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1-1 functions, Inverse & Trig Functions, Intro to Limits

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MATH 137
Jennifer Nissen

Lecture 5 11PreviouslySeptember2111827 AMExample 45Prove that if 0 2 then cossin 1DefinitionA function f is called onetoone 11 if it never takes the same value twiceThat is ifthen fx1 fx2Or if fx1fx2 then Example 51 Prove that fxwith domain Df1is 11ProofSuppose f ffor some x xDf12Then2 2x xx x xx 01122 12 22x x xxxx 01 21212xx xxxx01 21221What is BCxx1 xx01212BCsin Arclength of BATherefore x xor xx 11212Arc BAIf x x we are done12If xx 1 then since xx 1 Df1 x x 1121212Note BCBAarc BAsinBAIn both cases x x so f is onetoone12Notethus sinso1If we consider gxwith domain Dg0then gx is not 11 Intuitively arc BABEEAeg g2g25DA tanHorizontal Line TextSo arc BAtansotanA function f is 11 if and only if no horizontal line intersects its graph more than So cos sinonce cos NoteIncreasing and decreasing functions are 11Therefore cos for 0 Inverse Functions 1If f is 11 with domain DfA and range RfB then its inverse function fexists 1with domain B and range A and is defined by fyx if and only if fxy1To find f we solve fxy for x in terms of yExample 522Is the function gxx2 11 No2But fxx2 with domain Df0is 111Find f2yx22y2xx 1Since RfDf0then it must be1 ie fy 1 Df2 Rf MATH 137 Page 1
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