Section 3.7
Inverse Functions
Let f be a function with domain A. f is said to be one-to-one (has an inverse function) if no
two elements in A have the same image.
The inverse function reverses whatever the first function did.
The inverse of a function f is denoted by f
1
, read “f-inverse”.
Only one function below is 1-1, which one?
Domain
0
2
4
f
Range
-1
2
5
Domain
3
6
9
g
Range
-1
5
The Horizontal Line Test
Given the graph of a function, we can determine if that function has an inverse function by
applying the Horizontal Line Test.
A function f has an inverse function, f
more than one point.
1
, if there is no horizontal line that intersects the graph in
Example 1: Is the following graph the graph of a function that has an inverse function?
a.
b.
Section 3.7 – Inverse Functions
1
Example 2: Is the following function 1-1?
a. f ( x)  x  7
c. h( x) 
b. g ( x )  x  1
1
2
x 1
Domain and Range
The domain of f is the range of f
d. k ( x)  3 x 2  2
1
and the range of f is the domain of f
1
.
Example 3: Assume that the domain of f is all real numbers and that f is one-to-one. If
f (7) = 9, f (9) = -12, f (8) = 7, and f (-12) = 8, find:
a. f
1
(9)
b. f
1
(8)
c.
 f  f  (12)
d. f
1
f
1
(7) 
Section 3.7 – Inverse Functions
2
Property of Inverse Functions
Let f and g be two functions such that ( f  g )( x)  x for every x in the domain of g and
( g  f )( x)  x for every x in the domain of f then f and g are inverses of each other.
9
x  32 is used to convert
5
5
from x degrees Celsius to y degrees Fahrenheit. The formula g ( x)  ( x  32) is used to convert
9
from x degrees Fahrenheit to y degrees Celsius.
An example of real life inverse function is: The formula f ( x) 
Example 4: Verify the property of inverse functions for the following formulas.
9
5
f ( x)  x  32 and g ( x)  ( x  32)
5
9
So, we need to check to see if: ( f  g )( x)  x AND ( g  f )( x)  x
( f  g )( x)
( g  f )( x)
How to find the equation of the inverse function of a one-to-one function:
1.
2.
3.
4.
5.
Replace f(x) by y.
Exchange x and y.
Solve for y.
Replace y by f 1 ( x).
Verify. (i.e. check that ( f  g )( x)  x AND ( g  f )( x)  x )
Section 3.7 – Inverse Functions
3
Example 5: Find an equation for, f
function.
1
( x) , the inverse function of the following one-to-one
a. f ( x)  7 x  4
c. g ( x) 
1
x 1
Section 3.7 – Inverse Functions
4
d. g ( x) 
4
x2
e. f ( x) 
 3x  1
2x  5
Section 3.7 – Inverse Functions
5
Example 6: Given
if possible
4 for x
0, find its inverse
Difference Quotient:
,
Example 7: Find the difference quotient:
Section 3.7 – Inverse Functions
5
0
4
3
6