2.3 c inverse functions
TRANSCRIPT
Inverse Functions
1
Inverse Functions
If is a function consisting of ordered pairs , ,
then there is a relation called the ,
whose elements are the ord
inverse o
ered pair
f
s , .
f
f x y
y x
2
Inverse Functions
1
1
If is one-to-one, then the inverse of is
called the inverse function of denoted by
.
f f
y f x f y
f
f
x
3
Example 2.4.10
1
5 3Given ,
4
a. Find .
5 3
45 3
4
xf x
f x
xf x
xy
4
1
5 3
45 3
44 5 3
4 3 5
4 3
5
4 3
5
xy
yx
x y
x y
xy
xf x
5
1 1
1 1
b. Find and
4 3
5
4 35 3
5
44 3 3
44
4
f f x f f x
f f x f f x
xf
x
x
x
x
1
5 3
44 3
5
xf x
xf x
6
1 1
5 3
4
5 34 3
4
55 3 3
5
5
5
f f x f f x
xf
x
x
x
x
1
5 3
44 3
5
xf x
xf x
7
-2 -1 1 2
-2
-1
1
2
x
y
1
1
c. Show the graph of and in one
coordinate plane.
5 3
4
4 3
5
f f
xf x
xf x
x 0 -3/5
y 3/4 0
x 0 3/4
y -3/5 0
f
1f y x
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Properties
1
1
1
1 1
1.
2.
3. The graphs of and are symmetric
with respect to the line .
4.
Dom f Rng f
Rng f Dom f
f f
y x
f f x f f x x
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Example 2.4.1
1
Given 1 2,
a. Find .
1 2
1 2
1 2
g x x
g x
g x x
y x
x y
2
2
21
1 2
1 2
1 2
1 2
x y
x y
x y
g x x
10
1
21
1
1
b. Find the domain and range of
1 2
1 2
2 0 2,
2 1,
1,
2,
g
g x x
g x x
x Dom g
x Rng g
Dom g
Rng g
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-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
1
21
1
1
c. Show that the graph of and are
symmetric with respect to the line
1 2
2,
1 2
1,
2,
g g
y x
g x x
Dom g
g x x
Dom g
Rng g
x 2 3
y 1 2
y x
g
1g
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1 1
1 1
2
2
2
d. Show that
1 2
1 1 2 2
1 1
1 1
g g x g g x x
g g x g g x
g x
x
x
x
x
21
1 2
1 2
g x x
g x x
13
1 1
2
2
1 2
1 2 1 2
2 2
2 2
g g x g g x
g x
x
x
x
x
21
1 2
1 2
g x x
g x x
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