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CHAPTER 1 Basics
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EXAMPLE 122
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Let S be the set of all people and T be the set of all people Let f be the rule that assigns to each person his or her brother Is f a function
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SOLUTION In this case f is not a function For many people have no brother (so the rule makes no sense for them) and many people have several brothers (so the rule is ambiguous for them) EXAMPLE 123
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Let S be the set of all people and T be the set of all strings of letters not exceeding 1500 characters (including blank spaces) Let f be the rule that assigns to each person his or her legal name (Some people have rather long names; according to the Guinness Book of World Records, the longest has 1063 letters) Determine whether f : S T is a function
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SOLUTION This f is a function because every person has one and only one legal name Notice that several people may have the same name (such as JackArmstrong ), but that is allowed in the de nition of function You Try It: Let f be the rule that assigns to each real number its cube root Is this a function In calculus, the set S (called the domain of the function) and the set T (called the range of the function) will usually be sets of numbers; in fact they will often consist of one or more intervals in R The rule f will usually be given by one or several formulas Many times the domain and range will not be given explicitly These ideas will be illustrated in the examples below You Try It: Consider the rule that assigns to each real number its absolute value Is this a function Why or why not If it is a function, then what are its domain and range
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EXAMPLES OF FUNCTIONS OF A REAL VARIABLE
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EXAMPLE 124 Let S = R, T = R, and let f (x) = x 2 This is mathematical shorthand for the rule assign to each x S its square Determine whether f : R R
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is a function
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SOLUTION We see that f is a function since it assigns to each element of S a unique element of T namely its square
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Basics
Math Note: Notice that, in the de nition of function, there is some imprecision in the de nition of T For instance, in Example 124, we could have let T = [0, ) or T = ( 6, ) with no signi cant change in the function In the example of the name function (Example 123), we could have let T be all strings of letters not exceeding 5000 characters in length Or we could have made it all strings without regard to length Likewise, in any of the examples we could make the set S smaller and the function would still make sense It is frequently convenient not to describe S and T explicitly EXAMPLE 125 Let f (x) = + 1 x 2 Determine a domain and range for f which make
f a function
SOLUTION Notice that f makes sense for x [ 1, 1] (we may not take the square root of a negative number, so we cannot allow x > 1 or x < 1) If we understand f to have domain [ 1, 1] and range R, then f : [ 1, 1] R is a function Math Note: When a function is given by a formula, as in Example 125, with no statement about the domain, then the domain is understood to be the set of all x for which the formula makes sense You Try It: Let x + 4x + 3 What are the domain and range of this function g(x) = x2 EXAMPLE 126
Let f (x) =
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