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Warm Up
1
1 Supposef ( x) = x and g( x) = x + 2. Find
f o g.
2
f ( x) = x  1 and g ( x) = 3x
and g f
g o f x

 2
Suppose that
find
2
1
Suppose f ( x ) = x and g ( x ) =
. Find
x+2
f o g.
 f g  x
= f  g  x 
 1 
= f

x
+
2


=
=
1
x+2
1
x+2
Suppose that f ( x) = x  1
and
g ( x) = 3 x
find
2
g o f x
g o f x = g  f x


= g x 1
2
= 3x  1
2
= 3x  3
2
Suppose that f ( x) = x  1
and
g ( x) = 3 x
find
2
g
g
f  2 = g  f  2 
= g  (2)  1
= g  3
= (3)(3) = 9
2
f  2
REVIEW
Operations on Functions
Perform the indicated operation.
Addition: h
xf
=
x
+
g
x
f
x
=
7
x
+
1
g
x
=
x

1










a
.h
=
f+
g
h
xx
=
7
+
11
+
x



h
x=
8
x

b
.h
=
f
g
h
xx
=
7
+
11

x
+


h
x
=
6
x
+
2


c. h = fg
h
xxx
=
7
+
1

1






2
h
xxx
=
7
6

1

 
f
d. h=
g
7
x
+
1
h
x=

x

1
Subtraction: h
xf
=
x

g
x






xf
=
x
g
x
Multiplication: h





Division: hx=
f x
gx
x
=
g
fx



Composition: h


5
Inverse Functions
Section 7.8 in text
Many (not ALL) actions
are reversible

That is, they undo or cancel each other
• A closed door can be opened
• An open door can be closed
• $100 can be withdrawn from a
savings account
• $100 can be deposited into a savings
account
NOT all actions are
reversible

Some actions can
not be undone
• Explosions
• Weather
Mathematically, this basic
concept of reversing a
calculation and arriving at
an original result is
associated with an
INVERSE.
Actions and their inverses
occur in everyday life
Climbing up a ladder
 Inverse: Climbing down a ladder

Opening the door and turning on the
lights
 Inverse: Turning off the lights and
closing the door

A person opens a car door, gets in, and
starts the engine.
 Inverse: A person stops the engine,
gets out, and closes the car door.

Inverse operations can be
described using functions.
Multiply x by 5
 Inverse: Divide x by 5

Divide x by 20 and add 10
 Inverse: Subtract 10 from x and multiply
by 20

Multiply x by -2 and add 3
 Inverse: Subtract 3 from x and divide by -2

Notation

To emphasize that a function is an
inverse of said function, we use the
same function name with a special
notation.
Function, f(x)
-1
 Inverse Function of f(x) = f
(x)

As we noted earlier, not every
function has an inverse. So when
does a function have an inverse?
In words: Each different input
produces its own different output.
Graphically: Use the Horizontal Line
test.
Line Tests
Vertical Line Test on f:
determines if f is a function
f Function
f Not a Function
Horizontal Line Test on f:
determines if f -1 is a
function
Glencoe – Algebra 2
Chapter 7: Polynomial Functions
15
How about if the function is given
numerically or symbolically,how do you
determine its inverse?

Numerically:
interchange domain
(x) and range ( f(x) )

Symbolically:
• Interchange x and y
and solve for the
new y to obtain f-1(x)
fx
3
x5
=
Interchange
y
=
3
x

5 x
=
3
y

5
Solve
x
+
53
=y 1x+5=y
3 3
1 5

1
f 
x= x
+
3 3
Putting It All Together
with Examples
x
f(x)
-1
-2
0
1
1
2
2
1
3
-2
4
-6
Does this table
represent a
function?
 Does this function
have an inverse?
 Find the inverse

Example
x
f(x)
-3
10
-2
6
-1
4
0
1
2
-3
3
-10



Does this table
represent a
function?
Does this function
have an inverse?
Find the inverse
Example

f(x) =3x +5

f(x) = x3 + 1

Does this equation
represent a function?
How do you know?
Does this function have
an inverse?
How do you know?
Find the inverse
Confirm the inverse

Does this equation
represent a function?
How do you know?
Does this function have
an inverse?
How do you know?
Find the inverse
Confirm the inverse

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
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