Lesson 17

Writing Inverse Functions to Solve Problems

Lesson Narrative

In this lesson, students continue to expand their capacity to work with and interpret inverses of linear functions in various situations. Previously, students found inverses of functions that were defined using two variables. Here, they find inverses of functions given in function notation.

In earlier lessons, students worked with functions in which the quantities and the relationship between them were straightforward and well defined. In this lesson, students still engage with such functions, but they also work with a relationship that is less well defined. In the last activity, students analyze a data set, write a linear function to model the data, and use the model (including writing an equation that represents the inverse function) to solve problems. Along the way, students engage with different aspects of modeling (MP4).

Note that notation of the form \(f^{\text-1}\) is intentionally not used to denote inverse function at this point. This is so that students can focus their attention on the meaning of an inverse function rather than on learning a new notation. It is also to discourage students from thinking of finding an inverse function as a procedure.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. Consider making technology available.

Learning Goals

Teacher Facing

  • Find the inverse of a linear function given in function notation.
  • Write a linear function and an inverse function to model data and solve problems.

Student Facing

Let’s use inverse functions to solve problems.

Learning Targets

Student Facing

  • I can write a linear function to model given data and find the inverse of the function.
  • When given a linear function defined using function notation, I know how to find its inverse.

Glossary Entries

  • inverse (function)

    Two functions are inverses to each other if their input-output pairs are reversed, so that if one functions takes \(a\) as input and gives \(b\) as an output, then the other function takes \(b\) as an input and gives \(a\) as an output.

    You can sometimes find an inverse function by reversing the processes that define the first function in order to define the second function.