725 7.5 Inverse Circular Functions The following statements summarize the concepts of inverse functions. Review of Inverse Functions 1. In a one-to-one function, each x-value corresponds to only one y-value, and each y-value corresponds to only one x-value. 2. If a function ƒ is one-to-one, then ƒ has an inverse function ƒ-1. 3. The domain of ƒ is the range of ƒ-1, and the range of ƒ is the domain of ƒ -1. That is, if the point 1a, b2 lies on the graph of ƒ, then the point 1b, a2 lies on the graph of ƒ-1. 4. The graphs of ƒ and ƒ-1 are reflections of each other across the line y = x. 5. To find ƒ-11x2 for ƒ1x2, follow these steps. Step 1 Replace ƒ1x2 with y and interchange x and y. Step 2 Solve for y. Step 3 Replace y with ƒ-11x2. y x 0 (b, a) is the reflection of (a, b) across the line y = x. (b, a) (a, b) y = x x The graph of f –1 is the reflection of the graph of f across the line y = x. y = x f(x) = x3 – 1 f –1(x) = √x + 1 2 2 y 0 3 Figure 9 Figure 9 illustrates some of these concepts. Inverse Sine Function Refer to the graph of the sine function in Figure 11 on the next page. Applying the horizontal line test, we see that y = sin x does not define a one-to-one function. If we restrict the domain to the interval C - p 2 , p 2 D , which is the part of the graph in Figure 11 shown in color, this restricted function is one-to-one and has an inverse function. The range of y = sin x is 3-1, 14 , so the domain of the inverse function will be 3-1, 14, and its range will be C - p 2 , p 2 D . If the domain of g(x) = x2 is restricted so that x ≥ 0, then it is a one-to-one function. g(x) = x2, x ≥ 0 y x 2 0 4 Figure 10 NOTE Recall that we often restrict the domain of a function that is not one-to-one to make it one-to-one without changing the range. For example, the function g1x2 = x2, with its natural domain 1-∞, ∞2, is not one-to-one. However, if we restrict its domain to the set of nonnegative numbers 30, ∞2, we obtain a new function ƒ that is one-to-one and has the same range as g, 30, ∞2. See Figure 10. By interchanging the components of the ordered pairs of a one-to-one function ƒ, we obtain a new set of ordered pairs that satisfies the definition of a function. Recall that the inverse function of a one-to-one function ƒ is defined as follows. ƒ −1 = 5 1 y, x2 ∣ 1x, y2 belongs to ƒ 6 The special notation used for inverse functions is ƒ −1 (read “ƒ-inverse”). It represents the function created by interchanging the input (domain) and the output (range) of a one-to-one function. CAUTION Do not confuse the −1 in f −1 with a negative exponent. The symbol ƒ-11x2 represents the inverse function of ƒ, not 1 ƒ1x2.
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