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Section1.1Functions

V63.0121.006/016, CalculusI

January19, 2010

Announcements

I SyllabusisonthecommonBlackboardI OfficeHoursTBA

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Outline

Whatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicitySymmetry

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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.

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Outline

Whatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicitySymmetry

. . . . . .

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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Outline

Whatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicitySymmetry

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

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Outline

Whatisafunction?

Modeling

ExamplesoffunctionsFunctionsexpressedbyformulasFunctionsdescribednumericallyFunctionsdescribedgraphicallyFunctionsdescribedverbally

PropertiesoffunctionsMonotonicitySymmetry

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Monotonicity

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

.

..1

..0.5

..$0

..$52,115

..$100K

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Monotonicity

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

.

..1

..0.5

..$0

..$52,115

..$100K

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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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Symmetry

ExampleLet I(x) betheintensityoflight x distancefromapoint.

ExampleLet F(x) bethegravitationalforceatapoint x distancefromablackhole.

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PossibleIntensityGraph

..x

.y = I(x)

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PossibleGravityGraph

..x

.y = F(x)

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Definitions

Definition

I A function f iscalled even if f(−x) = f(x) forall x inthedomainof f.

I A function f iscalled odd if f(−x) = −f(x) forall x inthedomainof f.

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Examples

I Even: constants, evenpowers, cosineI Odd: oddpowers, sine, tangentI Neither: exp, log