Every human being has a blood type, and every human being has a DNA sequence. These are examples of functions, where a person is the input and the output is blood type or DNA sequence. These relationships are classified as functions because each person can have one and only one blood type or DNA strand. The difference between these functions is that many people have the same blood type, but DNA is unique to each individual. Can we map backwards? For instance, if you know the blood type, do you know specifically which person it came from? No, but, if you know the DNA sequence, you know exactly to which person it corresponds. When a function has a one-to-one correspondence, like the DNA example, then mapping backwards is possible. The map back is called the inverse function.
Determine Whether a Function Is One-to-One
In Section
3.1, we defined a function as a relationship that maps an input (contained in the domain) to exactly one output (found in the range). Algebraically, each value for
can correspond to only a single value for
. Recall the square, identity, absolute value, and reciprocal functions from our library of functions in Section
3.3.
All of the graphs of these functions satisfy the vertical line test. Although the square function and the absolute value function map each value of
to exactly one value for
, these two functions map two values of
to the same value for
. For example,
and
lie on both graphs. The identity and reciprocal functions, on the other hand, map each
to a single value for
, and no two
map to the same
. These two functions are examples of
one-to-one functions.
| DEFINITION� |
One-to-One Function |
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A function  is one-to-one if no two elements in the domain correspond to the same element in the range; that is,
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In other words, it is one-to-one if no two inputs map to the same output.
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EXAMPLE�1�
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Determining Whether a Function Defined as a Set of Points Is a One-to-One Function |
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For each of the three relations, determine whether the relation is a function. If it is a function, determine whether it is a one-to-one function.
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Just as there is a graphical test for functions, the vertical line test, there is a graphical test for one-to-one functions, the
horizontal line test. Note that a horizontal line can be drawn on the square and absolute value functions so that it intersects the graph of each function at two points. The identity and reciprocal functions, however, will intersect a horizontal line in at most only one point. This leads us to the horizontal line test for one-to-one functions.
| DEFINITION� |
Horizontal Line Test |
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If every horizontal line intersects the graph of a function in at most one point, then the function is classified as a one-to-one function. |
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 �EXAMPLE�2�
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Using the Horizontal Line Test to Determine Whether a Function Is One-to-One |
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For each of the three relations, determine whether the relation is a function. If it is a function, determine whether it is a one-to-one function. Assume that is the independent variable and is the dependent variable.
Solution
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(fails vertical line test) |
(passes vertical line test but fails horizontal line test) |
(passes both horizontal and vertical line tests) |
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Determine whether each of the functions is a one-to-one function.
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Another way of writing the definition of a one-to-one function is:
In the Your Turn following Example
2, we found (using the horizontal line test) that
is a one-to-one function, but that
is not a one-to-one function. We can also use this alternative definition to determine algebraically whether a function is one-to-one.
Words |
Math |
State the function. |
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Let there be two real numbers,  and  , such that  . |
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Subtract 2 from both sides of the equation. |
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is a one-to-one function. |
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Words |
Math |
State the function. |
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Let there be two real numbers, and , such that . |
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Subtract 1 from both sides of the equation. |
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Solve for . |
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 is not a one-to-one function. |
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�EXAMPLE�3�
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Determining Algebraically Whether a Function Is One-to-One |
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Determine algebraically whether the following functions are one-to-one:
Solution
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(a)�� |
Find  and  . |
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Let . |
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Add 2 to both sides of the equation. |
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Divide both sides of the equation by 5. |
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Take the cube root of both sides of the equation. |
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Simplify. |
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(b)�� |
Find and . |
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Let . |
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Solve the absolute value equation. |
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Inverse Functions
If a function is one-to-one, then the function maps each
to exactly one
, and no two
map to the same
. This implies that there is a one-to-one correspondence between the inputs (domain) and outputs (range) of a one-to-one function
. In the special case of a one-to-one function, it would be possible to map from the output (range of
) back to the input (domain of
), and this mapping would also be a function. The function that maps the output back to the input of a function
is called the
inverse function and is denoted
.
A one-to-one function
maps every
in the domain to a unique and distinct corresponding
in the range. Therefore, the inverse function
maps every
back to a unique and distinct
.
The function notations
and
indicate that if the point
satisfies the function, then the point
satisfies the inverse function.
For example, let the function
.
The inverse function undoes whatever the function does. For example, if
, then the function
maps any value
in the domain to a value
in the range. If we want to map backwards or undo the
, we develop a function called the inverse function that takes
as input and maps back to
as output. The inverse function is
. Note that if we input
into the inverse function, the output is
.
| DEFINITION� |
Inverse Function |
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If and  denote two one-to-one functions such that
then is the inverse of the function . The function  is denoted by (read “f-inverse”). |
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is used to denote the inverse of
. The
is not used as an exponent and, therefore, does not represent the reciprocal of
:
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Two properties hold true relating one-to-one functions to their inverses: (1) the range of the function is the domain of the inverse, and the range of the inverse is the domain of the function, and (2) the composite function that results with a function and its inverse (and vice versa) is the identity function
.
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EXAMPLE�4�
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Verifying Inverse Functions |
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Verify that  is the inverse of  .
Solution
Note the relationship between the domain and range of and .
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�EXAMPLE�5�
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Verifying Inverse Functions with Domain Restrictions |
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Verify that  , for  , is the inverse of  .
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Graphical Interpretation of Inverse Functions
In Example
4, we showed that
is the inverse of
. Let's now investigate the graphs that correspond to the function
and its inverse
.
Note that the point
lies on the function and the point
lies on the inverse. In fact, every point
that lies on the function corresponds to a point
that lies on the inverse.
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If the point  is on the function, then the point  is on the inverse. Notice the interchanging of the  and  . |
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Draw the line
on the graph. In general, the point
on the inverse
is the reflection (about
) of the point
on the function
.
In general, if the point
is on the graph of a function, then the point
is on the graph of its inverse.
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�EXAMPLE�6�
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Graphing the Inverse Function |
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Given the graph of the function , plot the graph of its inverse . |
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Solution
Because the points ,  ,  and  lie on the graph of  , then the points ,  ,  , and  lie on the graph of  . |
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Given the graph of a function , plot the inverse function. |
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We have developed the definition of an inverse function and described properties of inverses. At this point, you should be able to determine whether two functions are inverses of one another. Let's turn our attention to another problem: How do you find the inverse of a function?
Finding the Inverse Function
If the point
lies on the graph of a function, then the point
lies on the graph of the inverse function. The symmetry about the line
tells us that the roles of
and
interchange. Therefore, if we start with every point
that lies on the graph of a function, then every point
lies on the graph of its inverse. Algebraically, this corresponds to interchanging
and
. Finding the inverse of a finite set of ordered pairs is easy: simply interchange the
and
. Earlier, we found that if
, then
. But how do we find the inverse of a function defined by an equation?
Recall the mapping relationship if
is a one-to-one function. This relationship implies that
and
.
Let's use these two identities to find the inverse. Now consider the function defined by
. To find
, we let
, which yields
. Solve for the variable
.
Recall that
, so we have found the inverse to be
. It is customary to write the independent variable as
, so we write the inverse as
. Now that we have found the inverse, let's confirm that the properties
and
hold.
FINDING THE INVERSE OF A FUNCTION |
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Let be a one-to-one function. Then the following procedure can be used to find the inverse function if the inverse exists.
The same result is found if we first interchange and and then solve for in terms of .
Note the following:
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EXAMPLE�7�
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The Inverse of a Square Root Function |
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Find the inverse of the function  . State the domain and range of both and .
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 �Technology Tip� |
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Using a graphing utility, plot  ,  , and  .
Note that the function and its inverse are symmetric about the line .
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Solution
is a one-to-one function because it passes the horizontal line test. |
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(State the domain and range of both and .)
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Domain: 
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Range: 
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Domain:
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Range:
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The inverse of is  .
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Had we ignored the domain and range in Example 7, we would have found the inverse function to be the square function  , which is not a one-to-one function. It is only when we restrict the domain of the square function that we get a one-to-one function. |
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Check.
for all in the domain of . |
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for all in the domain of . |
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Note that the function and its inverse  for are symmetric about the line . |
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Find the inverse of the given function. State the domain and range of the inverse function.
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EXAMPLE�8�
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A Function That Does Not Have an Inverse Function |
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Find the inverse of the function  if it exists.
Solution
The function fails the horizontal line test and therefore is not a one-to-one function. Because is not a one-to-one function, its inverse function does not exist. |
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�EXAMPLE�9�
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Finding the Inverse Function |
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The function  , is a one-to-one function. Find its inverse.
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| �Technology Tip� |
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The graphs of  ,  , and  ,  , are shown.
Note that the function and its inverse are symmetric about the line .
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Solution
Step 1 |
Let  . |
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Step 2 |
Interchange and . |
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Step 3 |
Solve for . |
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Multiply the equation by  . |
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Eliminate the parentheses. |
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Subtract  from both sides. |
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Divide the equation by . |
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Step 4 |
Let  . |
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Note any domain restrictions on  . |
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The range of the function is equal to the domain of its inverse function. |
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The inverse of the function , is  .
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The function  , is a one-to-one function. Find its inverse.
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Note in Example
9 that the domain of
is
and the domain of
is
. Therefore, we know that the range of
is
, and the range of
is
.
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EXAMPLE�10�
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Finding the Inverse of a Piecewise-Defined Function |
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The function  , is a one-to-one function. Find its inverse.
Solution
From the graph of we can make a table with corresponding domain and range values.
Domain of
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Range of
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From this information we can also list domain and range values for .
Domain of  Range of
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Range of Domain of
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 on  ; find on .
Step 1 |
Let . |
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Step 2 |
Solve for in terms of . |
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Step 3 |
Solve for . |
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Step 4 |
Let . |
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 on  ; find on .
Step 1 |
Let . |
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Step 2 |
Solve for in terms of . |
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Step 3 |
Solve for . |
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Step 4 |
Let . |
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Step 5 |
The range of is
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Combining the two pieces yields a piecewise-defined inverse function.
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In the Your Turn following Example 2, we found (using the horizontal line test) that 
, Domain: 
, Domain: 
are repeated.
range of 
range of 
.
.
all map to 1 in the range, for instance.
are on the graph.
lie on the graph.
.
.
.
for 
.
for 
.
for 
.
for 
.
for 
.
for 
.
for 
.
for 
.
for 
.
is merely a reflection of this graph over the origin.
. Determine the inverse function 
; this now represents degrees Fahrenheit being turned into degrees Celsius.
for 
. The inverse function represents the conversion from degrees Fahrenheit to degrees Celsius.
. Determine the inverse function 
. What does the inverse function represent?
per boat for the first 10 boats and 
for each additional boat. Find the cost function 
as a function of the number of boats the club enters 
. The cost function is
, we calculate the inverse of each piece separately:
: Solve 
for 
. So, 
, for 
.
: Solve 
for 
. So, 
, for 
.
per minute for the first 10 minutes of a long-distance phone call and 
per minute every minute after that. Find the cost function 
per hour and the weekly number of hours worked per week 
of his earnings for taxes and Social Security, write a function 
that expresses the student's take-home pay each week. Find the inverse function 
. What does the inverse function tell you?
tells you how many hours the student will have to work to bring home 
. Then, the take home pay is given by
, solve 
for 
. So, 
.
per hour for the first 40 hours per week and time and a half for overtime. Write a piecewise-defined function that represents your weekly earnings 
where 
represents that patient's temperature in degrees Fahrenheit and 
represents the time of day in hours measured from 12:00 a.m. (midnight).
.
. For the range, note that 
is always increasing (being a translate of 
) and so, its minimum is 
and its maximum is 
. So, the range is the approximate interval 
.
.
is the range of 
?
a one-to-one function?
and 
lie on the graph of the function.
and 
lie on the graph of the inverse.
and 
.
.
. Then, the calculation will be valid.
, find the inverse function 
.
.
.
implies that the horizontal line test is violated.
.
. Then, 
also.
is the 
of a one-to-one function 
of the inverse 
since the 
is the 
. The portion in Quadrant I is given by
. The domain and range of both are 
.
.
a one-to-one function?
(that is, while the graph of 
, then they would have been.
