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Unformatted text preview: (5/31/07) Section 0.4 Inverse functions and logarithms Overview: Some applications require not only a function that converts a number x into a number y , but also its inverse , which converts y back into x . In this section we analyze inverse functions and discuss logarithms , which are the inverses of exponential functions. Topics: Inverse functions Changing variables Restricting domains Logarithms The common and natural logarithms Laws of logarithms Inverse functions The function of Figure 1 converts the year t into the number N = f ( t ) (millions) of cars that were registered in California at that time. (1) For example, starting with t = 1980 on the horizontal t-axis, moving vertically to the graph, and then horizontally to the vertical N-axis, we obtain N = f (1980) = 17 . 4. At the beginning of 1980 there were 17.4 million cars registered in California. t N (millions) 10 20 30 17 . 4 N = f ( t ) 1940 1960 1980 2000 N 10 17 . 4 30 t (year) 1940 1960 1980 2000 t = f 1 ( N ) (millions) FIGURE 1 FIGURE 2 The inverse of f , denoted f 1 and read f inverse, has the opposite effect. It converts the number N into the time t = f 1 ( N ) when there were N cars registered. We could use Figure 1 to study f 1 by having its variable (the independent variable) be on the vertical N-axis and its values on the horizontal t-axis. For example, we could start with N = 17 . 4 on the vertical axis, move horizontally to the graph, and then vertically to the horizontal t axis to obtain t = 1980. This gives f 1 (17 . 4) = 1980, which means that number of cars registered was 17.4 million at the beginning of 1980. (1) Data adapted from CALPERG Citizen Agenda , Vol. 18, No. 2, Los Angeles: CALPERG, 2000, p. 5. Because the symbol f- 1 also denotes the reciprocal 1 /f of f , its meaning must be determined from the context in which it is used. 1 p. 2 (5/31/07) Section 0.4, Inverse functions and logarithms This procedure for studying the inverse is, however, not convenient because we prefer to have the independent variable on the horizontal axis. To achieve this, we flip the drawing about the diagonal line that makes a 45 -degree angle with the positive x- and y-axes. This yields the graph t = f 1 ( N ) of the inverse function in Figure 2, where the variable N is on the horizontal axis and the values t of the function are on the vertical axis. The graph of f 1 in Figure 2 is the mirror image of the graph of f in Figure 1 with respect to the diagonal line. Here is a general definition: Definition 1 Suppose that y = f ( x ) is a function such that for each y in its range, the equation y = f ( x ) has one and only one solution x . Then x is the value of the inverse function x = f 1 ( y ) at y . Thus, x = f 1 ( y ) y = f ( x ) . (1) Question 1 Suppose that f has an inverse f 1 , that f (1) = 20, and that f 1 (30) = 2. What are the values of f 1 (20) and f (2)?...
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