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Table of Contents
Inverse Functions
Consider the functions, f (x) and g(x), illustrated by the mapping diagram.
1 6
8 7
64 9
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).
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
8 7
64 9
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).
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.
Table of Contents
Inverse Functions
Slide 4
x f (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
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. ,53 xxf
First, show that (f g)(x) = x.
Next, show that (g f)(x) = x.
(f g)(x) = 553 3 x 55 x = x.
(g f)(x) = 33 55 x 33 x = x.
Table of Contents
Inverse Functions
Slide 6
Try: Algebraically show that the one-to-one functions,
,8
3 xxgf (x) = 8x + 3, and are inverses of each
other.
(f g)(x) = 38
38
x
= x – 3 + 3 = x.
(g f)(x) =
.8
88
338x
xx
Table of Contents
Inverse Functions
Slide 7
A PROPERTY OF INVERSE FUNCTIONS
x f (x)
domain of f range of f
range of f - 1 domain of f - 1f - 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.
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) = x3 3 xxf
are shown along with the graph of y = x.
1
- 1
- 2 2
Table of Contents
Inverse Functions