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1.6 Functions

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Chapter 1, section 6 Functions notation: f: A B x in A, y in B, f(x) = y. concepts: –domain of f, –codomain of f, –range of f, –f maps A into B

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Chapter 1, section 6 Functions One to one, and onto functions f is 1 - 1 {injective} IFF – x, y in domain f, –(f(x) = f(y)) (x = y) –{So if we graph f, then a horizontal line will intersect f in at most one point} Example: –There is a 1 - 1 correspondence between SSN and DNA

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Chapter 1, section 6 Functions Surjective f is onto {surjective} IFF – y in codomain of f – x in domain of f, –such that f(x) = y

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Chapter 1, section 6 Functions Bijection-One to one Correspondence Note: the size of the domain and codomain are equal, since f gives a way of uniquely matching the elements from both sets.

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Chapter 1, section 6 Functions Inverse Functions If f is bijective, then the inverse function of f is: – y in range of f, f -1 (y) = x, where f(x) = y Example: Given SSN can get DNA but also, given DNA can get SSN

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Chapter 1, section 6 Functions Composition If g: A B, and f: B C, then the composition of f and g is (f g)(a) = f(g(a)) concepts: –floor and ceiling functions take real numbers to integers

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