SECTION 4.2
ONE-TO-ONE FUNCTIONS
INVERSE FUNCTIONS
INVERSE FUNCTIONS
There are some functions which
we almost intuitively know as
inverses of each other:
Cubing a number, taking the cube
root of a number.
Adding a value to a number,
subtracting that value from the
number.
INVERSE FUNCTIONS
These are inverses of each other
because one undoes the other.
Can a more complicated function
have an inverse?
x2
What
does
f(x)

do
tothe
variabl
x?
6
Adds two and divides by 6.
x2
f(x) 
6
What must the inverse of f(x) do to
its variable, x?
Multiply by 6 and subtract two.
x

2
f(x)

g(x)

6x
2
6
x
f(x)
x g(x)
Symbolically?
0
1/3
0
-2
Numerically?
1
1/2
1
4
Graphically?
4
1
-1
-8
-2
0
1/3
0
-8
-1
1/2
1
ANOTHER IMPORTANT
OBSERVATION
 4
f(x)
1
g(f(4)) = 4
1
In fact, the same thing
happens for any x-value.
g(x)
g(f(x)) = x
4
EXAMPLE:
f(x) 
3
x 6
Find the inverse of f(x) which we
refer to as f -1(x).
Then, check algebraically to
ensure that f(f -1(x)) = x.
f -1(x) = (x - 6) 3
Check that f (f -1(x)) = x
f (f (x)) 
-1
3
3
(x - 6)
=
x - 6
=
x
+
 6
6
RECALL:
DEFINITION
OF
FUNCTION
A set of ordered pairs
in which no two
ordered pairs have the
same first coordinate.
In other words:
FOR EVERY X, THERE IS ONLY ONE Y.
Consider the function f(x) = x 2
x f(x)
0
1
2
0
1
4
-1
-2
1
4
x
If this function had
an inverse, the
ordered pairs
would have to be
reversed.
y
0 0
1 1
4 2
1 -1
4 -2
DEFINITION OF
ONE-TO-ONE
FUNCTION
A set of ordered pairs in
which no two ordered
pairs have the same first
coordinate and no two
ordered pairs have the
same second coordinate.
In other words:
FOR EVERY X, THERE IS ONLY ONE Y.
FOR EVERY Y, THERE IS ONLY ONE X.
f(x) = x 2 is not a one-to-one function.
Thus, it has no inverse.
Recall a graphical test which
enables us to determine whether a
relation is a function.
“VERTICAL LINE TEST”
What kind of graphical test
would help us to determine
whether a function was one-toone?
“HORIZONTAL LINE TEST”
FINDING A FORMULA
FOR f -1(x)
Example:
5
f(x)

x-2
First of all, check to see if it is
one-to-one. Graph it!
5
f(x)

x-2
Now, find the formula:
5
y 
x -2
x 
5
y -2
x(y - 2)  5
x y - 2x = 5
x y = 2x + 5
y  2xx 5
EXAMPLE:
Find inverses for the two functions
below and graph them to see
symmetry.
2
x 2
g(x)

f(x) = 3x - 4
5
x

4
1
f (x)

3
g -1(x) does not exist.
RESTRICTING DOMAINS
Example: f(x) = x 2 - 4x
f(x) = x 2 - 4x + 4
-4
f(x) = (x - 2) 2 - 4
Vertex: (2, - 4) Domain: (x  2)
1
f
(
x
)

x

4

2
CONCLUSION OF SECTION 4.2