3rd Edition

ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

Concept explainers

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Textbook Question

Chapter 3.2, Problem 17E

Let A denote the set R \ { 0 , 1 } . Let i denote the identity function on A and define the functions f , g , h , s , r : A → A by

f ( x ) = 1 − 1 x , g ( x ) = 1 1 − x , h ( x ) = 1 x , r ( x ) = x x − 1 , s ( x ) = 1 − x .

Show that f ∘ g = i and g ∘ r = s . Complete the table, thereby showing that the composition of any two of the given functions is one of the given five, or the identity.
Which of the given six functions have inverses? Find (and identify) any inverses that exist.

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Chapter 3.2, Problem 16E

Chapter 3.2, Problem 18E

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Chapter 3 Solutions

Discrete Mathematics with Graph Theory

Ch. 3.1 - True/False Questions A function from a set A to a...Ch. 3.1 - Prob. 2TFQ Ch. 3.1 - Prob. 3TFQ Ch. 3.1 - Prob. 4TFQ Ch. 3.1 - Prob. 5TFQ Ch. 3.1 - True/False Questions Define f:ZZ by f(x)=x+2. Then...Ch. 3.1 - Prob. 7TFQ Ch. 3.1 - Prob. 8TFQ Ch. 3.1 - Prob. 9TFQ Ch. 3.1 - Prob. 10TFQ

Ch. 3.1 - Prob. 11TFQ Ch. 3.1 - Prob. 12TFQ Ch. 3.1 - Determine whether each of the following relation...Ch. 3.1 - 2. Suppose A is the set of students currently...Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Define g:ZB by g(x)=|x|+1. Determine (with...Ch. 3.1 - Define f:AA by f(x)=3x+5. Determine (with reasons)...Ch. 3.1 - 16. Define by . Determine (with reasons) whether...Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Define f:RR by f(x)=3x3+x. Graph f to determine...Ch. 3.1 - 21. (a) Define by . Graph g to determine whether g...Ch. 3.1 - Prob. 22E Ch. 3.1 - 23. Let a, b, c be real numbers and define by ....Ch. 3.1 - 24. For each of the following, find the largest...Ch. 3.1 - Prob. 25E Ch. 3.1 - Let S be a set containing the number 5. Let...Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.2 - True/False Questions The function defines by ...Ch. 3.2 - True/False Questions The function f:ZZ defines by...Ch. 3.2 - Prob. 3TFQ Ch. 3.2 - Prob. 4TFQ Ch. 3.2 - Prob. 5TFQ Ch. 3.2 - Prob. 6TFQ Ch. 3.2 - Prob. 7TFQ Ch. 3.2 - Prob. 8TFQ Ch. 3.2 - Prob. 9TFQ Ch. 3.2 - Prob. 10TFQ Ch. 3.2 - Let . Find the inverse of each of the following...Ch. 3.2 - 2. Define by . Find a formula for . Ch. 3.2 - Define f:(,0][0,) by f(x)=x2. Find a formula for...Ch. 3.2 - 4. Define by . Find a formula for . Ch. 3.2 - Prob. 5E Ch. 3.2 - Prob. 6E Ch. 3.2 - Show that each of the following functions f:AH is...Ch. 3.2 - Prob. 8E Ch. 3.2 - Prob. 9E Ch. 3.2 - Prob. 10E Ch. 3.2 - 11. Let and define functions by and . Find (a) ...Ch. 3.2 - Prob. 12E Ch. 3.2 - Prob. 13E Ch. 3.2 - Prob. 14E Ch. 3.2 - Prob. 15E Ch. 3.2 - Prob. 16E Ch. 3.2 - 17. Let A denote the set . Let i denote the...Ch. 3.2 - Prob. 18E Ch. 3.2 - Prob. 19E Ch. 3.2 - Prob. 20E Ch. 3.2 - Prob. 21E Ch. 3.2 - Prob. 22E Ch. 3.2 - Prob. 23E Ch. 3.2 - Prob. 24E Ch. 3.2 - Is the composition of two bijective functions...Ch. 3.2 - 26. Define by . (a) Find the values of . (b) Guess...Ch. 3.2 - Prob. 27E Ch. 3.2 - Prob. 28E Ch. 3.3 - True/False Questions If sets A and B are in...Ch. 3.3 - Prob. 2TFQ Ch. 3.3 - Prob. 3TFQ Ch. 3.3 - Prob. 4TFQ Ch. 3.3 - True/False Questions If A and B are finite sets...Ch. 3.3 - True/False Questions If the conditions of...Ch. 3.3 - Prob. 7TFQ Ch. 3.3 - Prob. 8TFQ Ch. 3.3 - Prob. 9TFQ Ch. 3.3 - Prob. 10TFQ Ch. 3.3 - Prob. 1E Ch. 3.3 - At first glance, the perfect squares 1, 4, 9, 16,...Ch. 3.3 - Prob. 3E Ch. 3.3 - Prob. 4E Ch. 3.3 - Prob. 5E Ch. 3.3 - Prob. 6E Ch. 3.3 - Prob. 7E Ch. 3.3 - Prob. 8E Ch. 3.3 - Prob. 9E Ch. 3.3 - Prob. 10E Ch. 3.3 - Prove that the notion of same cardinality is an...Ch. 3.3 - Prob. 12E Ch. 3.3 - Prob. 13E Ch. 3.3 - Prob. 14E Ch. 3.3 - Prob. 15E Ch. 3.3 - Prob. 16E Ch. 3.3 - Prob. 17E Ch. 3.3 - Prob. 18E Ch. 3.3 - Prob. 19E Ch. 3.3 - Prob. 20E Ch. 3.3 - Prob. 21E Ch. 3.3 - 22. Given an example of each of the following or...Ch. 3.3 - Prob. 23E Ch. 3.3 - Prob. 24E Ch. 3.3 - Prove that the points of a plane and the points of...Ch. 3.3 - Prob. 26E Ch. 3.3 - 27. (a) Show that if A and B are countable sets...Ch. 3.3 - Prob. 28E Ch. 3.3 - 29. Let S be the set of all real numbers in the...Ch. 3.3 - Let S be the set of all real numbers in the...Ch. 3.3 - Prob. 31E Ch. 3 - Define by . Determine whether f is one-to-one. Ch. 3 - Let f={(1,2),(2,3),(3,4),(4,1)} and...Ch. 3 - Prob. 3RE Ch. 3 - Prob. 4RE Ch. 3 - 5. Answer these questions for each of the given...Ch. 3 - Prob. 6RE Ch. 3 - Prob. 7RE Ch. 3 - Prob. 8RE Ch. 3 - Prob. 9RE Ch. 3 - Prob. 10RE Ch. 3 - Prob. 11RE Ch. 3 - Prob. 12RE Ch. 3 - Prob. 13RE Ch. 3 - Prob. 14RE Ch. 3 - Prob. 15RE Ch. 3 - Prob. 16RE Ch. 3 - Prob. 17RE Ch. 3 - Prob. 18RE Ch. 3 - Prob. 19RE Ch. 3 - Let S be the set of all real numbers in the...Ch. 3 - Prob. 21RE

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