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Unformatted text preview: Chapter 3: Functions The Definition ofa Function (3.1) 9? (Kg :3 lj ﬂ 0 IA} WW Let A and B be nonempty sets. A function from set A to B is a rule of correSpondence that assigns
to each in set A exactly one element in set B. ‘ Examples: ‘W‘kk Pet Weight [lbs]
i  Every element in set A has Exactly one elemetni
' one element in set E but Ford has no assignment ' . the car and thepet’s wetigh exam Il‘ ﬂu. The set A of inputs is called the domain of the function, and the set B of outputs is called the range.  Following are the examples of what can or can’t be a doniain value ' %w9 0U {:Wﬂxuﬁ W domain (Vex/aims) Rind  aForﬁﬁl—rnfﬁl‘ﬂnaequﬁorbiﬁkft+l, . GNUQ ME, (“"Wé) I ' Win ’JHQFM‘Q— Table
X consists ofdomainvalues, A“ Real/9 C 00 l
y consists of the range values QJ“ R ' C450 l (36 ) _ Find the domain of the following function: @ Interval notation: CF00 ’I 4,“ 1}) U 67590.5) 5% ﬂ. MﬁQ‘HU‘Q We @ a Wm; tam? *0 . Interval notatlon E 4 ) ‘
Set notation: E at 6\Q\ ‘b EFL’E Ex3. y: J3z+12, ”r —7:+12 . _ {1’1"}: 4.41 2. 0 ,  , ‘ U ,
Interval notatlon: €005 %\ U [ii‘ﬁ Set notation: {Hid “ht? “Wit A t The graph of a function (3.2] ‘ Definition: The graph of a function f in the xy plane consists of all those points (x, y] such that x is in the domain, and y = f x in the range _
. i /\ 9 (2C) : 3.04 mi L‘sv k How do you tell if a graph off represents a function?
“EH—m WI:— '1ne1ersects the graph in at most one point. (LL/AM 0\ ﬁn : § we
W350 ﬁ*"OYW{iﬂ/&O\ WNW To No)!“ a WONG“ graph in the xy plane represents y as a function of x if any vertical EXW Find if the following graphs represent a function. lustify your answer. Then find the
domain and range if it represénts a function. ‘ 32 The Graph ofa Function 153 'KEﬁﬂﬂﬁﬁ SE? 3.2 ._ 1n Exercises 5 and 6, use the vem‘cal line test to determine
edges I v» 4, speciﬁ: the ycoordinate ofthe point P on the whethe “Ch graph represents a function of x.
ear{he givenﬁmction. In each case. give an exact expres zz calculator approximation rounded to three decimal .y = x
dashed
volving'
places: ' In Exercises 744, the graph of a function is given. In each case,
specify the domain and the range 0 f the ﬁnction. (The axes are
marked aﬁcin oneunit intervals.) to the graph ofrhe function;
are marked affirm onew In In Exercises 1 5 and 16, refer
ﬁgure. (Assume that Ike axes 15. {3) Find F(« 5)
(I!) Find F(2).
(c) Is PU) positive?
(6) For which value of x is F(x) 3 —3'?
(e) Find F(2) — F(2).
16. {a} Find PM).
(11) Find F(—1).
(c) Is F(4) positive?
(a) For which value of x is F(x) = 5?
(c) Find HS) — F(—3). 154 Chapter 3 Functions 7. S. 9. ESL; 1
n Exercises 15 and 16, refer to the graph ofﬂze function F in III}:
ﬁgure. (Assume that the axes are marked affirm magunit irate ' ' 10.
11; ,' 15. (anFind F(—5)
= 4 EH —. i 01;} Find H2).
’ (c) Is F(1) positive?
. . ‘ , . . .. . '(d) For which value ofxis F(x) = —3?
, ;.  + 6. l Lﬂ (e) Find F(2) — F(—2).
 ‘ j 16. (3) Find F(4) .
(b) Find F("~1). \ ‘ {C} 13 F(—4) positive?
_ _ 1 3 (d) For which value of x is F (x) = 5?
I {13) Find F(5) » F(—3)_ Shapes of Graphs: Average Rate of Change (3.3) We will spend a lot of time on studying graphs with many shapes _0ing up and down in this
" . how to describe t'em precisely. ‘ . v . Gail i: g
Turni  points: P Q Rare callew. rning oints. Att " ur ing pomts, graph changes r
directi n from 1ncrease to decrease or decrease to increase. The coordinates with the E highest value' 1. e. point [b Q] is called the maximum point and the coordinates with the
lowest value i.e. point (c, R] is called the minimum goint. When you take much time will be devoted to actually figure out those poin . m Examples offunctions that do not contain maximum or minimum oint. ”That 15, it can
always be' 1ncreasing or decreasing. Ex: 312 x3 _ Average Rate of Change: Y3 E l )9] 1 X: a K;\0
Definition: The averagg whange of a function f(x] on [a, b] is the slope of thf
. . . ' A) ' ‘ I 8’ f[a)] and [b1 f(b)] Ify = f[X). the average: ateOf .a«r : Ex: The growth rate of a fetus 1n the mother’ S womb (by weight" 1n grams) is modeled by
the graph shown below beginning with the 25th week of gestation.
a] Calculate the average rate of change [slope o fthe secant line) between thew 25th week
and the 29th weeE: is the si?o:p::e:0f fﬁé secant line positive or negative? Discuss what @503“,
the slope means in this context.
b) Is the fetus gaining weight faster between the 25th and 29th week, or between the WQEK
32nd and 36Fh week? Compaie the s Opes of both secant lines and discuss. 5‘0 g
W15 
1’5  4 61¢— 61) Mmemc 0T WMC': 7.2% Y1
YZMX, Pole/r4136 T414: a? wage: \ _, J. , 1
3ﬁ1\=%(l):2(15l__% tl~q:_L\w‘ OAUDT?( ‘38} 1:: ".2—(3) #4 : 1%F4:‘4 492:9 . . A  ‘ I IA I I ‘ I M®“'”“WQUWW;' A Ar ‘ ':  A Definition of Difference Quotient: Average rate of change over an interval with variable
end points. Formula: Difference Quotient of function f(X+h)f(X)_ _ f(X+h)"f(X)
(x+h)~— x h 3". Grapzh SLOYQ (3112 W U”?
(weigh) WSW?) W3?) ‘
”Vii
E15”) Hon} 0,14%quth
,_... MM ti a: ﬂu: nZ— *¥CX+M)"‘
W f(X)= a K1+yh+mh¥hL n L Zf+47k+lh74ﬁz+ LEWLﬂDZhQ
‘ 'L , .—. «2m 4% MIL: , 74 a 13:2“ .: Observation between the ave age rate of change and the difference uotientl e same 4K +214
E function: ’Jﬁ“
'E \/ Techniques in Graphing: (3.4) Vocabulary: Translation : shift of a known graph either right or left [horizontal], or up and down (vertical) C. X“), .2 0 :27 x51, Vail/M?!» VQFHORU \f 2’? Ex: Graphy=lgc Fla21 y =x+2l Graph y=lxl+2 y 9 1‘12 Hvriwflnl Type °ftransiati°n= “Verna
.2 gkiH ‘ ‘ ‘ ‘
8 ”$4. l5 lh CDT' on mot as 'n Silo. W " 04 the am: I'm L H 3%.: ~  we» in mi. mirror image of a known graph abo t the Xax15 or the Ex: Graph y =x2 y =9x2 Type of reﬂection: ‘O—a? [S (2" {14‘ I echo? Methods of Combining Functions: Iterations [3.5) g(x)= x 2——3x+8 :Gwm u OWL 31 *8) Law?) 5% Composing GEL (80(1): _ Domain of f+g, fg, and (f) (g) :set of all inputs x belonging to both the domain of f and the domain ofg. Domain of i = set of all inputs x belonging to both f an g except those values of x which ‘
' g .  W
make g[x) = 0 '  " A' “ ' .. ”ginn   . '.QI~.;_‘.".alm.m_im_i.a_...,.m ‘5 ' ' A   :' A ‘ Methods of Combining Functions: Iterations [3.5) Let's consider two functions: f(x) = 2x —1, g(x) = 95.2 — 3x__+ 8 RM "F300 _, : Licgﬂyé’i) L
’0 was) : 14+? im—MWRQ’QRD ./. 1' V
’X+§_’j'§ I T" (Wu) “"‘ (13%”) “51“”: Multipiying: RX), 3 ._. ’ 9)C,L) F; m XLigx ﬁg 7 ‘
_Ci%~ 1)” Xm~3x+%) ; 2X2: (O’Bﬂexéﬁg : +542 "*2
1—,: 2x3 _ Uer—HCM __. 9 W : 03331:] effigfg, and (Mg) %_Wbem§0.n ' ' . —
iaomaln GEE) DOW/Wan 0‘: 8:: (w 60) 00) era ”DOWN 0P 6: (R 00 05)
> _ CH?) :00) ' Domain of — = set of all inputs x belonging to both an ; xcept those values of x which ‘
Raﬁ @ ~—,—‘—"—'—" in? _ . _ make g[x) = 0 ”ff—F; OW“: :Dblimik 0:? 8:0 . ( MM?“ aﬁ
THC/‘01 f ‘ Nﬂkﬂ §WQLﬁbKiS1HWdfa
\L . 2) Emeﬁe a T6 Finite Domain of f o = set of all those inputs x [in the domain of g! for which g[x) is in the _ MWIFV
I domain 015% " 1—H ' Ex: Let f(x) 2:4, g(x)=\/; Find the domain of the following expressions: Domain 9‘? 1C“ Aii 3L,)€+.’F2 " \ Domain of f(x): Domain of g[x): ...
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 Spring '08
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