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Hawkes Learning Systems:
College Algebra
Section 4.6: Inverses of Functions
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Objectives
o Inverses of relations.
o Inverse functions and the horizontal line test.
o Finding inverse function formulas.
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Inverses of Relations
1
Let R be a relation. The inverse of R, denoted R , is
the set
R 1   b, a  |  a, b   R.
In other words, the inverse of a relation is the set of
ordered pairs of that relation with the first and second
coordinates of each exchanged.
Consider the relation R  1, 7  ,  2, 0 .
The inverse of R is R 1   7,1 ,  0, 2 .
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Example: Inverses of Relations
Determine the inverse of the relation. Then graph the relation and its
inverse and determine the domain and range of both.
R   4, 1 ,  3, 2  ,  0,5  Dom  R    4, 3,0 , Ran  R    1, 2,5
R 1   1,4  ,  2, 3 ,  5,0  Dom  R 1   1, 2,5 , Ran  R 1   4, 3,0
In the graph to the left, R is in blue and
its inverse is in red. R consists of three
ordered pairs and its inverse is simply
these ordered pairs with the coordinates
exchanged. Note: the domain of the
relation is the range of its inverse and
vice versa.
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Example: Inverses of Functions
Determine the inverse of the relation. Then graph the relation and its
inverse and determine the domain and range of both.
y  x2
Dom  R    ,  , Ran  R    0,  
2
R   x, y  | y  x 
1
1
Dom
R

0,

,
Ran
R
  ,  


1
2
R   x, y  | x  y 
 
 
In this problem, R is described by the
given equation in x and y. The inverse
relation is the set of ordered pairs in R
with the coordinates exchanged, so we
can describe the inverse relation by just
exchanging x and y in the equation, as
shown at left.
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Inverses of Relations
Note:
A relation and its inverse are mirror images of one
another (reflections) with respect to the line y  x.
Even if a relation is a function, its inverse is not
necessarily a function.
Verify these two facts against the previous examples.
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Inverse Functions and the Horizontal Line Test
In practice, we will only be concerned with whether or
not the inverse of a function f , denoted f 1, is itself a
function. Note that f 1 has already been defined: f 1
stands for the inverse of f , where we are making use of
the fact that a function is also a relation.
Caution!
1
1
f does not stand for , when f is a function!
f
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Inverse Functions and the Horizontal Line Test
The Horizontal Line Test
Let f be a function. We say that the graph of f passes
the horizontal line test if every horizontal line in the
plane intersects the graph no more than once.
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Inverse Functions and the Horizontal Line Test
One-to-One Functions
A function f is one-to-one if for every pair of distinct
elements x1 and x2 in the domain of f, we have
f  x1   f  x2 . This means that every element of the
range of f is paired with exactly one element of the
domain of f.
Note: If a function is one-to-one, it will pass the
horizontal line test.
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Inverse Functions and the Horizontal Line Test
Tip!
1
The inverse f of a function f is also a function if and
only if f is one-to-one.
In Example 1 you have R   4, 1 ,  3,2  ,  0,5  with
Dom  R   4, 3,0 , and Ran  R   1,2,5, then R is oneto-one, so its inverse must be a function. But, if you notice in
2
Example 2, the graph of y  x is a parabola and obviously
fails the horizontal line test. Thus, R is not one-to-one so its
inverse is not a function.
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Example: Inverse Functions and the
Horizontal Line Test
f  x  x  4
Does f  x  have an inverse function?
No.
We can see by graphing this
function that it does not pass the
horizontal line test, as it is an
open “V” shape. By this, we know
that f is not one-to-one and can
conclude that it does not have an
inverse function.
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Example: Inverse Functions and the
Horizontal Line Test
g  x    x  5
3
Does g  x  have an inverse function?
Yes.
We know that the standard cube
shape passes the horizontal line
test, so g has an inverse function.
We can also convince ourselves
of this fact algebraically:
x1  x2  x1  5  x2  5
3
3
  x1  5    x2  5 
 g  x1   g  x2  .
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Finding Inverse Function Formulas
To Find a Formula for f 1
Let f be a one-to-one function, and assume that f is
defined by a formula. To find a formula for f 1, perform
the following steps:
1. Replace f  x  in the definition of f with the variable y.
The result is an equation in x and y that is solved for y.
2. Interchange x and y in the equation.
3. Solve the new equation for y.
4. Replace the y in the resulting equation with f 1  x  .
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Finding the Inverse Function Formulas
For example, to find the inverse function formula for
the function f  x   5x  1 1. Replace f  x  with y.
y  5 x  1 2. Interchange x and y.
x  5 y  1 3. Solve for y.
5 y  1  x
5 y  x 1
x 1
4. Replace y with f  x  .
y
5
x 1
1
f  x 
5
1
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Finding the Inverse Function Formulas
If you noticed, finding the inverse function formula for
f  x   5x  1 with the defined algorithm was a
relatively long process for how simple the function is.
Notice that f follows a sequence of actions: first it
multiplies x by 5, then it adds 1. To obtain the inverse
of f we could “undo” this process by negating these
actions in the reverse order. So, we would first subtract
1 and then divide by 5:
x 1
1
f  x 
.
5
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Example: Finding Inverse Function Formulas
Find the inverse of the following function.
f  x    x  3  1
5
1
5
f 1  x    x  1  3
We can always find the inverse function formula by
using the algorithm we defined. However, this function
is simple enough to easily undo the actions of f in
reverse order. The application of the algorithm would
be:
5
5
f  x    x  3  1
 y  3  x  1
y   x  3  1
y  3  5 x 1
5
y  5 x 1  3
x   y  3  1
5
  y  3  1  x
5
f
1
 x    x  1
1
5
As you might notice, for this particular function,
“undoing” the actions of f in reverse order is much
simpler than applying the algorithm.
3
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Example: Finding Inverse Function Formulas
Find the inverse of the following function.
x  1 We will apply the
algorithm to find
g  x 
3 x  2 the inverse.
3xy  y  2 x  1
x  1 Substitute y for
y
y  3x  1  2 x  1
g  x .
3x  2
y  1 Interchange x
2 x  1
x
y
and
y.
3y  2
3x  1
3 y  2 x  y  1
3xy  2 x  y  1
Solve for y.
2 x  1
g  x 
3x  1
1
Substitute
g 1  x 
for y.
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Finding Inverse Function Formulas
The graph of a relation and its inverse are mirror
images of one another with respect to the line y  x.
This is still true of functions and their inverses.
3
f ( x)   x  1  2, its inverse
Consider the function
1
f 1 ( x)   x  2  3  1 and the graph of both:
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Finding Inverse Function Formulas
Note: the key characteristic of the inverse of a function
is that it “undoes” the function. This means that if a
function and its inverse are composed together, in
either order, the resulting function has no effect on any
allowable input; specifically:
f  f 1  x    x for all x  Dom  f 1  , and
f 1  f  x    x for all x Dom  f  .
For reference, observe the graph on the previous slide.
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Finding Inverse Function Formulas
Consider the function f  x 
x  1


2
and its inverse
3
f 1  x   3x  1 (you should verify that this is the inverse of f ).
Below are both of the compositions of f and f 1 :
ff
1
 x  
f






3x

3
x

3x  1
 
3x  1  1
3x
3

3
2
2
2


x

1


1
f  f  x   f 

 3 


  x  12 
 3
 1
 3 


 x  1  1
  x  1  1

x
2