Functions and their Applications

8/12/2019 Functions and their Applications

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I.2 FUNcnoNsANDTHEIRROpERTTEsWhatyou'll learnaboutr FunctionDefinit ion nd Notationr Domainand Ranger Continuityr Increasing nd DecreasingFunctionsr Boundednessr Local nd AbsoluteExtremar Symmetryr Asymptotesr EndBehavior. . . andwhyFunctionsnd raphsorm hebasisfor understandinghe mathematics

FunctionDefinitionandNotationMathematics and its applications abound with exawhich quantitativevariablesare related o each other.tion of functions is ideal for that purpose.A function iscept; f it were not, history would havereplaced t withHere is the definition.

$E{fi*l* "1 Functionsand

DefinitionFunction,Donrain, nd RangeA funCtion rom a setD to a set R is a rule that assin D a uniqueelement n R. The set D of all input vof the function, and thesetR of all output values s function.


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8z C}|APTER Functions ndGraphs

: EXAnfiPLEDefininga functioni Does the formula : xz definey as a function of r?,I SOrUTrOlr; V"r, y is a function of x. In fact, we can write the foi tion:./(x) : -x2.When a number r is substituted ntoi of., will be the output, and there is no ambiguity ai ' ls. f.l*

Another useful way to look at functions is graphifunctiony =/(r) is the set of all points (r,f (*)), xmatch domain values along the x-axis with theil rangto get the orderedpairs that yield the graphof y : f(x

EXAIfiPtE Seeing functiongraphicaOf the threegraphsshown n Figure 1.11,which is tion? How canyou tell?


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WHntnsour ontn?iVhenmoving rom a numericalmodel o analgebraic odelwe wil l oftenusea function;o approximate atapairs hat by hem-selves iolate ur definit ion.n Figure .12 e:an see hat several airsof data points ai l:hevertical ine est,andyet the l inear unc-:"onapproximateshe dataquitewell.

SFenON"e Functionsnd Th

Domain ndRangeWe will usually define functions algebraically, givingterms of the domain variable. The rule, however,does story without some considerationof what the domain acFor example, we can define the volume of a sphereas a by the formula

4V(r) :52'rr (Note that this is "V of y''-notThis formula is defined for all real numbers,but the vodefined for negative r values. So, if our intention werefunction, we would restrict the domain to be all r > 0.

VerticalLineTeptA graph(setof points(x, y)) in the.ry-planedefinesif and only if no vertical ine intersectshe graph n m


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84 CF{APTER FunctionsnctGraphs

$$iiIf*if

5upportGraphicallyWe cansupportour answersn (a) and (b) grahouldnorplotpoinrswhere he unc; )rr"io"o3l)gt":jt::'fi:::lj. orv : f , ', 3-Figu(b) Ttre graphof t : {x/(x- 5) @gure 1.13r > 0, as expecred, ur shows"" ;;;";; fi: 5. This line, a form "r gr"ph* r",;"";;;bedhould otbe here.gnoringit," r"",rr"is,

"rl^0""Gf re graph f y : 16 ^1r::,(Figurel.t3c) showr thealgebraicxpression:ll reall rrO.rr. ifknowing hat ir ttr. r.ngtt,of asideof ut iunet"l"c

J-I l r , r r


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SECTION.2 Functions nd Thei

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u,5l by -3,3lFrcunr 1.14 tregraph fy : 2/x. ls : 0 in herange?It appears hat x :0 is not in the domain (as expected,bnator cannot be zero). It alsoappears hat the range consibersexcept 0.ConfirmAlgebraicallyWe confirm that 0 is not in the range by trying to solve 2)


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86 CHAPTER FunctionsndGraphs

Continuous at all.r Removablediscontinuity Remo


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SlCnOl{ t.Z FunctionsndTheirPro

This graPh also has a removableJir"oo'tittoitY at x:

a' If we ateffi;;int;ehavioi of this function/"forl uifo"t close o a' we atestill not'ass,ireffi;, the /(x) .values will be"i"'"rt"'o::i.i*,Ill'inTiflloesn't even exlst' r;;; we could defrne/(a) ln sYcha*"t ^-1" pttg

thehoie andmake/con-tinuous at a'

Here is a discontinuitY that it i:t."-""rlot". It is a jump discontinuity.;;;;tt" there is more than just ahole at"r"-J)-; U"r" is a jump in function val-i,", irtn,makeshegan Tlol;{}e toa' (c))'no

Removable disco


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88 CffAP?t|l t Functionsand Graphs

CnoosrucVrrwrnc WlroowsMost viewingwindowswil l showa vert icall ine or the funct ion n Figure .16.t i ssometimes ossibleo choose viewingwindow in which he vert ical inedoesnotappear, s we did n Figure .16.

EXAMPLI ldentifyingpointsof discoJudging from the graphs,which of the following fthat are discontinuousat -{ : 2? Are any of the discsotuTlotuFigure 1.16showsa function hat s undefinedatxtinuous there.The discontinuity at x : 2 is not remThe function graphed n Figure 1 17 s a quadraticpis aparabola,a graph hat has no breaksbecause ts dnumbers. t is continuous or all;r.The functiongraphedn Figure 1.18 s not definedbe contiduousthere. The graph looks like the graphexq)pt that there s a hole where the point (2, 4) shoable discontinuity.

gt{'eg


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EXPTOtATtOil

AU:r on a Clrcuuron

SECTIOI{1.2 Functionsand


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9o CHAnEn I FunctionsndGraphs

E)(AfipH 6 Analyzinga function for indecreasing ehaviorFor each function, ten the intervars on which it isvalson which it is decreasins.(a l f (x) :(x*2\2sotuTtotSolveGraphically(a)We see rom the- raptrin fiSure 1 20 that is dand ncreasing nl-2, oo). Notice hatwe incluDon't worry that this setsup somecontradictat -2, because e onrytalk about unctionsncrintervals,and 2 is noi an nterval.)


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SECTfOlf1.2 FunctionsandT

BoundednessThe concept of boundednesss fairly simple to undersand algebraically. We will move directly to the algebraicvating the conceptwith some ypical graphs(Figure I'2

Not boundedaboveNot boundedbelow

Not boundedaboveBoundedbelow

Bounded aboveNot bounded below

above and below.lcune1.22 Some xamplesf graphs oundedndnotbounded


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92 CHAPTER FunctionsndGraphs

Flcun: 1.24 The raph uggestshat/hasa ocalmaximum tp,a localminimum tQ,anda ocalmaximum tR.

Thus, -4 is a lower bound for w(x) : 3x2- 4.we leavethe verification thatp is boundedas an

localandAbsoluteExtremaMany graphsare characterizedby peaks and valleyincreasingto decreasingand vice versa.The extremelocal extrema) anbe characterized seither ocal mndistinction can be easily seengraphically. Figure 7.24local extrema:local maxima at points p andR and a This is another function concept that is easier todescribealgebraically.Notice that a local maximummaximum value of a function; it only needs o be thfunction on some iny interval.

{f,f,t,i*f

DefinitionLocal ndAbsoluteExtrema


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Symmetry ith respecto they-axisExample:/(x) YzGraphically

v

SECTfON.t Functions ndT

SymmetqyIn the graphical sens'e, he word "symmet4r" in mathtially the samemeaning as t does in art: The picfure (i"looks the same" when viewed in more than one wayabout mathematical symmetry is that it can be charactealgebraically as well. We will be looking at three partictry, each of which can be spotted easily from a graph, aalgebraic formula, onceyou know what to look for. Sinamong the three models (graphical, numerical, and algemphasize in this section, we will illustrate the variothree ways, side-by-side.

Numerically AlgFor all r in the df(x)


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94 CHAPTER FunctionsndGraohs

Symmetry ith respecto theoriginExample:/(x)2r:Graphicallyv

Frcunr 1.28 rhe graphookshesameupside-downs t does ightside-up.

Numerically For all x in

Functionsn odd) are

-?_a

1

1zJ

-27-8-1

18

27

EXAMPTECheckingunctionsor s


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t-s,5lbY -4,alFtcunr 1.3OThis raph oes otappearto besymmetric ith respecto either hey-axis r the origin, owe conjecturehata isneither ven orodd.

SECTfOI{.2 FunctionsndT

Since this identity is true for all x, the function/is ind(b) SolveGriphicallyThegraphical olutions shownConfirmAlgebraicallyWeneed o verify that

in Figure 1.30.

sex)+ s(x) nd ex)+ -sGs(-r) :( -r) t-2(-x)-2:x2+2x-2

s(x):x2-2x-2sk) : -x2 + 2x-r 2Sos(-x) + s@)andg(-x)+ -sG).Weconcludehatg is neitherodd nor even.(c)SolveGraphicallyThegraphical olutions shownn Figure1.31.

: l-t ,: I :


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96 {g{&*TSn "l Functionsand Graphs

ilu;umer.33 he graphoff(x) : 2x2/(a x2)with theasymptotesshown sdashedines.

Ffiri-i* r*The line y : b is a horizontal asymptote of rhey : f (x) if l(x) approaches limit of b asx apprIn limit notation;

r \-f t*l : u or "ri11/(x)The |i1e*.- - o it a vertical asymptoteof the gra,;f*iilillJ,o#**'s a imitr+oo r oo

J52_f txl : -f oo or lim /(x)

KK&ffiP$"SS dentifyimghe asymptoIdentify any horizontal or vertical asymptotesof tx

lti

t:i


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98 CHAPTERI Functions and Graphs

In Exercises _4,determiru,u un"tiont .ffi;"fi;f'*irf;|,:"**ra determines In Exercises5_2g, tatelocalminimum,a rocalrJi:t9tonwhich rhe unctin. ," ,:1T:,nwhichheunction, d;;;;;25. yr .y=yf i j3.)c:2y2 2'Y=*z-+t

In Exercises-g,usetheo' a = 72 -

thecurve s thegraph"r. ;il':,[fetest o determinewherher

5.gk


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SECTION.2 FunctionsndTh

In Exercises 47-54, statewhether the function is' odd' even'or

#;;S"u.pp"rt graphically and confirm algebraically'

a7. (x) :2xaqs. f (x) : \F*2sr. (x) : -x2+0.03x+5 t2' f@ : f +O'O4xz 3

68. Can a graPh have two horizontal asmost graphshave at ^o* oo" horizon. possible for a graph to ttuu" more tha' i;itt*t;* tonJtion' have graphs witha8.S(x): x3 350. ( .x) :T+7s+. ("): l

In Exercises55-62' use a method of your choice to find all

O"-."i"f *A verticalasymptotes of the function'

the corresponding

horizontalasymptote?lx3+ 1l(a)/(x) : -s -7 (b)(c) t(x) vx ' -+

69. Cana graph ntersect ts ownvertGraPhhetunction/(x) = l-fl +(a) Thegraphof this functiondoesnurY-Ptot". ExPlainwhY t doesnot(b) Showhow you canadda singleandgetagraph rhatdoes ntersect(c) s thegraph n (b) thegraphof a

zo.Writ ingtoLearnExplainwhyagthan wo horizontal symptotes'

953.(x) :2x3 - 3x

x- l56. q\x)58.q@): r'5'

A60.p(x) 7 + t52.k):fu


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CHAPIERI Functionsnd Grophs

75. Decreasing Function Which function is decreasing?(a) Outdoor temperature as a function of time(b) The Dow Jones Industrial Average as a function of time(c)Air pressure in the Earth's atmosphere as a function ofaltitude(d)World population since 1900 as a function of time(e)Water pressure in the ocean as a function of depth

76. lncreasing or Decreasing Which function cannotbe classi-fied as either increasing or decreasing?(a)Weight of a lead brick as a function of volume(b)Height of a ball that has been tossedupward as a func-tion of time(c)Time of travel from Buffalo to Syracuse as a function ofdriving speed(d)Area of a squareas a function of side length(e)Height of a swinging pendulum as a function of time

(a) Considering GPA (y) as a fu(x), is it increasing,decreasing(b) Make a table showingthe cmove down the list. (SeeExplo(c) Make a table showing the chdown the list. (This is AAy.) Cin GPA as a function of percening, decreasing,constant,or no(d) In general,what can you sagraph if y is an increasing uncdecreasing function of x?(e) Sketch the graph of a functiodecreasing function of x and Aof .x.

79. Group Activity Sketch (freehawith domain all real numbers ting conditions:(a)/is continuous for all x;(b)/is increasing n (-o, 0] a


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xtending he ldeasB. A function that is bounded above has an infinite number of

upper bounds,but there s always aleast upper bound,i'e.,an upper bound that is lessthan all the others.This leastupper bound may or may not be in the rangeoff For eachof the following functions, find the least upper bound andrcll whether or not it is in the range of the function.(a)"f(x) :2-0.8x2

^1x'O)s(x) 3+x,(c lh(x) :+(d)p(x)=2sin(x)

A. v(elqk) :;r+;_ + |

SECTfON.3 Twelv

84. Writing to Learn A continuous funreal numbers. f/(-1) : 5 and/(l)must have at least one zero in the ingeneralizes o a property of continuthe IntermediateValue Theorem.)

85. Proving a Theorem Prove that thefunction with domain all real numbthe origin.

eo.Findingthe angeGraphhe unthewindow-6, 6l by -2,2).(a) What is the apparent horizontal (b) Based on your graph, determine(c)Show lgebraicallyhat 1 =#thus confirming your conjecture n

Functions and their Applications

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