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Inverse Functions

The function, f ( x ), takes the domain values of 1, 8 and 64 and produces the corresponding range values of 6, 7, and 9. f.

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Inverse Functions

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  1. The function, f (x), takes the domain values of 1, 8 and 64 and produces the corresponding range values of 6, 7, and 9. f The function, g(x), "undoes" f (x). It takes the f (x) range values of 6, 7, and 9 as its domain values and produces as its range values, 1, 8, and 64 which were the domain values of f (x). 6 1 8 7 64 9 g Inverse Functions Consider the functions, f (x) and g(x), illustrated by the mapping diagram.

  2. f domain of f (x) range of f (x) 6 1 8 7 64 9 g g range of g(x) domain of g(x) Inverse Functions The mapping diagram with the domains and ranges of f (x) and g(x) labeled is shown. If there exists a one-to-one function, g(x), that "undoes" f (x) for every value in the domain of f (x), then g(x) is called the inverse function off (x) and is denoted f- 1(x). Also note here f (x), "undoes" g(x). Similarly, the one-to-one function f (x) is called the inverse function of g (x) and is denoted g- 1(x). Slide 2

  3. DEFINITION: One-to-one functions, f (x) and g(x), are inverses of each other if (f g)(x) = x and (g f)(x) = x for all x-values in the domains of f (x) and g(x). First what is done is inside parentheses. This means the f function takes as its domain value, x,and produces the range value, f (x). Inverse Functions To see why the definition is written this way, recall (g f)(x) = g(f (x)), so (g f)(x) = x can be rewritten as g(f (x)) = x . The g function then takes this range value of the f function,f (x) as its domain valueand produces x (the original domain value of the f function as its range value. This is illustrated on the next slide. Slide 3

  4. range value of f function, f (x) domain value of f function, x f x f (x) g range value of g function, x domain value of g function, f (x) Inverse Functions Slide 4

  5. Example: Algebraically show that the one-to-one functions, and g(x) = (x – 5)3, are inverses of each other. Inverse Functions First, show that (f g)(x) = x. (f g)(x) = = x.  Next, show that (g f)(x) = x. (g f)(x) = = x.  Slide 5

  6. Try: Algebraically show that the one-to-one functions, f (x) = 8x + 3, and are inverses of each other. (f g)(x) = = x – 3 + 3 = x.  (g f)(x) =  Inverse Functions Slide 6

  7. The domain of a function, f, is the range of its inverse, f- 1. The range of a function, f, is the domain of its inverse, f- 1. domain of f range of f f x f (x) f - 1 range of f- 1 domain of f- 1 Inverse Functions A PROPERTY OF INVERSE FUNCTIONS Slide 7

  8. The graphs of a function, f, and its inverse, f- 1, are symmetric across the line y = x. For example, the graphs of and f- 1(x) = x3 1 are shown along with the graph of y = x. - 2 2 - 1 Inverse Functions ANOTHER PROPERTY OF INVERSE FUNCTIONS Slide 8

  9. Inverse Functions END OF PRESENTATION Click to rerun the slideshow.

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