inverses & one-to-one
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Copyright © 2011 Pearson Education, Inc. Slide -1
CHAPTER 6
INVERSES & ONE-to-ONE Functions
Copyright © 2011 Pearson Education, Inc. Slide -2
Inverse Functions
Example
Also, f [g(12)] = 12. For these functions, it can be shown that
for any value of x. These functions are inverse functions of each other.
12)]12([i.e.129681
)96(
96128)12(
.81
)( and 8)(Let
fgg
f
xxgxxf
xxfgxxgf )]([and)]([
Copyright © 2011 Pearson Education, Inc. Slide -3
• Only functions that are one-to-one have inverses.
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).
Copyright © 2011 Pearson Education, Inc. Slide -4
One-to-One Functions
Example Decide whether each function is one-to-one.
(a) (b)
Solution
(a) For this function, two different x-values produce two different y-values.
(b) If we choose a = 3 and b = –3, then 3 –3, but
124)( xxf 225)( xxf
one.-to-one is),()( Since .124124
and 44 then , that Suppose
fbfafba
baba
one.-to-onenot is therefore),3()3( so ,4)3(25)3(and4325)3( 22
fffff
Copyright © 2011 Pearson Education, Inc. Slide -5
Horizontal Line Test
Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions.
(a) (b)
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one.
Not one-to-one One-to-one
Copyright © 2011 Pearson Education, Inc. Slide -6
Inverse Functions
Exampleare inverse functions of each other.
Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if
. ofdomain in the every for ))((and
, ofdomain in the every for ))((
fxxxfg
gxxxgf
1)( and 1)( that Show 33 xxgxxf
xxxxfgxfg
xxxxgfxgf
3 33 3
33
11)]([))((
1111)]([))((
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Finding an Equation for the Inverse Function
.
Finding the Equation of the Inverse of y = f(x)
For a one-to-one function f defined by an equation
y = f(x), find the defining equation of the inverse as follows. (Any restrictions on x and y should be considered.)
1. Interchange x and y.
2. Solve for y.
3. Replace y with f -1(x).
Copyright © 2011 Pearson Education, Inc. Slide -8
Example of Finding f -1(x)
Example Find the inverse, if it exists, of
Solution
.5
64)(
xxf
Write f (x) = y.5
64 xy
Interchange x and y.5
64 yx
Solve for y.
465645
x
y
yx
Replace y with f -1(x).4
65)(1 x
xf
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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.
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Finding the Inverse of a Function with a Restricted Domain
Example Let
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.
The range of f is the domain of f -1, so its inverse is
).( Find.5)( 1 xfxxf
55
55
2
2
xyyx
yxxy
.0,5)( 21 xxxf
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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 © 2011 Pearson Education, Inc. Slide -12
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 21B 2 G 7 L 12 Q 17 V 22C 3 H 8 M 13 R 18 W 23D 4 I 9 N 14 S 19 X 24E 5 J 10 O 15 T 20 Y 25
Z 26
Solution BE VERY CAREFUL would be encoded as7 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.