Section 4: Inverse Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 8.4 Inverse Functions 803 Version: Fall 2007 8.4 Inverse Functions As we saw in the last section, in order to solve application problems involving expo- nential functions, we will need to be able to solve exponential equations such as 1500 = 1000 e 0 . 06 t or 300 = 2 x . However, we currently don’t have any mathematical tools at our disposal to solve for a variable that appears as an exponent, as in these equations. In this section, we will develop the concept of an inverse function, which will in turn be used to define the tool that we need, the logarithm, in Section 8.5. One-to-One Functions Definition 1. A given function f is said to be one-to-one if for each value y in the range of f , there is just one value x in the domain of f such that y = f ( x ) . In other words, f is one-to-one if each output y of f corresponds to precisely one input x . It’s easiest to understand this definition by looking at mapping diagrams and graphs of some example functions. l⚏ Example 2. Consider the two functions h and k defined according to the mapping diagrams in Figure 1 . In Figure 1 (a), there are two values in the domain that are both mapped onto 3 in the range. Hence, the function h is not one-to-one. On the other hand, in Figure 1 (b), for each output in the range of k , there is only one input in the domain that gets mapped onto it. Therefore, k is a one-to-one function. 1 2 3 h 1 2 3 4 k (a) (b) Figure 1. Mapping diagrams help to determine if a function is one-to-one. l⚏ Example 3. The graph of a function is shown in Figure 2 (a). For this function f , the y -value 4 is the output corresponding to two input values, x = 1 and x = 3 (see the corresponding mapping diagram in Figure 2 (b)). Therefore, f is not one-to-one. Graphically, this is apparent by drawing horizontal segments from the point (0 , 4) on the y -axis over to the corresponding points on the graph, and then drawing vertical segments to the x -axis. These segments meet the x -axis at 1 and 3 . Copyrighted material. See: 1
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804 Chapter 8 Exponential and Logarithmic Functions Version: Fall 2007 x y f 4 3 1 1 3 4 f (a) (b) Figure 2. A function which is not one-to-one. l⚏ Example 4. In Figure 3 , each y -value in the range of f corresponds to just one input value x . Therefore, this function is one-to-one. Graphically, this can be seen by mentally drawing a horizontal segment from each point on the y -axis over to the corresponding point on the graph, and then drawing a vertical segment to the x -axis. Several examples are shown in Figure 3 . It’s apparent that this procedure will always result in just one corresponding point on the x -axis, because each y -value only corresponds to one point on the graph. In fact, it’s easiest to just note that since each horizontal line only intersects the graph once, then there can be only one corresponding input to each output.
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