Section 3.1 One-To-One and Inverse Functions
§1 One-To-One
A one-to-one function is so named because of a special property it has. Recall the definition of a
function. A function is a relation in which for every x-value there is exactly one y-value. Basically, this
means that there can no repeating x-values. A one-to-one function is a special type of function in which
there can be no repeating x- and y- values. You can tell if a function is one-to-one a variety of ways. If
the elements of the domain and range are given, you have to make sure that each x-value corresponds
to exactly one y-value. Graphically, the function must pass the horizontal line test. This is similar to the
definition of the vertical line test. Remember, a function always passes the vertical line test. So, a
function is one-to-one if its graph also passes the horizontal line test. This means that there will be no
two same x-values AND no two same y-values.
§2 Inverse Functions
Why is this important? It’s important because a function can have an inverse only if it is one-to-one. The
inverse of a function is denoted by f −1 ( x). The relationship between a function and its inverse is as
−1
−1
follows: the domain of f ( x) is the range of f ( x ) , and the range of f ( x) is the domain of f ( x ) .
So lets think about this carefully. What happens when we take the composite of a function and its
inverse? Look at the diagram below.
So it turns out that the composite of a function and its inverse is simply x! This is how to show
algebraically that two functions are inverses of each other. If we take the composite function and the
result is x, then we can safely conclude that the functions are inverses of each other. For example, take
the two functions f ( x=
) 2 x + 3 and g ( x) =
x −3
. We can make a table of values for each.
2
x
-2
-1
0
1
2
f(x)
-1
1
3
5
7
x
-1
1
3
5
7
g(x)
-2
-1
0
1
2
Do you see any relationship here? The domain of f ( x) is the range of g ( x) , and the range of f ( x) is
the domain of g ( x) ! Hence we can conclude that f and g are inverses of each other. How can we
show this algebraically? Take the composition function. You should do it both ways. For example, lets try
x −3
 + 3 = x − 3 + 3 = x , Similarly, if you do
 2 
( f  g )( x ) . You end up with ( f  g )( x ) = 2 
x)
( g  f )( x ) , you end up with ( g  f )( =
3
( 2 x + 3) −=
2
2x
= x . Hence they are inverses of each
2
other.
PRACTICE
1) Given that f ( x) is one-to-one and f (4) = −2, what is f −1 ( −2 ) ?
(
2) Given that f ( x) is one-to-one, what is f −1  f
) ( −3) ? What is ( f  f ) ( −3) ?
−1
§3 The Graph Of Inverse Functions
There is also a way to tell graphically if two functions are inverses of each other. They are mirror images
of each other across the line y=x. Note that the domain and range of inverse functions are switched. This
simply means that the x and y values are switched! So, the graphs should be mirror images across the
line y=x. Try graphing the two lines from the previous example!
PRACTICE
3) Use the graph below to sketch the graph of the inverse function.
§4 How To Find the Inverse Of A Function
If you are given a function and are asked to find its inverse, there is very simple method to do so. First,
let y = f ( x ) . This way we get the expression in terms of x and y. Next, switch the values of x and y. So
every x becomes a y, and every y becomes an x. Finally, solve for y. The resulting expression should be
the inverse function!
You should always check your result, that is, take the composite function to see if the result is x.
For example, say we want to find the inverse function of f ( x=
) 4 x + 3. We should first let y = f ( x ) ,
so we get =
y 4 x + 3. Then we switch the x’s and the y’s to end up with =
x 4 y + 3. Now we solve for y
to end up with y =
f −1 ( x ) =
x −3
. This should represent the inverse function. So if f ( x=
) 4 x + 3, then
4
x −3
. You can verify that these are inverses of each other either by graphing (not
4
recommended) or by taking the composite function and getting x (recommended).
PRACTICE
4) Find the inverse of f ( x)= 3 − 2 x and verify that they are inverses of each other by showing that
( f  f )( x) = x .
−1
5) Find the inverse of f ( x) =
1
and verify that they are inverses of each other by showing that
x−2
( f  f ) ( x ) = x . Also find the domain of each.
−1
6) Find the inverse of f ( x) =
2x + 3
and find the domain of each.
x−2
7) Find the inverse of f ( x) = x 2 − 1, x > 0 . Sketch the graph of each, and find the domain and range of
each.