436 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Inverse Functions Certain pairs of one-to-one functions “undo” each other. For example, consider the functions g1x2 = 8x + 5 and ƒ1x2 = 1 8 x - 5 8 . We choose an arbitrary element from the domain of g, say 10. Evaluate g1102. g1x2 = 8x + 5 Given function g1102 = 8 # 10 + 5 Let x = 10. g1102 = 85 Multiply and then add. Now, we evaluate ƒ1852. ƒ1x2 = 1 8 x - 5 8 Given function ƒ1852 = 1 8 1852 - 5 8 Let x = 85. ƒ1852 = 85 8 - 5 8 Multiply. ƒ1852 = 10 Subtract and then divide. Starting with 10, we “applied” function g and then “applied” function ƒ to the result, which returned the number 10. See Figure 3. Tests to Determine Whether a Function Is One-to-One 1. Show that ƒ1a2 = ƒ1b2 implies a = b. This means that ƒ is one-to-one. (See Example 1(a).) 2. In a one-to-one function, every y-value corresponds to no more than one x-value. To show that a function is not one-to-one, find at least two x-values that produce the same y-value. (See Example 1(b).) 3. Sketch the graph and use the horizontal line test. (See Example 2.) 4. If the function either increases or decreases on its entire domain, then it is one-to-one. A sketch is helpful here, too. (See Example 2(b).) 10 Function g(x) = 8x + 5 f(x) = 85 Function 10 1 8 5 8 x – Figure 3 These functions contain inverse operations that “undo” each other. As further examples, confirm the following. g132 = 29 and ƒ1292 = 3 g1-52 = -35 and ƒ1-352 = -5 g122 = 21 and ƒ1212 = 2 ƒ122 = - 3 8 and ga- 3 8b = 2
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