Section 4.1 - Inverse Functions

Section Objectives

  1. Identify one-to-one functions.
  2. Determine whether two functions are inverses.
  3. Find the inverse of a function.



One-to-One Functions

A function is one-to-one if different input values always produce different output values. In other words, a function is one-to-one if no -values are reused.



Examples





A function is one-to-one if and only if its graph passes the horizontal line test. (Remember that the graph of any function must pass the vertical line test.)









Inverse Functions

The inverse of a function is a new function that undoes the action of the original function.



Examples



Formal definition of Inverse: The functions and are inverses if for all in the domain of and for all in the domain of .


When is the inverse of , we write .



If exists, then



The last two bullet points above reiterate the fact that inverse functions undo the action of each other: if and only if .




VERY IMPORTANT IDEA: If is a point on the graph of , then is a point on the graph of . Since is the reflection of about the line , it follows that the graph of can be obtained from that of by reflecting it about the line .



Examples










Finding Inverses

For some functions, we can use algebra to determine inverse functions. Essentially, to find an inverse function we "undo" the original function.


Given ,

  1. Find the domain of and verify that is one-to-one.
  2. Solve for in terms of .
  3. Interchange and to get .
  4. Define the domain of to be the range of .


Examples