Inverse Functions

Definition: A function is called a one-to- one function if and only if different x values in the domain (different
inputs) have different y values in the range (different outputs). This means that if are both in the domain,
then

Examples:

1. Which of the fol lowing is a function and if a function, is it one-to one?

2.   is not one to one, because:

3.

3. Let the functions be given by the graphs below . Which one is one-to- one?
 

Horizontal Line Test : A function is one-to-one if every horizontal line intersects the graph of the function in at most one
point.

Examples:

Theorem:

1. All increasing and decreasing functions are one-to-one.

2. All linear functions are one-to-one.

Examples:

The Inverse Function: Let be a function. The converse of this function is obtained by writing x in terms of
y , that is to say

Examples: Find the converse of the following functions

Note: In examples 3 and 5 above the converses are NOT functions. In example 6, the converse of one function is a
function, whereas the converse of the other one is not.

Question: When would the converse of a function be a function as well?

Answer: If the function is one-to-one then the converse of it would also be a function. In this case the converse
is called the inverse function. We use the symbol for the inverse function.
Finding the Inverse of a Function: If a function is one-to-one, then to find ,

Step 1. Write the function as

Step 2. Solve for x in terms of y.

Step 3. Inter change the role of x and y. The result id the inverse function.

Example:

Theorem:

a) A functions and its inverse exchange domain and range. This means that if is the
domain and is the range of , then is the domain and is the range of

b) and

Graph of the Inverse Functions: The graph of and are symmetric relative to the line .

Examples: