevariable x definesafunction. Inthiscase, thedomainisunderstoodtobethelargestsetof x whichaftersubstitution, givearealnumber.

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Example

Let f(x) =x + 1x− 1

. Findthedomainandrangeof f.

SolutionThedenominatoriszerowhen x = 1, sothedomainisallrealnumbersexceptingone. Asfortherange, wecansolve

y =x + 1x− 1

=⇒ x =y + 1y− 1

Soaslongas y ̸= 1, thereisan x associatedto y.

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Example

Let f(x) =x + 1x− 1

. Findthedomainandrangeof f.

SolutionThedenominatoriszerowhen x = 1, sothedomainisallrealnumbersexceptingone. Asfortherange, wecansolve

y =x + 1x− 1

=⇒ x =y + 1y− 1

Soaslongas y ̸= 1, thereisan x associatedto y.

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No-no’sforexpressions

I CannothavezerointhedenominatorofanexpressionI Cannothaveanegativenumberunderanevenroot(e.g.,squareroot)

I Cannothavethelogarithmofanegativenumber

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Piecewise-definedfunctions

ExampleLet

f(x) =

{x2 0 ≤ x ≤ 1;

3− x 1 < x ≤ 2.

Findthedomainandrangeof f andgraphthefunction.

SolutionThedomainis [0, 2]. Therangeis [0, 2). Thegraphispiecewise.

...0

..1

..2

..1

..2

.

.

.

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Piecewise-definedfunctions

ExampleLet

f(x) =

{x2 0 ≤ x ≤ 1;

3− x 1 < x ≤ 2.

Findthedomainandrangeof f andgraphthefunction.

SolutionThedomainis [0, 2]. Therangeis [0, 2). Thegraphispiecewise.

...0

..1

..2

..1

..2

.

.

.

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Functionsdescribednumerically

Wecanjustdescribeafunctionbyatableofvalues, oradiagram.

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Example

Isthisafunction? Ifso, whatistherange?

x f(x)1 42 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, therangeis {4, 5, 6}.

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Example

Isthisafunction? Ifso, whatistherange?

x f(x)1 42 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, therangeis {4, 5, 6}.

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Example

Isthisafunction? Ifso, whatistherange?

x f(x)1 42 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, therangeis {4, 5, 6}.

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Example

Isthisafunction? Ifso, whatistherange?

x f(x)1 42 43 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, therangeis {4, 6}.

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Example

Isthisafunction? Ifso, whatistherange?

x f(x)1 42 43 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, therangeis {4, 6}.

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Example

Isthisafunction? Ifso, whatistherange?

x f(x)1 42 43 6

.

. .

..1

..2

..3

. .4

. .5

. .6

Yes, therangeis {4, 6}.

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Example

Howaboutthisone?

x f(x)1 41 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

No, thatone’snot“deterministic.”

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Example

Howaboutthisone?

x f(x)1 41 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

No, thatone’snot“deterministic.”

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Example

Howaboutthisone?

x f(x)1 41 53 6

.

. .

..1

..2

..3

. .4

. .5

. .6

No, thatone’snot“deterministic.”

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Inscience, functionsareoftendefinedbydata. Or, weobservedataandassumethatit’sclosetosomenicecontinuousfunction.

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Example

HereisthetemperatureinBoise, Idahomeasuredin15-minuteintervalsovertheperiodAugust22–29, 2008.

...8/22

..8/23

..8/24

..8/25

..8/26

..8/27

..8/28

..8/29

..10

..20

..30

..40

..50

..60

..70

..80

..90

..100

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Functionsdescribedgraphically

Sometimesallwehaveisthe“picture”ofafunction, bywhichwemean, itsgraph.

.

.

Theoneontherightisarelationbutnotafunction.

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Functionsdescribedgraphically

Sometimesallwehaveisthe“picture”ofafunction, bywhichwemean, itsgraph.

.

.

Theoneontherightisarelationbutnotafunction.

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Functionsdescribedverbally

Oftentimesourfunctionscomeoutofnatureandhaveverbaldescriptions:

I Thetemperature T(t) inthisroomattime t.I Theelevation h(θ) ofthepointontheequationatlongitude

θ.I Theutility u(x) I derivebyconsuming x burritos.

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OutlineWhatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicity

ClassesofFunctionsLinearfunctionsOtherPolynomialfunctionsOtherpowerfunctionsRationalfunctionsTrigonometricFunctionsExponentialandLogarithmicfunctions

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Monotonicity

ExampleLet P(x) betheprobabilitythatmyincomewasatleast$x lastyear. Whatmightagraphof P(x) looklike?

.

..1

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Monotonicity

ExampleLet P(x) betheprobabilitythatmyincomewasatleast$x lastyear. Whatmightagraphof P(x) looklike?

.

..1

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Monotonicity

Definition

I A function f is decreasing if f(x1) > f(x2) whenever x1 < x2foranytwopoints x1 and x2 inthedomainof f.

I A function f is increasing if f(x1) < f(x2) whenever x1 < x2foranytwopoints x1 and x2 inthedomainof f.

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Examples

ExampleGoingbacktotheburritofunction, wouldyoucallitincreasing?

ExampleObviously, thetemperatureinBoiseisneitherincreasingnordecreasing.

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Examples

ExampleGoingbacktotheburritofunction, wouldyoucallitincreasing?

ExampleObviously, thetemperatureinBoiseisneitherincreasingnordecreasing.

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OutlineWhatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicity

ClassesofFunctionsLinearfunctionsOtherPolynomialfunctionsOtherpowerfunctionsRationalfunctionsTrigonometricFunctionsExponentialandLogarithmicfunctions

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ClassesofFunctions

I linearfunctions, definedbyslopeanintercept, pointandpoint, orpointandslope.

I quadraticfunctions, cubicfunctions, powerfunctions,polynomials

I rationalfunctionsI trigonometricfunctionsI exponential/logarithmicfunctions

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Linearfunctions

Linearfunctionshaveaconstantrateofgrowthandareoftheform

f(x) = mx + b.

ExampleInNewYorkCitytaxiscost$2.50togetinand$0.40per 1/5mile. Writethefare f(x) asafunctionofdistance x traveled.

AnswerIf x isinmilesand f(x) indollars,

f(x) = 2.5 + 2x

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Linearfunctions

Linearfunctionshaveaconstantrateofgrowthandareoftheform

f(x) = mx + b.

ExampleInNewYorkCitytaxiscost$2.50togetinand$0.40per 1/5mile. Writethefare f(x) asafunctionofdistance x traveled.

AnswerIf x isinmilesand f(x) indollars,

f(x) = 2.5 + 2x

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Linearfunctions

Linearfunctionshaveaconstantrateofgrowthandareoftheform

f(x) = mx + b.

ExampleInNewYorkCitytaxiscost$2.50togetinand$0.40per 1/5mile. Writethefare f(x) asafunctionofdistance x traveled.

AnswerIf x isinmilesand f(x) indollars,

f(x) = 2.5 + 2x

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OtherPolynomialfunctions

I Quadraticfunctions taketheform

f(x) = ax2 + bx + c

Thegraphisaparabolawhichopensupwardif a > 0,downwardif a < 0.

I Cubicfunctions taketheform

f(x) = ax3 + bx2 + cx + d

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OtherPolynomialfunctions

I Quadraticfunctions taketheform

f(x) = ax2 + bx + c

Thegraphisaparabolawhichopensupwardif a > 0,downwardif a < 0.

I Cubicfunctions taketheform

f(x) = ax3 + bx2 + cx + d

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Otherpowerfunctions

I Wholenumberpowers: f(x) = xn.

I negativepowersarereciprocals: x−3 =1x3.

I fractionalpowersareroots: x1/3 = 3√x.

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Rationalfunctions

DefinitionA rationalfunction isaquotientofpolynomials.

Example

Thefunction f(x) =x3(x + 3)

(x + 2)(x− 1)isrational.

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TrigonometricFunctions

I SineandcosineI TangentandcotangentI Secantandcosecant

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ExponentialandLogarithmicfunctions

I exponentialfunctions(forexample f(x) = 2x)I logarithmicfunctionsaretheirinverses(forexample

f(x) = log2(x))