388 CHAPTER 6 Algebra, Graphs, and Functions The set of values that can be used for the independent variable is called the domain of the function, and the resulting set of values obtained for the dependent variable is called the range. The domain and range for the function = c n 1.29 are illustrated in Fig. 6.41 below. Domain Function c 5 1.29n Range 2 0 1 3 4. . . . . . 0.00 1.29 2.58 3.87 5.16 Figure 6.41 Definition: Function A function is a special type of relation where each value of the independent variable corresponds to exactly one value of the dependent variable. Example 1 Is the Relation Also a Function? Determine whether the given relation is also a function. a) { } − (1, 4), (2, 3), (3, 5), ( 1, 3), (0, 6) b) { } −( 1, 2), (4, 2), (3, 1), (2, 6), (3, 5) Solution In both parts (a) and (b), each of the ordered pairs is of the form x y ( , ). The x represents the independent variable and the y represents the dependent variable. a) Each x corresponds with a unique y. For example, the value = x 1 corresponds only with the value = y 4. The value = x 2 corresponds only with the value = y 3, and so on. Since each value of the independent variable corresponds to a unique value of the dependent variable, this relation is also a function. b) Notice that the value = x 3 corresponds to both = y 1 and = y 5. Therefore, each value of the independent variable does not correspond to a unique value of the dependent variable. Thus, this relation is not a function. 7 Now try Exercise 11 y 4 5 3 2 1 23 25 24 2221 2423 25 1 2 3 4 5 x 21 y 5 2x 2 1 22 Figure 6.42 When we graphed equations of the form + = ax by c in Section 6.6, we determined that they were straight lines. For example, the graph of = − y x2 1 is illustrated in Fig. 6.42. Is the equation = − y x2 1 a function? To answer this question, we must ask, Does each value of x correspond to a unique value of y? The answer is yes; therefore, this equation is a function. For the equation = − y x2 1, we say that “y is a function of x” and write = y f x( ). The notation f x( ) is read “f of x.” When we are given an equation that is a function, we may replace the y in the equation with f x( ) because f x( ) represents y. Thus, = − y x2 1 may be written = − f x x ( ) 2 1. To evaluate a function for a specific value of x, replace each x in the function with the given value, then evaluate. For example, to evaluate = − f x x ( ) 2 1when = x 8, we do the following. = − = − = − = f x x f ( ) 2 1 (8) 2(8) 1 16 1 15
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