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# Class Note for MATH 1432 at UH

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

Lecture 1Section 71 One To One Functions Inverses Jiwen He 1 OneToOne Functions 11 De nition of the OneToOne Functions What are One To One Functions Geometric Test V I f 5 not onetoone m1 m2 f S nneytorone Horizontal Line Test o If some horizontal line intersects the graph of the function more than once7 then the function is not onetoone o If no horizontal line intersects the graph of the function more than once7 then the function is onetoone What are One To One Functions Algebraic Test snmimerturmwe nxzz w r scner39orone De nition 1 A function f is said to be oneitoione or injective if fz1 implies 961 96 Lemma 2 The function f is oneitoione and only Vxhvxzy 961 7E 962 impli 951 f952 Examples and CounterExamples Examples 3 o 31 7 5 is lto ll o fz z is not 1 to 1l o 13 is lto ll fz g is 1401 o I 7 z n gt 0 is not lto ll Proof 0 311 7 5 312 7 5 11 12 In general am 7 b a f 0 is lto ll o 12 l 7l2 f7ll In general I n even is not lto ll o 1 13 11 12 In general I n odd is lto ll o i i 11 12 In general 1 n odd is lto ll o 0 7 0 0 1 71 In general lto l of f and 9 does not always imply lto l of f g 1 12 Properties of OneTo One Functions Properties Properties If f and g are oneto one then f o g is oneto onel me f 09I1 f MW e fyI1 f9r2 9901 DH 912 11 12 Examples 4 o 313 7 5 is oneto one since f g o u Where gu Su 7 5 and 13 are oneto onel o 31 7 53 is oneto one since f g o u Where gu u3 and 31 7 5 are oneto onel o 9175 is oneto one since f g o u Where gu i and 313 7 5 are oneto onel 13 Increasing Decreasing Functions and One ToOneness IncreasingDecreasing Functions and OneToOneness De nition 5 o A function f is strictly increasing if VII 17V1 27 11 lt 12 implies lt o A function f is strictly decreasing if VII 17V1 27 11 lt 12 implies gt Theorem 6 Functions that are increasing or decreasing are onetoone Proof For 11 f 12 either 11 lt 12 or 11 gt 12 ans so7 by monotonicity7 either fzl lt or gt 16127 thus f 1 Sign of the Derivative Test for OneToOneness Theorem 7 o If gt 0 for allz then f is increasing thus 0ne t00ne o If lt 0 for all I then f is decreasing thus onetoone Examples 8 o zg z is oneto one7 since 312 7 gt 0 for all z o 7157213721 is oneto one7 since 7514 7 612 7 2 lt 0 for all z o z7ncosz is oneto one7 since I 7 sinz 2 0 and 0 only at z g 216 2 Inverse Functions 21 De nition of Inverse Functions What are Inverse Functions x ff1x O f 1 De nition 9 Let f be a onetoone function The inverse of f7 denoted by f l7 is the unique function With domain equal to the range of f that satis es fquot1 x for all x in the range of Warning DON7T Confuse f 1 With the reciprocal of f7 that is7 With The 7 177 in the notation for the inverse of f is not an exponent f 1 does not mean 1fx Example Example 10 o 963 gt f 1x x 3 Proof 0 By de nition7 f 1 satis es the equation z for all 1 0 Set y f 1z and solve z for y fyr y31 y1 0 Substitute f 1z back in for y 1041 I In general7 161 I n odd a filw zln Example yA 5 96 my 0 fz39575 gt f 1z Example 11 Proof 0 By definition7 f 1 satis es x Vac 0 Set 3 f 1z and solve x for y fyz 3 3y75z 5 y7x7 0 Substitute f 1z back in for y 1 5 1 7 7 7 f 7 31 3 D In general7 1 b fzazb ay o gt f 1xxig 22 Properties of Inverse Functions Undone Properties 1 f o f7 Idkw Df 173f xfULMM f l f 1x ff1 0 f Lip 73f 1 DU f x o f 1 o X f 1fx x Theorem 12 By de nition f 1 satis es x for all x in the range of It is also true that f71 x for all x in the domain of Proof 0 V95 6 Df set 3 Since 3 E ff71 y 3 ff 1fx f96 o f being onet0one implies x 1 Graphs of f and f 1 y A 10 Graphs of f and f 1 The graph of f 1 is the graph of f re ected in the line 3 95 Example 13 Given the graph of f7 sketch the graph of f4 Solution First draw the line 3 95 Then re ect the graph of f in that line Corollary 14 f is continuous 3 50 2395 f l 23 Differentiability 0f Inverses Differentiability of Inverses l X f 1x flm x x Theorem 15 f W 7 07 y f96 Proof 0 Vy E Df 1 396 E st 3 By definition7 f 1fxx 5 f lmx f 1 fxf x1 11 o If f 0 then PM it a r fz W L f 1 Example Example 16 Let 13 Calculate f 19l Solution Note that 312 gt 0 thus f is onetoonel Note that PM y may To calculate f 1y at y 9 nd a number 1 st 9 fz9 zsz9 12 2 7 f 2 25 Since fQ 322 then f 19 Note that to calculate f 1y at a speci c y using PM ff 1W o y fa V we only need the value of 1 st y not the inverse function f l Which may not be known explicitlyl Daily Grades Daily Grades ll z 16711 z a not exist b I c 2i 13 16711 7 a not exist b 1 c 3i I2 16711 z a not exist b 1 c 4i 31 7 3 f 1l a not exist b 3 c Out line 12 Contents 1 OneToOne Functions 1 11 De nition i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1 12 Properties i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 2 1 3 Monotonicity i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 3 2 Inverses 3 21 De nition i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 3 22 Properties i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 8 2 3 Differentiability i i i i i i i i i i i i i i i i i i i i i i i i i i i i 11 13

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