2.7 INVERSE FUNCTIONS
Begin with a function:
1, 6, 2, 6, 3, 8, 4, 9, 5, 8
Interchange the domain (x’s) and range (y’s):
6, 1, 6, 2, 8, 3, 9, 4, 8, 5
The result is called the inverse. Is the inverse also a function?
Your turn. Begin with a function:
1, 3, 2, 5, 3, 7, 4, 9, 5, 11
Interchange the x’s and the y’s. Write the result here:
The result is the inverse of the function. Is the inverse also a function?

One-to-One Functions
After interchanging the x- and y-values, if the resulting relation is also a function, then
the original function is called a one-to-one function.
If a function is one-to-one, then it has an inverse!

The Horizontal Line Test
There is an easy geometric test to determine whether a function has an inverse function.
The test is called the horizontal line test. A function has an inverse if no horizontal line
intersects the graph of the function at more than one point.
a)
b)

Notation: If a function f x  has in inverse, then the inverse is denoted as
f 1 x  .

Finding the inverse of a function.
To find the inverse of a function, do the following:
1.
2.
3.
4.
5.
Use the Horizontal Line Test to determine whether the function has an inverse.
Replace f x  with y .
Interchange the roles of x and y.
Solve for y.
Replace y with f 1 x  in the new equation.
Example 1
Find the inverse of each:
a)
f ( x)  3x  2
b)
f ( x)  x 2  2
c)
f ( x)   x  2 
d)
f ( x)  x 3  1
e)
f x  
f)
f x   2 x  3
g)
f ( x) 
3
5  3x
2
2x  3
x 1
Algebraically, two functions f x  and g x are inverses of each other if f g x   x
and g  f x   x .
Example 2
Verify that the functions are inverses of each other.
a)
f x  
x 8
7
g x   7 x  8
b)
f x   8  x 3
g x   3 8  x
Example 3
Which of the following represents functions that have inverse functions?
6
5
4
3
2
1
-6
-4
-2
2
4
6
8
-1
-2
-3
-4
-5
5
4
3
2
1
-6
-4
-2
2
4
6
8
-1
-2
-3
-4
5
4
3
2
1
-6
-4
-2
2
-1
-2
-3
-4
4
6
8
Graphs of f and f 1 .
The graph of a function and its inverse are reflections of each other around the line y = x.
Example 4
The graph of f x  is given. Sketch the graph of f
1
x  .
7
6
5
4
3
2
1
8
6
4
2
2
1
2
3
4
4
6
Example 5
The graph of f x  is given. Sketch the graph of f
1
x  .
6
5
4
3
y= x+4
2
1
-6
-4
-2
2
4
6
-1
-2
-3
-4
Example 6 – If a function f ( x) has an inverse, and f (2)  8 , then what is f 1 (8) ?
Example 7 – Decide whether the given functions are inverses:
x
-1
-5
-3
0
f ( x)
-5
1
-8
-2
x
-5
1
-8
-2
g ( x)
-1
-5
-3
0
8
Example 8 Use the graph of f to create a table of values for the given points. Then
create a second table that can be used to find f 1 , and sketch the graph of f 1 on the
same axis as the given graph.
Example 9
Below is the graph of f 1  x  . Use the graph to complete the tables that follow.
x
f

-4
-3
-2
0
1
4
x
f  x
-3
-2
-1
0
1
2
x
-4
-2
0
4
-2
0
1
-3
-2
0
1
-4
-3
0
4
1
f f
 x
1
 x 
x
 f  f 1   x 
x
 f  f  x
-3
1
x
f
1
 x