Avon High School
Section: 2.7
ACE COLLEGE ALGEBRA II - NOTES
Inverse Functions
Mr. Record: Room ALC-129
Day 1 of 1
Inverse Functions
Definition of the Inverse of a Function
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 .
Example 1
The function g is the inverse of the function f
and is denoted by f 1 (read “f inverse”).
Thus, f  f 1 ( x)   x and f 1  f ( x)   x. The
domain of f is equal to the range of f 1 , and
vice versa.
Verifying Inverse Functions
Show that each function is the inverse of the other.
x7
f ( x)  4 x  7 and g ( x) 
4
Finding the Inverse of a Function
Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows:
1. Replace f ( x) with y in the equation for f ( x) .
2. Interchange x and y.
3. Solve for y. If this equation does not define y as a function of x, the function f does not
have an inverse function and this procedure ends. If this equation does define y as a
function of x, the function y has an inverse.
4. If f has an inverse function, replace y in step 3 by f 1 ( x) . We can verify our result by
showing that f  f 1 ( x)   x and f 1  f ( x)   x.
Example 2
Finding the Inverse of a Function
Find the inverse of each of the following functions:
a. f ( x)  2 x  7
b. f ( x)  4 x3  1
c. f ( x) 
3
1
x
Horizontal Line Test and One-to-One Functions
Investigation: Proceed in finding the inverse of the function f ( x)  x 2 .
Make a list of ordered pairs with x values such that x | x  2, 1,1, 2 . What do these ordered pairs
say about the result of your “inverse?”
The Horizontal Line Test For Inverse Functions
A function f has an inverse that is a function f 1 , if there is no horizontal line that intersects the
graph of the function f at more than one point.
Note: A graph that passes the horizontal line test is considered “one-to-one” and therefore will
have an inverse.
Example 3
Applying the Horizontal Line Test
Which of the following represent a function that has an inverse function?
a.
b.
c.
Graphs of f and f -1
There is an important relationship between the graph of a one-to-one function f and its inverse, f -1
Since the definition of an inverse suggests that if (a, b) is a point on f, then (b, a) is a point on f -1.
This will result in an interesting type of symmetry.
Example 4
Graphing the Inverse Function
Use the graph of f given to the right, to sketch its inverse f -1 on
the same coordinate plane.
Example 5
Finding the Inverse of a Domain-Restricted Function
Find the inverse of f ( x)  x 2  1 if x  0 . Graph
f and f 1 on the same coordinate plane to the right.