Building Functions - Reverse to Inverse

Learning Resource Type

Lesson Plan

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This lesson will provide an introduction to finding the inverse of a function or a relation. Through a combination of teacher-led instruction and collaboration, students will discover a method for finding the inverse of a function or relation. The use of an online graphing calculator will aid students with their discovery.

This lesson results from the ALEX Resource Gap Project.

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

MA19.PRE.28
Find inverse functions.

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.

Primary Learning Objectives

Students will identify the order of operations in an algebraic statement and will be able to reverse (or undo) the operations.

Students will be able to find domain and range values of a function and will recognize that the domain and range values of the inverse function will be in reverse order (range becomes domain and vice versa).

Students will be able to write the inverse of a function after being given a function.

Procedures/Activities

Before:

1. Show students a gift that needs to be wrapped.

2. Place the gift in a box.

3. Measure and cut the paper for the box.

4. Wrap the box

5. Tie a ribbon around the box.

6. Now ask the students to list the steps to unwrap the gift. Make sure they list the steps for wrapping the gift in reverse order.

If you choose not to wrap the gift yourself you can show the students this video on wrapping gifts.

Explain to students that they will be doing the same thing with functions in today's lesson.

During:

1. Group students into pairs and distribute computers. Have them go to Desmos online graphing calculator.

2. Begin with a simple function such as f(x) = 2x² + 3. Have students graph the function.

3. Ask students to list the order of operations necessary to evaluate the function when x = 2 and to find the value when x=2. Graph the point on Desmos.

4. Next, ask the students to find the point on the graph when x=0. You will want to remind them that the x or 0 represents the input or the domain value and the answer or y represents the output or the range. Have them graph this point on Desmos.

5. Now tell the students that they will be finding the inverse of the function. Ask them to write down operations that would undo the original function in reverse order. For example: The original order of operations was square input, multiply by 2, and add 3. Remind them that to unwrap the gift, we must do the last thing (take off the ribbon) first. Ask them what they would do to reverse add 3. (subtract 3) So our first step in the inverse is to subtract 3 from x. (x-3) Next, ask what they would do to reverse or undo multiply by 2. (divide by 2) ((x-3)/2) Finally ask what they would do to reverse or undo square a number. (take the square root). f^-1(x)=(+/-)sqrt((x-3)/2) Be sure to emphasize the notation for the inverse.

6. Now have them graph the inverse function on Desmos.

7. Explain that if the inverse function reverses what the original function does we should be able to start with the original output or range value to obtain the original input value or domain value. Ask them to begin with 11 as an input for the inverse and see if they can get 2 and -2 for answers. (You may need to remind some students that when you take the square root of a value the answer may be positive or negative.)

8. Have students find the answer for the input value of 3 and ask them to plot all 3 points on Desmos.

9. Students should see that the points on the inverse are simply the points from the original function in reverse order. Point this out if they do not see it.

10. Point out that the inverse function is a reflection of the original function over the line y=x.

11. Give students 5 other problems to work on. Allow them to collaborate to build confidence. The teacher will be available to circulate the room to help pairs who are struggling.

After:

Each student will be given an exit ticket to complete. This will provide a quick summative assessment for the teacher to identify concepts that need to be reinforced the next day or students who may need extra instruction.

Assessment Strategies

Formative - Teachers will assess student progress as they circulate around the room while students are graphing. Teachers can address any misconceptions or concerns.

Summative - Students will complete an Exit Ticket before leaving class. This will be a problem where students must find the inverse of a function. Teachers will be able to quickly check for misconceptions and concerns.

Acceleration

Advanced students can be encouraged to find the inverse of functions that have more complex operations.

Advanced students can be asked to determine using the vertical line test if the inverse of a function is also a function. They can then be asked to determine from the original graph if the inverse will also be a function (leading them to discover the horizontal line test which will be covered in another lesson).

Intervention

Students who have difficulty with the order of operations can view a tutorial and take a short quiz.

Peer tutoring while working in pairs can also benefit struggling students.

Total Duration

31 to 60 Minutes

Background/Preparation

Students should be familiar with the terminology associated with functions and relations: domain, range, input, and output.

Students should know the order of operations for working mathematical statements.

Students should be familiar with the Desmos online graphing calculator.

Teachers should have a small item suitable for wrapping, a box for the item, gift wrap, tape, and ribbon.

Teachers should be familiar with the Desmos online graphing calculator. Click here for a short tutorial.

Teachers should make one copy per student of the Exit Ticket activity (2 tickets per page).

Materials and Resources

Teachers: Computer connected to a projector

One copy per student of Exit Ticket activity (2 tickets per page)

A copy of Inverse Function Practice Problems

Students: One computer with internet access per pair of students

Exit Ticket Activity (2 tickets per page)

Technology Resources Needed

Desmos Online Graphing Calculator

Approved Date

2017-06-20

Owner (Author)

mrswhite