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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.
. . . . . .
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
. . . . . .
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
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