Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Definition: The composition of n functions f i : A i %A i+1 is defined by
(f 1 °f 2 °…°f n )(a) = f 1 (f 2 (…(f n (a)…)),
where a,A 1 .
Definition: If f: A%B, then the inverse of f, denoted f -1 : B%A exists if and only
if (b,B )a,A (f(a) = b ! f -1 (b) = a).
Examples:
• Let A = [0,1] 1 R, B = [0,2] 1 R.
o f(a) = a 2 and g(a) = a+1. Then f+g: A%B and f*g: A%B.
o f(a) = 2*a and g(a) = a-1. Then neither f+g: A%B nor f*g: A%B.
• Let B = A = [0,1] 1 R.
o f(a) = a 2 and g(a) = 1-a. Then f+g: A%A and f*g: A%A. Both
compound functions are bijections.
o f(a) = a 3 and g(a) = a 1/3 . Then g°f(a): A%A is a bijection.
• Let A = [-1, 1] and B=[0, 1]. Then
o f(a) = a 3 and g(a) = {x>0 | x= a 1/3 }. Then g°f(a): A%B is onto.
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Definition: The graph of a function f is {(a,f(a)) | a,A}.
Example: A = {0, 1, 2, 3, 4, 5} and f(a) = a 2 . Then
(a) graph(f,A)
(b) an approximation to graph(f,[0,5])
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