Section 3.6
One-to-one Functions; Inverse Functions
Objectives
1. Understanding the Definition of a One-to-one Function
2. Determining if a Function is One-to-one Using the Horizontal Line Test
3. Understanding and Verifying Inverse Functions
4. Sketching the Graphs of Inverse Functions
5. Finding the Inverse of a One-to-one Function
Later in this section we will examine a process for finding the inverse of a function, but keep in mind the
following. We can find the inverse of many functions using this process, but that inverse will not always be a
function. In this text we are only interested in inverses that are functions, so we will first develop a test to
determine if a function has an inverse function. When the word inverse is used throughout the remainder of this
section, we will assume that we are referring to the inverse that is a function.
Objective 1: Understanding the Definition of a One-to-one Function
Definition
One-to-one Function
A function f is one-to-one if for any values a  b in the domain of f ,
f (a )  f (b) .
The definition above suggests that a function is one-to-one if for any two different input values (domain values)
the corresponding output values (range values) must be different. An alternate definition says that if two range
values are the same, f (u )  f (v) , then the domain values must be the same, that is u  v .
Alternate Definition of a One-to-one Function
A function f is one-to-one if for any two range values f (u ) and f (v ) , f (u )  f (v) implies that u  v .
The function sketched below on the left is one-to-one because for any two distinct x-values in the
domain  a  b  , the function values or range values are not equal  f (a)  f (b)  . In the sketch on the right we
see that the function y  g ( x) is not one-to-one since we can easily find two different domain values which
correspond to the same range value.
y  f ( x)
g (a )  g (b)
f (b )
a
a
b
f (a)
y  g ( x)
b
Objective 2: Determining if a Function is One-to-one Using the Horizontal Line Test
Any horizontal line intersects the graph of y  f ( x) in at most one place, while we can find many horizontal
lines that intersect the graph of y  g ( x) more than once. This gives us a visual example of how we can use
horizontal lines to help us determine from the graph if a function is one-to-one. Using horizontal lines to
determine if the graph of a function is one-to-one is known as the horizontal line test.
y  f ( x)








 
y  g ( x)
The Horizontal Line Test
If every horizontal line intersects the graph of a function f at most once, then
f is one-to-one.
Objective 3: Understanding and Verifying Inverse Functions
We are now ready to ask ourselves the question, “Why should we be concerned with one-to-one functions?”
Answer: Every one-to-one function has an inverse function!
Definition
Inverse Function
Let f be a one-to-one function with domain A and range B. Then f 1 is the inverse
function of f with domain B and range A. Furthermore, if f (a )  b then f 1 (b)  a .
According to the definition of an inverse function, the domain of f is exactly the same as the range of f 1 and
the range of f is the same as the domain of f 1 . The figure below illustrates that if the function f assigns a
number a to b then the inverse function will assign b back to a. In other words if the point  a, b  is an ordered
pair on the graph of f, then the point  b, a  must be on the graph of f 1 .
Domain of f
Range of f
Range of f
1
Domain of f
f
•a
•b
f 1
1
Do not confuse f
1
1
1
. The negative 1 in f
is NOT an exponent!
f ( x)
with
Inverse functions “undo” each other. For example, it can be shown that if f ( x)  x3 then the inverse of f is
f 1 ( x)  3 x . Note that f  2    2   8 and f 1 8  3 8  2 .
3
The function f takes the number 2 to 8 while f 1 takes 8 back to 2. Observe what happens if we look at the
composition of f and f 1 and the composition of f 1 and f at specified values:
xvalue
f
f



same as
x value
f 1  8  f f 1 (8)  f  2   8 and
xvalue
1
f
  2  f
1
 f (2)  
same as
x value
f 1 8  2
Because of the “undoing” nature of inverse functions, we get the following composition cancellation
equations:
Composition Cancellation Equations


f f 1 ( x)  x for all x in the domain of f 1 and
f 1  f ( x)   x for all x in the domain of f
These cancellation equations can be used to show whether two functions are inverses of each other or not. We
can see from our example above that if f ( x)  x3 and f 1 ( x)  3 x then


f f 1 ( x)  f
 x  x
3
3
3
 x and f 1  f ( x)   f 1  x3   3 x3  x
Objective 4: Sketching the Graphs of Inverse Functions
If f is a one-to-one function then we know that it must have an inverse function, f 1 . Given the graph of a
one-to-one function f , we can obtain the graph of f 1 by simply interchanging the coordinates of each ordered
pair that lies on the graph of f . In other words, for any point  a, b  on the graph of f , the point  b, a  must
lie on the graph of f 1 .
Notice below on the left that the points  a, b  and  b, a  are symmetric about the line y  x . Therefore, the
graph of f 1 is a reflection of the graph of f about the line y  x . The figure on the right shows the graph of
f ( x)  x3 and f 1 ( x)  3 x . You can see that if the functions have any points in common, they must lie along
the line y  x .
f ( x)  x3
The graph of
y  f 1 ( x)
The graph of f ( x )  x
and f
a one-to-one
function and
its inverse.
1
( x) 
3
3
x
f 1 ( x)  3 x
y  f ( x)
yx
yx
Objective 5: Finding the Inverse of a One-to-one Function
We know that if a point ( x, y ) is on the graph of a one-to-one function, then the point ( y , x ) is on the graph of its
inverse function. We can use this information to develop a process for finding the inverse of a function
algebraically simply by switching x and y in the original function to produce its inverse function.
To find the inverse of a one-to-one function, replace f ( x) with y, interchange the variables x and y then solve
for y. This is the function f 1 ( x) .
Inverse Function Summary
1.
f 1 exists if and only if the function f is one-to-one
2.
The domain of f is the same as the range of f 1 and the range of f
3.
is the same as the domain of f 1 .
To verify that two one-to-one functions f and g are inverses of each other we must use
the composition cancellation equations to show that f  g ( x)   g  f ( x)   x .
4.
The graph of f 1 is a reflection of the graph of f about the line y  x . That is, for any
5.
point  a, b  that lies on the graph of f, the point  b, a  must lie on the graph of f 1 .
To find the inverse of a one-to-one function, replace f ( x) with y, interchange the
variables x and y then solve for y. This is the function f 1 ( x) .