Precalc0108to0109-page25

Precalc0108to0109-page25 - of f ; f has a left inverse f is...

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(Section 1.9: Inverses of One-to-One Functions) 1.9.15 Observe that the red graph for f and the brown graph for f ± 1 below are reflections about the line y = x . FOOTNOTES 1. Identity functions and compositions of inverse functions. There are technically different identity functions on different domains. (See Footnote 3 below and Section 1.1, Footnote 1.) Let f be an invertible function that maps from domain X to codomain Y ; i.e., f : X ± Y . If f is invertible, then f is onto , meaning that the range of f is the codomain Y . f ± 1 maps from Y to X ; i.e., f ± 1 : Y ² X . • Let I X be the identity function on Dom f () , which is X . I X : X ± X . • Let I Y be the identity function on Dom f ± 1 () , which is Y . I Y : Y ± Y . • Then, f ± 1 ± f = I X , and f ± f ± 1 = I Y . • If g is a function such that g ± f = I X , then g is a left inverse
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Unformatted text preview: of f ; f has a left inverse f is one-to-one. For example, let X = 1, 2 { } , Y = 10, 20, 30 { } , f = 1,10 ( ) , 2, 20 ( ) { } , and g = 10, 1 ( ) , 20, 2 ( ) , 30, 2 ( ) { } . Then, g is a left inverse of f . If h is a function such that f h = I Y , then h is a right inverse of f ; f has a right inverse f is onto. For example, let X = 1, 2, 3 { } , Y = 10, 20 { } , f = 1, 10 ( ) , 2, 20 ( ) , 3, 20 ( ) { } , and h = 10,1 ( ) , 20, 2 ( ) { } . Then, h is a right inverse of f , although h is not a left inverse of f . If f is one-to-one and onto, then f has a unique inverse function that serves as both a unique left inverse and a unique right inverse....
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This document was uploaded on 12/29/2011.

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