Sets and Functions Help (page 2)
Introduction to Sets and Functions
A set is a collection of objects. We denote a set with a capital roman letter, such as S or T or U. If S is a set and s is an object in that set then we write s S and we say that s is an element of S. If S and T are sets then the collection of elements common to the two sets is called the intersection of S and T and is written S ∩ T. The set of elements that are in S or in T or in both is called the union of S and T and is written S ∪ T.
A function from a set S to a set T is a rule that assigns to each element of S a unique element of T. We write f : S → T.
Let S be the set of all people who are alive at noon on October 10, 2004 and T the set of all real numbers. Let f be the rule that assigns to each person his or her weight in pounds at precisely noon on October 10, 2004. Discuss whether f : S → T is a function.
Indeed f is a function since it assigns to each element of S a unique element of T. Notice that each person has just one weight at noon on October 10, 2004: that is a part of the definition of “function.” However two different people may have the same weight —that is allowed.
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?
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).
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.
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 “Jack Armstrong”), but that is allowed in the definition 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 . 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?
Examples of Functions of a Real Variable
Let S = , T = , and let f ( x ) = x2. This is mathematical shorthand for the rule “assign to each x S its square.” Determine whether f : → is a function.
We see that f is a function since it assigns to each element of S a unique element of T —namely its square.
Math Note: Notice that, in the definition of function, there is some imprecision in the definition of T . For instance, in Example 1.24, we could have let T = [0, ∞) or T = (−6, ∞) with no significant change in the function. In the example of the “name” function (Example 1.23), 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.
Let . Determine a domain and range for f which make f a function.
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 , then f : [−1, 1] → is a function.
Math Note: When a function is given by a formula, as in Example 1.25, 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
What are the domain and range of this function?
Determine whether f is a function.
Notice that f unambiguously assigns to each real number another real number. The rule is given in two pieces, but it is still a valid rule. Therefore it is a function with domain equal to and range equal to . It is also perfectly correct to take the range to be (−4, ∞), for example, since f only takes values in this set.
Math Note: One point that you should learn from this example is that a function may be specified by different formulas on different parts of the domain .
You Try It: Does the expression
define a function? Why or why not?
Let . Discuss whether f is a function.
This f can only make sense for x ≥ 0. But even then f is not a function since it is ambiguous. For instance, it assigns to x = 1 both the numbers 1 and −1.
Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.
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