SECTION 5.2 One-to-One Functions; Inverse Functions 281 5.2 One-to-One Functions; Inverse Functions Now Work the ‘Are You Prepared?’ problems on page 290. • Functions (Section 2.1, pp. 61–73) • Increasing/Decreasing Functions (Section 2.3, pp. 89–90, 93) • Rational Expressions (Section A.5, pp. A35–A42) • Properties of Rational Functions (Section 4.5, pp. 236–244) PREPARING FOR THIS SECTION Before getting started, review the following: DEFINITION One-to-One A function is one-to-one if any two different inputs in the domain correspond to two different outputs in the range.That is, if x1 and x2 are two different inputs of a function f, then f is one-to-one if ( ) ( ) ≠ f x f x . 1 2 In Words A function is not one-to-one if two different inputs correspond to the same output. OBJECTIVES 1 Determine Whether a Function Is One-to-One (p. 281) 2 Determine the Inverse of a Function Defined by a Mapping or a Set of Ordered Pairs (p. 283) 3 Obtain the Graph of the Inverse Function from the Graph of a One-to-One Function (p. 285) 4 Verify that a Function Defined by an Equation Is an Inverse Function (p. 286) 5 Find the Inverse of a Function Defined by an Equation (p. 287) 1 Determine Whether a Function Is One-to-One Section 2.1 presented five ways to represent a function: (1) verbally, (2) with a mapping, (3) as a set of ordered pairs, (4) with a graph, and (5) with an equation. For example, Figures 6 and 7 illustrate two different functions represented as mappings. The function in Figure 6 shows the correspondence between states and their populations (in millions). The function in Figure 7 shows a correspondence between animals and life expectancies (in years). Figure 6 Indiana Washington South Dakota North Carolina Oklahoma State 6.9 7.8 0.9 11.0 4.1 Population (in millions) Figure 7 Dog Cat Duck Goat Pig Rabbit Animal 11 10 7 Life Expectancy (in years) Suppose several people are asked to name a state that has a population of 0.9 million based on the function in Figure 6. Everyone will respond “South Dakota.” Now, if the same people are asked to name an animal whose life expectancy is 11 years based on the function in Figure 7, some may respond “dog,” while others may respond “cat.” What is the difference between the functions in Figures 6 and 7? In Figure 6, no two elements in the domain correspond to the same element in the range. In Figure 7, this is not the case: Different elements in the domain correspond to the same element in the range. Functions such as the one in Figure 6 are given a special name.
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