Find inverse functions.
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.
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.
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.
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 have to do the last thing (take off the ribbon) first. Ask them what would they 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 are able to 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. Allow them to collaborate with each other to build confidence. The teacher will be available to circulate around the room to help pairs that are struggling.
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.
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.
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).
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.
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).
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)