222 CHAPTER 2 Graphs and Functions –4 –2 1 0 H H is not a function. x-values y-values Figure 19 1 –2 3 2 4 F x-values y-values F is a function. As the correspondence below shows, relation G is not a function because one first component corresponds to more than one second component. 51, 26 x-values of G 51, 2, 36 y-values of G In relation H the last two ordered pairs have the same x-value paired with two different y-values (-2 is paired with both 1 and 0), so H is a relation but not a function. In a function, no two ordered pairs can have the same first component and different second components. Different y-values H= 51-4, 12, 1-2, 12, 1-2, 026 Not a function Same x-value S Now Try Exercises 11 and 13. Relations and functions can also be expressed as a correspondence or mapping from one set to another, as shown in Figure 19 for function F and relation Finally, we can describe a relation or function using a rule that tells how to determine the dependent variable for a specific value of the independent variable. The rule may be given in words: for instance, “the dependent variable is twice the independent variable.” Usually the rule is an equation, such as the one below. Dependent variable y = 2x Independent variable H from Example 1. The arrow from 1 to 2 indicates that the ordered pair 11, 22 belongs to F—each first component is paired with exactly one second component. In the mapping for relation H, which is not a function, the first component -2 is paired with two different second components, 1 and 0. Because relations and functions are sets of ordered pairs, we can represent them using tables and graphs. A table and graph for function F are shown in Figure 20. x y 1 2 -2 4 3 4 x y 0 (3, 4) (–2, 4) (1, 2) Graph of F Figure 20 EXAMPLE 1 Deciding Whether Relations Define Functions Determine whether each relation defines a function. F = 511, 22, 1-2, 42, 13, 426 G= 511, 12, 11, 22, 11, 32, 12, 326 H= 51-4, 12, 1-2, 12, 1-2, 026 SOLUTION Relation F is a function because for each different x-value there is exactly one y-value. We can show this correspondence as follows. 51, -2, 36 x-values of F 52, 4, 46 y-values of F
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