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Inverse Functions

f. 6. 1. 8. 7. 64. 9. Inverse Functions. Consider the function f illustrated by the mapping diagram. The function f takes the domain values of 1, 8 and 64 and produces the corresponding range values of 6, 7, and 9. f. 6. 1. 8. 7. 64. 9. g. Inverse Functions.

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Inverse Functions

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  1. f 6 1 8 7 64 9 Inverse Functions Consider the function f illustrated by the mapping diagram. The function ftakes the domain values of 1, 8 and 64 and produces the corresponding range values of 6, 7, and 9.

  2. f 6 1 8 7 64 9 g Inverse Functions Now consider the function gillustrated in the mapping diagram. The function g"undoes" function f. 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).

  3. 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) are labeled is shown.

  4. 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 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 of f (x) and is denoted f- 1(x).

  5. Inverse Functions DEFINITION: Let f and g be functions where f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f. Then function g is the inverse of function f, and is denoted f-1(x)

  6. Inverse Functions To see why the definition is written this way, considerg(f (x)) = x. The part that is done first is inside parentheses. This means the function f takes as its domain value, x, and produces the range value, f (x). The function g then takes this range value of the f function, f (x), as its domain value and produces x (the original domain value of the ffunction) as its range value.

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

  8. 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. 

  9. 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

  10. 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

  11. 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

  12. Inverse Functions END OF PRESENTATION

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