In Section 3-2-2, we briefly introduced the concept of inverse functions. Recall that for a one-to-one function , we can define the inverse function . If we think of as a process that takes some input and produces some output , then providing as an input to produces the original input , and vice versa. Symbolically, we wrote that and .

We learned several important principles, which we summarize below.

  • A function has an inverse function if and only if there exists a function that undoes the work of : that is, there is some function for which for each in the domain of , and for each in the range of . We call the inverse of , and write .
  • A function has an inverse function if and only if the graph of passes the Horizontal Line Test.
  • A function has an inverse function if and only if is a one-to-one function.
  • When has an inverse, we know that writing “” and “” are two different perspectives on the same statement.
  • If is a point on the graph of , then is a point on the graph of .
  • The graph of is the graph of reflected across the line .
  • The domain of is the range of and the range of is the domain of .
  • If is the inverse of , then is the inverse of .

In this section, we’ll explore inverse functions more in-depth.