64 CHAPTER 2 Functions and Their Graphs Determining Whether a Relation Given by a Set of Ordered Pairs Is a Function For each relation, state the domain and range. Then determine whether the relation is a function. (a) {( )( )( )( )} 1, 4 , 2, 5 , 3, 6 , 4, 7 (b) ( ) ( ) ( ) ( ) { } 1, 4 , 2, 4 , 3, 5 , 6, 10 (c) ( ) ( ) ( ) ( ) ( ) { } − − − 3, 9 , 2, 4 , 0, 0 , 1, 1 , 3, 8 Solution EXAMPLE 3 (a) The domain of this relation is { } 1, 2, 3, 4 , and its range is { } 4, 5, 6, 7 . This relation is a function because there are no ordered pairs with the same first element and different second elements. (b) The domain of this relation is { } 1, 2, 3, 6 , and its range is { } 4, 5, 10 . This relation is a function because there are no ordered pairs with the same first element and different second elements. (c) The domain of this relation is { } − − 3, 2, 0, 1 , and its range is { } 0, 1, 4, 8, 9 . This relation is not a function because there are two ordered pairs, ( ) −3, 9 and ( ) −3, 8 , that have the same first element and different second elements. Solution (a) The domain of the relation is {Sweet tea, Hamburger, Decaf coffee, } Double Cheeseburger, McChicken , and the range of the relation is { } $1, $2, $3 . The relation in Figure 8 is a function because each element in the domain corresponds to exactly one element in the range. (b) The domain of the relation is { } Sedentary, Light, Moderate, Intense , and the range of the relation is { } 2370, 2716, 3061, 3407 . The relation in Figure 9 is a function because each element in the domain corresponds to exactly one element in the range. (c) The domain of the relation is { } 122, 201, 240, 284 , and the range of the relation is { } 5, 8, 12, 15, 20 . The relation in Figure 10 is not a function because there is an element in the domain, 240, that corresponds to two elements in the range, 12 and 20. If a gestation period of 240 days is selected from the domain, a single life expectancy cannot be associated with it. Now Work PROBLEM 19 We may also think of a function as a set of ordered pairs ( ) x y , in which no ordered pairs have the same first element and different second elements. The set of all first elements x is the domain of the function, and the set of all second elements y is its range. Each element x in the domain corresponds to exactly one element y in the range. In Example 3(b), notice that 1 and 2 in the domain both have the same image in the range.This does not violate the definition of a function; two different first elements can have the same second element.The definition is violated when two ordered pairs have the same first element and different second elements, as in Example 3(c). Now Work PROBLEM 23 Up to now we have shown how to identify when a relation is a function for relations defined by mappings (Example 2) and ordered pairs (Example 3). But relations can also be expressed as equations. It is important to understand that it is okay for two (or more) different elements in the domain to correspond to the same element in the range, as in Example 2(a). A relation is not a function when an element in the domain corresponds to more than one element in the range, as in Example 2(c).
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