Pre-Calculus
Section 1.9 Inverses of Functions
Name:_______________________
Notes 1-7
Complete the tables for the following functions. Then graph on the coordinate plane. f ( x )

2 x

2 g ( x )

1
2 x

1 x


0
1
2
2
1 y x



0
2
6
4
2 y

1
What do you notice about the tables?
Now, graph y = x.
What do you notice about the graphs?
The functions f ( x )

2 x

2
and f ( x )

1 
1 x

1
are _______________ of each other.
2
Section 1.9 : Figure 1.92, Illustration
From now on, we will denote the inverse of f(x) as:
Properties of Inverses a.
Domain and Range:
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The Domain of f(x) = Range of ________.
1 | 19
The Range of f(x) = Domain of _____
The line of symmetry: ______________
If the point (x, y) is on the function_________________________________
If f ( g ( x ))
 x and g ( f ( x ))
 x , then __________________________________________

Example Set 1: Determine whether the following functions are inverses. a. f ( x )

9
 x
2 g ( x )

9
 x
2
b. f ( x ) g ( x )

4 x
 x

4

1
1
c. f ( x g ( x )
)
 3
 x
3 x

1

1
Example Set 2
1
. Sketch the graphs of 2. Sketch g
 x

4
and f

2 x

3 and f

1
Show that they are inverses of each other.
  
1
2
 x

3

Definition:
Examples Set 3: Are the following graphs one-to-one? g

1
 
 x
2 
4
, x

0 and
and show they are inverses of each
other.

 What do you notice about the one-to-one graphs?
 How can you tell a function is one-to-one by its graph?
 What does this have to do with inverses?
If a function is one-to-one, then the function has an inverse function.
Example Set 4: Do the following functions have inverse functions? x 2 4 5 7 f
 
0 3 0 2
1.) Use horizontal line test to determine if an inverse exists
2.) Switch the ‘x’ and ‘y.’ Solve for y
3.) Write f

1
( x ) instead of y
4.)
Verify your answer.
x
-2
-1
0
10
20
3 f ( x )
4
1
0
1
4
9
Example Set 5: Find the inverse of the following functions. a. f
 
4 x

9 b.
   x
2 
2, x

0 c. f ( x )

5

3 x
2
Write the equation of each function’s inverse. Then graph the function and its inverse. a.
f ( x )

3 x

6
b. f ( x )
 x
2 
2 ,
x > 0 c. f ( x )

( x

2 )
2 
1
, x < 0