Sullivan PreCalculus
Section 4.2
Inverse Functions
Objectives of this Section
• Determine the Inverse of a Function
• Obtain the Graph of the Inverse Function From the Graph
of the Function
• Find the Inverse Function f -1
A function f is said to be one-to-one if, for
any choice of numbers x1 and x2, x1  x2, in
the domain of f, then f (x1)  f (x2).
Which of the following are one - to - one
functions?
{(1, 1), (2, 4), (3, 9), (4, 16)}
one-to-one
{(-2, 4), (-1, 1), (0, 0), (1, 1)}
not one-to-one
Theorem Horizontal Line Test
If horizontal lines intersect the graph of a function
f in at most one point, then f is one-to-one.
Use the graph to determine whether the following
function is one - to - one.
y
(0,0)
f ( x)  2 x 2
x
Use the graph to determine whether the following
function is one - to - one.
y
1
f (x) 
x
(1,1)
x
(-1,-1)
Theorem: A function that is increasing on an interval I is
one-to-one on I.
Theorem: A function that is decreasing on an interval I
is one-to-one on I.
2
f
(
x
)

x
Example: The function
in increasing on
the interval to the right of zero. Therefore, it is one-toone on that interval. Verify this statement with the
horizontal line test.
Let f denote a one-to-one function y = f (x).
The inverse of f, denoted f -1, is a function
such that f -1(f (x)) = x for every x in the
domain f and f (f -1(x)) = x for every x in the
domain of f -1.
In other words, the function f maps each x
in its domain to a unique y in its range.
The inverse function f -1 maps each y in
the range back to the x in the domain.
Domain of f
Range of f
f
f 1
Range of f
1
Domain of f
Domain of f  Range of f 1
Range of f  Domain of f
1
1
The graph of a function f and the graph of its
inverse are symmetric with respect to the line y=x.
6
f
4
f -1
2
2
y=x
0
2
2
4
6
Finding the inverse of a function.
Since the domain of f is the range of f -1 and
visa versa, we find the inverse of f by
interchanging x and y.
5
Example: Find the inverse of f ( x ) 
,x  3
x3
5
y
x3
5
x
y3
5
x
y3
xy  3x  5
3x  5
y
x
xy  3 x  5
3x  5
f ( x) 
x
1
To determine if two given functions are
inverses, we use the definition of inverse
functions. If two functions are inverses, then
f -1(f (x)) = x and f (f -1(x)) = x.
3x  5
5
Verify f ( x ) 
and g ( x ) 
x
x3
are inverses.
5 

f ( g ( x ))  f 

 x  3
5 
5 


3
 5
  5 3
 x  3
 x  3
x3



5
5
x3
x3
x3
5 

3
 5
 x  3
x  3 15  5 x  3 5x





x
5
x3
5
5
x3
5
3x  5
5
x

g ( f ( x ))  g 
 


 x  3x  5
3x  5
x
3
3
x
x
5x
5
x

 x
3x  5  3x
5