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 f

Inverse Functions Now consider the function g illustrated in the mapping diagram. The function g "undoes" function f f g 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).

Inverse Functions The mapping diagram with the domains and ranges of f (x) and g(x) are labeled is shown. g f g domain of f (x)range of f (x) 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). g f g domain of f (x)range of f (x) range of g(x)domain of g(x)

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)

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

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

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

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

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

Inverse Functions ANOTHER PROPERTY OF INVERSE FUNCTIONS 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) = x 3 are shown along with the graph of y = x

Inverse Functions