442 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Important Facts about Inverses 1. If ƒ is one-to-one, then ƒ -1 exists. 2. The domain of ƒ is the range of ƒ -1, and the range of ƒ is the domain of ƒ-1. 3. If the point 1a, b2 lies on the graph of ƒ, then 1b, a2 lies on the graph of ƒ -1. The graphs of ƒ and ƒ -1 are reflections of each other across the line y = x. 4. To find the equation for ƒ -1, replace ƒ1x2 with y, interchange x and y, and solve for y. This gives ƒ -11x2. y = x2 −4.1 −6.6 4.1 6.6 x = y2 Despite the fact that y = x2 is not one-to-one, the calculator will draw its “inverse,” x = y2. Figure 12 Some graphing calculators have the capability of “drawing” the reflection of a graph across the line y = x. This feature does not require that the function be one-to-one, however, so the resulting figure may not be the graph of a function. See Figure 12. It is necessary to understand the mathematics to interpret results correctly. 7 Graphs of ƒ and ƒ -1 are shown in Figures 10 and 11. The line y = x is included on the graphs to show that the graphs of ƒ and ƒ -1 are mirror images with respect to this line. S Now Try Exercise 75. f –1(x) = x2 – 5, x $ 0 y 5 0 –5 y = x –5 5 x f(x) = !x + 5, x $ – 5 Figure 10 f(x) = !x+5, x ≥ −5 y = x −10 −16.1 10 16.1 f –1(x) = x2 − 5, x ≥ 0 Figure 11 An Application of Inverse Functions to Cryptography A one-to-one function and its inverse can be used to make information secure. The function is used to encode a message, and its inverse is used to decode the coded message. In practice, complicated functions are used. EXAMPLE 9 Using Functions to Encode and Decode a Message Use the one-to-one function ƒ1x2 = 3x + 1 and the following numerical values assigned to each letter of the alphabet to encode and decode the message BE MY FACEBOOK FRIEND. A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 K 11 L 12 M 13 N 14 O 15 P 16 Q 17 R 18 S 19 T 20 U 21 V 22 W 23 X 24 Y 25 Z 26
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