438 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions By the definition of inverse function, the domain of ƒ is the range of ƒ−1, and the range of ƒ is the domain of ƒ−1. See Figure 5. CAUTION Do not confuse the −1 in ƒ−1 with a negative exponent. The symbol ƒ-11x2 represents the inverse function of ƒ, not 1 ƒ1x2 . Domain of f f X x y Y Range of f –1 Range of f Domain of f –1 f –1 Figure 5 EXAMPLE 4 Finding Inverses of One-to-One Functions Find the inverse of each function that is one-to-one. (a) F = 51-2, 12, 1-1, 02, 10, 12, 11, 22, 12, 226 (b) G= 513, 12, 10, 22, 12, 32, 14, 026 (c) The table in the margin shows the number of hurricanes recorded in the North Atlantic during the years 2014–2018. Let ƒ be the function defined in the table, with the years forming the domain and the numbers of hurricanes forming the range. SOLUTION (a) Each x-value in F corresponds to just one y-value. However, the y-value 2 corresponds to two x-values, 1 and 2. Also, the y-value 1 corresponds to both -2 and 0. Because at least one y-value corresponds to more than one x-value, F is not one-to-one and does not have an inverse. (b) Every x-value in G corresponds to only one y-value, and every y-value corresponds to only one x-value, so G is a one-to-one function. The inverse function is found by interchanging the x- and y-values in each ordered pair. G-1 = 511, 32, 12, 02, 13, 22, 10, 426 Notice how the domain and range of G become the range and domain, respectively, of G-1. (c) Each x-value in ƒ corresponds to only one y-value, and each y-value corresponds to only one x-value, so ƒ is a one-to-one function. The inverse function is found by interchanging the x- and y-values in the table. ƒ-11x2 = 516, 20142, 14, 20152, 17, 20162, 110, 20172, 18, 201826 The domain and range of ƒ become the range and domain of ƒ-1. S Now Try Exercises 47, 51, and 53. Equations of Inverses The inverse of a one-to-one function is found by interchanging the x- and y-values of each of its ordered pairs. The equation of the inverse of a function defined by y = ƒ1x2 is found in the same way. Year Number of Hurricanes 2014 6 2015 4 2016 7 2017 10 2018 8 Data from National Oceanic and Atmospheric Administration.
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