Building Functions: Inverse Functions from Tables and Graphs

Learning Resource Type

Lesson Plan

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This lesson will lead students on a guided discovery to find the inverse of a function given the graph or a table of values. Students will relate the inverse of a graph to finding the reflection of the graph over the line y=x. They will identify characteristics of functions whose inverses are also functions (One-to-One Functions) and will be introduced to the horizontal line test. Students will also apply their knowledge of a graph to a table of values to determine if the table represents a One-to-One Function.

This lesson results from the ALEX Resource Gap Project.

Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.28

Find inverse functions.

UP:MA19.PRE.28

Vocabulary

  • Inverse
  • Domain/output of a relation/function
  • Range/output of a relation/function
  • Horizontal line test (one-to-one)
  • Inverse Function
  • Composition
  • Invertible Function
  • Non-Invertible Function
  • Restricting the Domain

Knowledge

Students know:
  • The domain and range of a given relation or function.
  • Algebraic properties.
  • Symmetry about the line y = x.
  • Techniques for composing functions.
  • The composition of a function and its inverse is the identity function.
  • When (x,y) is a point on an invertible function, (y,x) is a point on the inverse.
  • In order for a function to have an inverse function, the original function must have a one-to-one correspondence.

Skills

Students are able to:
  • Find the inverse of function.
  • Accurately perform algebraic properties to find the inverse.
  • Accurately identify restrictions on a non-invertible function that allow it to be invertible.
  • Accurately find the composition of two functions.

Understanding

Students understand that:
  • The graphical and algebraic relationship between a function and its inverse.
  • The process of finding the inverse of a function.
  • The inverse of a function interchanges the input and output values from the original function.
  • The inverse of a function must also be a function to exist and the domain may need to be restricted to make this occur.
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.28c

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Primary Learning Objectives

Students will create the graph of the inverse of a function and use the vertical line test to determine if the inverse is also a function. 

They will determine if a function is a One-to-One Function using a variety of methods (inspection of the graph of the inverse, using the Horizontal Line Test, looking for repeated f(x) values in the table of values of the original function).

Procedures/Activities

Before:  

Time allotment: 15 minutes

1.  Direct students to the Math Is Fun Symmetry Artist.

2.  Ask them to create their masterpiece. Give the students 3 or 4 minutes to play around and become familiar with the website.  

3. Now instruct students to select the y=x reflection symmetry and tell them to create a quick drawing using this setting.  

4.  Have the students share a description of their drawings. (You want them to notice that the image is a reflection over the line.) You may choose to use an online bulletin board like Padlet to allow students to copy and paste a screenshot of their pictures along with a description of how the drawing was created. You are looking for a discussion of the reflection.  

5.  Discuss student descriptions, being sure to ask students to recall the method for reflecting over the line y=x. 

During: 

Time allotment: 30 - 40 minutes 

1.  Explain to students that they will be finding the inverse of a function from its graph.

2.  Show students the graph of y=x^3. Ask them to decide if the graph represents a function. 

3.  Remind students that to find an inverse you reverse the input and the output.  

4.  Have the students locate at least 3 points on the graph (you can lead them to choose (-2, -8), (0,0) and (2,8).

5.  Next, ask them to use the points from step 4 to find points on the inverse and to sketch the inverse.

6.  Ask students to imagine drawing the graph on the Symmetry Artist website. They should see that the sketch of the inverse would be the same drawing.  

7.  Now ask them to determine if the sketch of the inverse is also a function.  

8. Repeat steps 3-7 for the graph of y=x^2.  Make sure that students realize that not all functions have inverse functions (the graphs of the inverse also pass the vertical line test).

9.  Allow students to work in pairs or groups to complete the Is the Inverse a Function? Worksheet. Students will sort functions into two categories: those that have inverse functions and those that don't. They will make a list of characteristics for the functions with inverse functions as well as for those that do not have inverse functions. You may need to lead students to the discovery that the repeated y-values on the inverse of non-functions were repeated x-values on the original function.  

10.  Allow students to present their findings. If students do not discover the horizontal line test for determining if a function is a one-to-one function (it has an inverse function) share this with them.  

11.  Now ask student pairs/groups to apply their findings to a table of values. Ask them to explain how to find the points on the inverse and how would they know if the function was a one-to-one function.  

12.  Circulate among the groups to address problems and misconceptions.

13.  Allow students to present their findings.

After:

Time allotment: 5-10 minutes

Give students the Quick Check exit slip. They will find the inverse of the given functions and decide if the function is one-to-one.

Assessment Strategies

Formative:  

Teachers will assess student understanding of reflecting over the line y=x through the discussion in the before activity.  

Teachers will circulate the classroom while students are working on the discovery part of the lesson (Is the Inverse a Function? Worksheet). Teachers will be able to talk to student pairs/groups to correct misconceptions and problems.

Summative:  

A Quick Check exit slip will be collected to assess student understanding.

Acceleration

Advanced students can be given function rules (for example: f(x)=(x+3)^2) and be asked to produce the graph and then determine if the function is One-to-One. They can find the characteristics of rules that are One-to-One Functions vs. those that are not.  Using these characteristics, they can describe which rules produce One-to-One Functions.

Intervention

If students need extra practice on finding the inverse, extra practice can be provided. You may need to provide struggling students with both the graph and the table of values of a function.

Total Duration

31 to 60 Minutes

Background/Preparation

Teacher:

Be familiar with the Math is Fun Symmetry Artist website.

If using an online bulletin board, such as Padlet, have the board and access code prepared. (Padlet is an online bulletin board where students use their own devices (even phones) to post their ideas or thoughts. Individual student ideas are displayed together and can be viewed by the entire class. Student names are not associated with their thoughts so it can be a way to get students to speak out without fear of being wrong. Teachers can become the moderator approving only appropriate comments.) 

Make copies of Is the Inverse a Function? Worksheet for students. (one copy per student)

Make copies of Inverse Function Quick Check. (one copy per student)

Students:

Students need to be familiar with the process of reflecting an image across the line y=x.  

Students should know how to find the inverse of a function (input becomes output and vice versa).

Students should also be familiar with the vertical line test.

Materials and Resources

Teacher:

Projector

Chart Paper/White Board/Online Bulletin Board to display student discussion ideas

Graphs of y=x^3 and y=x^2

Is the Inverse a Function? Worksheet (One copy per student)

Inverse Function Quick Check (One copy per student)

Students:

Computer/Device with access to the internet

Math is Fun Symmetry Artist

Is the Inverse a Function? Worksheet

Inverse Function Quick Check 

Technology Resources Needed

Math is Fun Symmetry Artist

Padlet (optional)

Teacher:

Projector and interactive whiteboard

Students:

Computer/Device with access to the internet

Approved Date

2017-06-30

Owner (Author)

mrswhite
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