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The real domains of the ratio functions are subsets of the real domains for the sum, product, and difference functions We have to look at each ratio function and check to see where the denominators are equal to 0: The denominator of ( f1 / f2) becomes 0 when x = 0 The denominator of ( f2 / f1) becomes 0 when x = 1 The denominator of ( f1 / f3) becomes 0 when x = 0 The denominator of ( f3 / f1) becomes 0 when x = 1 The denominator of ( f2 / f3) becomes 0 when x = 0 The denominator of ( f3 / f2) becomes 0 when x = 0

On the basis of these observations, we can create one final list: The real domain of ( f1 / f2) is the set of all real numbers except 0 The real domain of ( f2 / f1) is the set of all real numbers except 1

The real domain of ( f1 / f3) is the set of all strictly positive real numbers The real domain of ( f3 / f1) is the set of all nonnegative real numbers except 1 The real domain of ( f2 / f3) is the set of all strictly positive real numbers The real domain of ( f3 / f2) is the set of all strictly positive real numbers

It is better to go to yellow alert several times, and return to green each time, than it is to not be forthcoming about problems, and then jump to red alert, confronting the project sponsor with a problem he or she did not even know was coming At least, that is true in a well-managed company Sometimes, unfortunately, we have to choose a di erent approach to work with a di cult project sponsor Some executives do not keep in touch with the project, and others have a tendency to jump in and micromanage at the rst sign of trouble We can educate executives by simply telling them what green, yellow, and red status mean And, in our emails, we say exactly what we want, when we want it In a yellow alert, identify the problem and the solution you are working on.

This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Examine the relation illustrated in Fig 11-10 Suppose that for every element x in set X, there exists at most one element y in set Y Is this relation an injection Is it a surjection Is it a bijection Note that the range of the relation is the entire co-domain Is that true of all relations As described here, is this relation a function Explain each answer

m n = 7 (r - c,)

Figure 11-10 Illustration for Problem 1 2 Imagine a relation in which the domain X is the set of all positive rational numbers, while the range Y is the set of all positive integers Let s call the independent variable x and the dependent variable y Suppose that for any x in set X, the relation rounds x up to the next larger integer to obtain the corresponding element y in set Y Is this relation an injection Is it a surjection Is it a bijection Is it a function of x Explain each answer 3 Suppose that we reverse the action of the relation described in Problem 2 Let the domain X be the set of all positive integers, while the range Y is the set of all positive rationals Suppose that for any value of the independent variable x in set X, the relation