Inverse Functions

Inverse functions bring together the concepts we have learned about composite functions and domain and range, so make sure you are confident with those sections before you dig in. A function must be one-to-one to have an inverse. If you are uncertain how to determine whether a function is one-to-one, it may be a good idea to revisit that concept.

Inverse Functions

Learning Objectives

In this section, you will:

  • Verify inverse functions.
  • Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
  • Find or evaluate the inverse of a function.
  • Use the graph of a one-to-one function to graph its inverse function on the same axes.

A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.

If some physical machines can run in two directions, we might ask whether some of the function "machines" we have been studying can also run backwards. Figure 1 provides a visual representation of this question. In this section, we will consider the reverse nature of functions.

Diagram of a function and would be its inverse.
Figure 1 Can a function "machine" operate in reverse?

Source: Rice University, https://openstax.org/books/college-algebra/pages/3-7-inverse-functions
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