437 4.1 Inverse Functions In particular, for the pair of functions g1x2 = 8x + 5 and ƒ1x2 = 1 8 x - 5 8 , ƒ1g1222 = 2 and g1ƒ1222 = 2. In fact, for any value of x, ƒ1g1x22 = x and g1ƒ1x22 = x. Using the notation for composition of functions, these two equations can be written as follows. 1ƒ∘ g21x2 = x and 1g∘ ƒ21x2 = x The result is the identity function. Because the compositions of ƒ and g yield the identity function, they are inverses of each other. Inverse Function Let ƒ be a one-to-one function. Then g is the inverse function of ƒ if 1ƒ° g2 1x2 =x for every x in the domain of g, and 1g° ƒ2 1x2 =x for every x in the domain of ƒ. The condition that f is one-to-one in the definition of inverse function is essential. Otherwise, g will not define a function. x y ƒ(x) = x3 – 1 Figure 4 EXAMPLE 3 Determining Whether Two Functions Are Inverses Let functions ƒ and g be defined respectively by ƒ1x2 = x3 - 1 and g1x2 = 23 x + 1. Is g the inverse function of ƒ? SOLUTION As shown in Figure 4, the horizontal line test applied to the graph indicates that ƒ is one-to-one, so the function has an inverse. Because it is oneto-one, we now find 1ƒ∘ g21x2 and 1g∘ ƒ21x2. 1ƒ∘ g21x2 = ƒ1g1x22 = A 23 x + 1 B 3 - 1 = x + 1 - 1 = x 1g∘ ƒ21x2 = g1ƒ1x22 = 23 1x3 - 12 + 1 = 23 x3 = x Since 1ƒ∘ g21x2 = x and 1g∘ ƒ21x2 = x, function g is the inverse of function ƒ. S Now Try Exercise 37. A special notation is used for inverse functions: If g is the inverse of a function ƒ, then g is written as ƒ −1 (read “f-inverse”). ƒ1x2 = x3 - 1 has inverse ƒ -11x2 = 23 x + 1. See Example 3.
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