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Unformatted text preview: 60 | | | | CHAPTER 1 FUNCTIONS AND MODELS Not all functions possess inverses. Let’s compare the functions and whose arrow diagrams are shown in Figure 1. Note that never takes on the same value twice (any two inputs in have different outputs), whereas does take on the same value twice (both 2 and 3 have the same output, 4). In symbols, but Functions that share this property with are called one-to-one functions. DEFINITION A function is called a one-to-one function if it never takes on the same value twice; that is, If a horizontal line intersects the graph of in more than one point, then we see from Figure 2 that there are numbers and such that . This means that is not one-to-one. Therefore we have the following geometric method for determining whether a function is one-to-one. HORIZONTAL LINE TEST A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLE 1 Is the function one-to-one? SOLUTION 1 If , then (two different numbers can’t have the same cube). Therefore, by Definition 1, is one-to-one....
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- Spring '09
- Inverse function, Injective function, horizontal line, 10 m