Curriculum  /  Math  /  11th Grade  /  Unit 1: Linear Functions and Applications  /  Lesson 5

Linear Functions and Applications

Lesson 5

Math

Unit 1

11th Grade

Lesson 5 of 13

Objective

Find the inverse of contextual and non-contextual situations algebraically.

Common Core Standards

Core Standards

  • F.BF.B.4.A — Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x—1) for x ? 1.
  • F.BF.B.4.B — Verify by composition that one function is the inverse of another.
  • F.BF.B.4.C — Read values of an inverse function from a graph or a table, given that the function has an inverse.

Foundational Standards

  • A.CED.A.4

Criteria for Success

  1. Use inverse notation to describe an inverse function. 
  2. Describe how "solving for a variable" is like finding the inverse in terms of another variable. 
  3. Describe why the range of a function is the domain of the inverse of the function. 
  4. Define whether a linear function is "invertible"—that is, whether the inverse of the function is a function. 
  5. Identify functions, from a table of values and in a scatterplot, that are and are not invertible.

Tips for Teachers

  • This lesson has a focus on the mechanics of finding the inverse of a function algebraically but has thin context provided for Anchor Problem #1. Be sure to have a balance of contextual and non-contextual problems throughout this lesson. 
  • In future lessons, students will verify inverse by composition, but this is not a criterion for success for this lesson. 
  • This lesson has components that extend beyond F-BF.4a into F-BF.4d. If you are not teaching an advanced Algebra 2 course, focus on the algebraic process of writing an inverse function rather than on whether the function is invertible or not. 
  • The target task has a decimal coefficient. In this lesson, highlight the usefulness of fractions when finding the inverse of functions. 
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Anchor Problems

25-30 minutes

Problem 1

Complete the table below to determine the process for finding the inverse of a function. The price of hats is written as a function of the number of hats, $$h$$, produced. 

Guiding Questions

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Problem 2

Below is a graph of a relation, $$f$$, represented as individual points. 

Explain why the relation $$f$$ is “not invertible.” 

Guiding Questions

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Target Task

5-10 minutes

The function, $$f$$, defines the cost of supplies for a lemonade stand as a function of the number of cups of lemonade you expect to sell. 

$$f(x) = .5x + 10$$

Write  $$f^{-1}$$algebraically and define the domain and range for both $$f$$ and $$f^{-1}$$ in the context of the problem. 

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where nonlinear functions are presented (like absolute value functions) in graphical or tabular form and students are asked whether they are invertible or not. 
  • Include contextual problems even though context was not a part of the anchor problems. Feel free to use contextual problems from the two previous lessons, written in function notation, and ask students to find the inverse. 
  • Include problems where two functions are given and students need to determine whether the relations are inverse of one another.

Next

Write a system of functions for contextual situations and solve algebraically. Describe the solutions in context of the problem.

Lesson 6
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Linear Functions

Topic B: Systems of Functions and Constraints

Topic C: Piecewise Functions

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