4.2 A One­to­One Functions
January 09, 2009
4.2A One­to­One Functions; Inverse Functions
Objectives: • Determine whether a function is one­to­one
• Obtain the graph of the inverse function from the graph of the function
• Find the inverse of a function defined by an equation
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4.2 A One­to­One Functions
January 09, 2009
A function is one­to­one if each input has its own output. How to check:
• Every x value has its own y value
• The graph passes the horizontal line test.
Examples:
#1
{(0, 0), (1, 1), (2, 16), (3, 81)}
#2
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4.2 A One­to­One Functions
January 09, 2009
Determine whether each given function is one­to­one:
1. Domain
Jeffrey
Benjamin Carolyn Elizabeth
3.
Range
Liz
Ben
Carol
Jeff
2. {(1, 4), (2, 5), (3, 6), (4, 6)}
4.
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4.2 A One­to­One Functions
January 09, 2009
If a function f is one­to­one, then it has an inverse function f ­1.
Domain of f = Range of f ­1; Range of f = Domain of f ­1.
The graphs of f and f ­1 are symmetric with respect to the line y = x.
Example:
f(x)
f ­1(x)
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4.2 A One­to­One Functions
January 09, 2009
The graph of a one­to­one function is given.
Draw the graph of the inverse function f ­1.
For convenience, the graph of y = x is also given.
(2, 5)
(0, 1)
(­3, 0)
(­4, ­2)
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4.2 A One­to­One Functions
January 09, 2009
To verify that f ­1 is the inverse of f, show that f ­1(f(x)) = x for every x in the domain of f and f(f ­1(x)) = x for every x in the domain of f ­1.
Given:
f(x) = 4x ­ 8
f ­1(x) = x/4 + 2
Show:
f ­1(f(x)) = x
f(f ­1(x)) = x
f ­1(4x ­ 8) = x
f(x/4 + 2) = x
4x ­ 8 + 2 = x
4(x/4 + 2) ­ 8 = x
4 x + 8 ­ 8 = x
x ­ 2 + 2 = x
x = x
x = x
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4.2 A One­to­One Functions
January 09, 2009
HOMEWORK: page 267 (10, 12, 14, 16, 19 ­ 22, 32, 34, 37, 42, 43, 45)
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