4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Warm­up
Given the following functions:
1. 2. 1
4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
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4.2 One­to­One Functions and Inverse Functions 2011
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4.2 One­to­One Functions and Inverse Functions 2011
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4.2 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 One­to­One Functions and Inverse Functions 2011
April 20, 2011
On the graph, sketch an example of a function.
Why do you know that it is a function?
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Functions can be categorized as one­to­one or many­to­one.
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 One­to­One Functions and Inverse Functions 2011
April 20, 2011
EXAMPLES
One­to­one
Many­to­one
Not a function
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10
­10
­10
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Mapping Functions
One­to­one
each x has only one y
Many­to­one
is the image of two x's
Not a function
is paired with two y's
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
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 One­to­One Functions and Inverse Functions 2011
April 20, 2011
If a function f is one­to­one, then it has an inverse function f ­1.
The graphs of f and f ­1 are symmetric with respect to the line y = x.
Example:
f(x)
f ­1(x)
Domain of f = Range of f ­1
Range of f = Domain of f ­1
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Graph the inverse function:
x y
­1 ­5
0 ­3
1 ­1
2 1
3 3
x
­5
­3
­1
1
3
y
­1
0
1
2
3
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Graph the inverse function:
x
­1
0
1
2
3
y
­5
­3
­1
1
3
x
­5
­3
­1
1
3
y
­1
0
1
2
3
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
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 One­to­One Functions and Inverse Functions 2011
April 20, 2011
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 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Finding the Inverse Function:
Given: f(x) = 2x + 1
x ≠ 1
x ­ 1
Step 1: Interchange the variables x and y.
y = 2x + 1
x = x ­ 1
2y + 1
y ­ 1
Step 2: Solve for y.
(y ­ 1) x = 2y + 1
y ­ 1
(y ­ 1)
xy ­ x = 2y + 1
xy ­ 2y = x + 1
y(x ­ 2) = x + 1 y = x + 1
x ­ 2
Step 3: Re­write as f ­1(x)
f ­1(x) = x + 1
x ­ 2
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Step 4: Check the result showing that f ­1(f(x)) = x and f(f ­1(x)) = x
f(x) = 2x + 1
x ≠ 1
x ­ 1
f ­1(x) = x + 1
x ­ 2
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
Step 5: Find the domain and range of f and f ­1.
HINT: Remember Domain of f = Range of f ­1 and Domain of f ­1 = Range of f.
f(x) = 2x + 1
x ≠ 1
x ­ 1
f ­1(x) = x + 1
x ­ 2
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4.2 One­to­One Functions and Inverse Functions 2011
April 20, 2011
HOMEWORK: page 267 (10, 12, 14, 16, 19 ­ 22, 32, 34, 37, 42, 43, 45, 48 ­ 58 even & no graphing, 76)
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