CHAPTER 5 SECTION 7
Chapter 5: One-to-One and Inverse Functions
Learning Objectives:
A. Identify one-to-one functions
B. Explore inverse functions using ordered pairs
C. Find inverse functions using an algebraic method
D. Graph a function and its inverse
E. Solve applications of inverse functions
A. Identify One-to-One Functions
One-to-One Functions
A function f is one-to-one if every element in the range corresponds to only one element of the
domain. In symbols, if f  x1   f  x2  then x1  x2 , or if x1  x2 , then f  x1   f  x2  .
Horizontal Line Test
If every horizontal line intersects the graph of a function in at most one point, the function is
one-to-one.
SECTION 7 CHAPTER 5
Example: Determine whether each graph or relation shown depicts a function. If so, determine
whether the function is one-to-one.
a.
b.
c.
B. Inverse Functions and Ordered Pairs
Definition: An inverse function for f is one that takes the result of elements of the range, and
returns the original domain element.
Inverse Functions
If f is a one-to-one function with ordered pairs  a, b  .
1. f 1 is a one-to-one function with ordered pairs  b, a  .
2. The range of f will be the domain of f 1 .
3. The domain of f will be the range of f 1 .
Caution: The notation f 1  x  is simply a way of denoting an inverse function and has nothing
to do with exponential properties. In particular, f 1  x  does not mean
1
.
f ( x)
CHAPTER 5 SECTION 7
Example: Find the inverse function of each one-to-one function given:
a.
 2, 7 ,  1,0 , 0,1 , 1,2 ,  2,9
1
b. f  x   x  3
4
C. Find Inverse Functions using an Algebraic Method
Finding an Inverse Function
1. Replace f  x  with y.
2. Interchange x and y.
3. Solve the new equation for y.
4. The result gives the inverse function: substitute f 1  x  for y.
Example: State the domain and range of f  x   1 
the inverse function, and state its domain and range.
2
, then use the algebraic method to find
x 3
SECTION 7 CHAPTER 5
In cases where a given function is not one-to-one, we can sometimes restrict the domain to create
a function that is, and then determine the inverse.
Example: Given f  x    x  1 , restrict the domain to create a one-to-one function, then find
4
f 1  x  . State the domain and range of both resulting functions.
Verifying Inverse Functions
If f is a one-to-one function, then the function f 1 exists
and satisfies
and
.
Example: Use the algebraic method to find the inverse function for f ( x) 
Then verify the inverse you found is correct.
2x  5
.
7
CHAPTER 5 SECTION 7
D. The Graph of a Function and Its Inverse
Graphing a function and its inverse on the same axes reveals the graphs are reflections across the
line y = x.
Example: Given the graph shown below, draw a graph of the inverse function.
Functions and Inverse Functions
1. If the graph of a function passes the horizontal line test, the function is one-to-one.
2. If a function f is one-to-one, the function f 1 exists.
3. The domain of f is the range of f 1 , and the range of f is the domain of f 1 .
4. For a function f and its inverse f 1 ,  f o f 1   x   x and
f
1
o f  x  x .
5. The graphs of f and f 1 are symmetric about the line y = x.
E. Applications of Inverse Functions
Example: The volume of a can of chicken soup with a fixed height of 6 in. is given by
V ( x)  6 x2 , where V(x) represents the volume in cubic inches for a can with a radius of x inches.
a. Find the amount of soup in can with a radius of 2 in. that is filled to the brim.
b. Find V 1  x  , and discuss what the input and output variables represent.