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Week 1, Math 3

# Week 1, Math 3 - Chapter 3 Functions The Definition ofa...

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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 non-empty 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 x-y 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‘s-v 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 x-y 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 'K-Eﬁﬂﬂﬁﬁ SE? 3.2 ._ 1n Exercises 5 and 6, use the vem‘cal line test to determine edges I v» 4, speciﬁ: the y-coordinate 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 one-unit intervals.) to the graph ofrhe function; are marked affirm one-w 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 mag-unit irate ' ' 10. 11; ,' 15. (an-Find 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?!» VQ-FHORU \f 2’? Ex: Graphy=lgc| Fla-21 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 X-ax15 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, f-g, 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' “ -' .. ”gin-n - - . '.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:] effigf-g, 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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Week 1, Math 3 - Chapter 3 Functions The Definition ofa...

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