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4.1 Inverse Functions
Copyright © 2007 Pearson Education, Inc.
Slide 5-1
5.1 One-to-One Functions
A function f is a one-to-one function if, for
elements a and b from the domain of f,
ab
•
implies
f (a)  f (b).
Only functions that are one-to-one have
inverses.
Copyright © 2007 Pearson Education, Inc.
Slide 5-2
5.1 One-to-One Functions
Example Decide whether the function is one-to-one.
(a) f ( x)  4 x  12
(b) f ( x)  25  x 2
Solution
(a) For this function, two different x-values produce
two different y-values.
Suppose that a  b, then  4a  4b and
 4a  12  4b  12. Since f (a)  f (b), f is one - to - one.
(b) If we choose a = 3 and b = –3, then 3  –3, but
f (3)  25  32  4 and f (3)  25  (3) 2  4,
so f (3)  f (3), therefore f is not one - to - one.
Copyright © 2007 Pearson Education, Inc.
Slide 5-3
5.1 The Horizontal Line Test
If every horizontal line intersects the graph of a
function at no more than one point, then the function
is one-to-one.
Example Use the horizontal line test to determine
whether the graphs are graphs of one-to-one functions.
(a)
(b)
Not one-to-one
Copyright © 2007 Pearson Education, Inc.
One-to-one
Slide 5-4
5.1 Inverse Functions
Let f be a one-to-one function. Then, g is the
inverse function of f and f is the inverse of g if
( 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 .
Example Show that f ( x)  x3  1 and g ( x)  3 x  1
are inverse functions of each other.
( f  g )( x)  f [ g ( x)]   x  1  1  x  1  1  x
3
3
( g  f )( x)  g[ f ( x)]  3 x 3  1  1  3 x 3  x
Copyright © 2007 Pearson Education, Inc.
Slide 5-5
5.1 Finding an Equation for the Inverse
Function
• Notation for the inverse function f -1 is read
“f-inverse”
Finding the Equation of the Inverse of y = f(x)
1. Interchange x and y.
2. Solve for y.
3. Replace y with f -1(x).
Any restrictions on x and y should be considered.
Copyright © 2007 Pearson Education, Inc.
Slide 5-6
5.1 Example of Finding f -1(x)
Example Find the inverse, if it exists, of
4x  6
f ( x) 
.
5
4x  6
Write f (x) = y.
y
Solution
5
4y  6
Interchange x and y.
x
5
Solve for y.
5x  4 y  6
5x  6
y
4
5x  6
1
Replace y with f -1(x).
f ( x) 
4
Copyright © 2007 Pearson Education, Inc.
Slide 5-7
5.1 The Graph of f -1(x)
• f and f -1(x) are inverse functions, and f (a) = b for
real numbers a and b. Then f -1(b) = a.
• If the point (a,b) is on the graph of f, then the point
(b,a) is on the graph of f -1.
If a function is one-to-one,
the graph of its inverse f -1(x)
is a reflection of the graph of
f across the line y = x.
Copyright © 2007 Pearson Education, Inc.
Slide 5-8
5.1 Finding the Inverse of a Function
with a Restricted Domain
Example
Let f ( x)  x  5. Find f 1 ( x).
Solution
Notice that the domain of f is restricted
to [–5,), and its range is [0, ). It is one-to-one and
thus has an inverse.
y  x5
x  y5
x2  y  5
y  x2  5
The range of f is the domain of f -1, so its inverse is
f 1 ( x)  x 2  5, x  0.
Copyright © 2007 Pearson Education, Inc.
Slide 5-9
5.1 Important Facts About Inverses
1. If f is one-to-one, then f -1 exists.
2. The domain of f is the range of f -1, and the
range of f is the domain of f -1.
3. If the point (a,b) is on the graph of f, then the
point (b,a) is on the graph of f -1, so the graphs
of f and f -1 are reflections of each other
across the line y = x.
Copyright © 2007 Pearson Education, Inc.
Slide 5-10
5.1 Application of Inverse Functions
Example Use the one-to-one function f(x) = 3x + 1 and the
numerical values in the table to code the message BE VERY
CAREFUL.
A 1
F 6
K 11
P 16
U 21
B 2
G 7
L 12
Q 17
V 22
C 3
H 8
M 13
R 18
W 23
D 4
I 9
N 14
S 19
X 24
E 5
J 10
O 15
T 20
Y 25
Z 26
Solution BE VERY CAREFUL would be encoded as
7 16 67 16 55 76 10 4 55 16 19 64 37
because B corresponds to 2, and f(2) = 3(2) + 1 = 7,
and so on.
Copyright © 2007 Pearson Education, Inc.
Slide 5-11