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You ask, Do the commutative, associative, distributive, and other rules of arithmetic and algebra work with functions in the same ways as they do with numbers and variables The answer is a qualified yes All the rules of addition, subtraction, multiplication, and division of functions are identical to the rules for arithmetic or algebra involving numbers or variables, as long as we heed the cautions outlined earlier in this section
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Define the real-number domains of all the sum, difference, product, and ratio functions we ve found in this section
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We found the real-number domains (which we can call the real domains for short) of the functions f1, f2, and f3 earlier in this chapter Here they are again, for reference: The real domain of f1 (x), which subtracts 1 from x, is the set of all reals The real domain of f2 (x), which squares x, is the set of all reals The real domain of f3 (x), which takes the nonnegative square root of x, is the set of all nonnegative reals The real domains of the sum, difference, and product functions are the intersections of these Let s list them: The real domains of ( f1 + f2), ( f1 f2), and ( f1 f2) are the set of all real numbers The real domains of ( f2 + f1), ( f2 f1), and ( f2 f1) are the set of all real numbers The real domains of ( f1 + f3), ( f1 f3), and ( f1 f3) are the set of all nonnegative real numbers The real domains of ( f3 + f1), ( f3 f1), and ( f3 f1) are the set of all nonnegative real numbers The real domains of ( f2 + f3), ( f2 f3), and ( f2 f3) are the set of all nonnegative real numbers The real domains of ( f3 + f2), ( f3 f2), and ( f3 f2) are the set of all nonnegative real numbers
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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
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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
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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
Practice Exercises
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
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