1. Table of Contents Inverse Functions Consider the functions, f (x) and g(x), illustrated by the mapping diagram. 1 6 87 649 f The function, f (x), takes the domain values of 1, 8 and 64 and produces the corresponding range values of 6, 7, and 9. g 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).

2. Table of Contents Inverse Functions Slide 2 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 of f (x) and is denoted f - 1 (x). g 1 6 87 649 f g domain of f (x)range of f (x) range of g(x)domain of g(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).

3. Table of Contents Inverse Functions Slide 3 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). The g function 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. 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). 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. This is illustrated on the next slide.

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

5. Table of Contents Inverse Functions Slide 5 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. 

6. Table of Contents Inverse Functions Slide 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) = 

7. Table of Contents Inverse Functions Slide 7 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.

8. Table of Contents Inverse Functions Slide 8 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. 1 - 1 - 22

9. Table of Contents Inverse Functions