I. Curricular Framework

Concept:

Properties

Essential Question:

How would you characterize inverse functions?

Bridge:

Visual Mapping

Content:

Inverse Functions

Outcomes:

II. Standards Aligned

III. Instruction and Assessment

1. Connect: Connecting to the Concept Experientially

Objective: To reduce math anxiety by allowing the students to become aware of their everyday use of inverse functions. To simulate an everyday experience which illustrates the concept of inverse functions. To further illustrate inverse functions by shifting into the language of simple arithmetic.

Activity: Choose three students to participate in a minidrama. One will be the student, one will be the mother of the student, and the third will be the brother of the student. Have them line up in a row, with the mother on one end, the student in the middle, and the brother on the other end. Tell them that the student has no money at present, that the brother owes the student $5, but the student owes the mother $5. This situation is probably familiar to the students. Then have the brother give $5 to the student, making the student $5 richer. Making the student $5 richer is the FUNCTION. Point out to the class that the student now has $5 more than when the drama started. Now have the mother reach over and take the $5 away from the student. The student is back to where s/he started from. Making the student $5 poorer is the INVERSE FUNCTION. Point out that the student started with no money and now has no money again. Therefore, an INVERSE FUNCTION undoes the action of the FUNCTION. This point is the main concept of inverse functions and should be stressed throughout the lesson. Next, have the students sit down and tell the class to close their eyes. Have them choose a number between 1 and 10. Then have them add 54 to that number. Wait, and then have them subtract 54 from their new number. Now have them open their eyes. Tell them that although you cannot guess the exact number they picked, you CAN tell them that it is the same number they started with. They will not be very impressed, so have them close their eyes again, and tell them to start with the same number they had before, and add 120 to it. Wait, then have them subtract 80 from the new number. Tell them that this number is different from their old number. In other words, adding one number, and subtracting a different number are not inverse functions. One more time, have them close their eyes, and start with their old number once again, tell them to multiply it by 4. Wait, and then have them divide their new number by 2. This number will not be the same as the one they started with. Again, multiplication and division are inverse functions only when you multiply and divide by the same constant.

Assessment: Quality of class participation.

2. Attend: Attending to the Connection

Objective: To have the students analyze their experience in order to see if they understand the basic concept of inverse functions. This is done by having them submit examples of inverse functions.

Activity: Organize the students into groups of four, and tell them they have to come up with examples of inverse functions, both from the activities just done, and from functions they have studied previously (either in lower level classes, or perhaps earlier in precalculus). Make sure that their examples are complete: they cannot say multiplication and division, they must say multiplication and division by the same constant. Once the groups have finished, have each group put its list of functions/inverse functions on the board. Combine the lists into a single list, pointing out which pairs of functions/inverse functions are not correct, and why. At this point, the teacher may want to list other pairs of functions/inverse functions that the class might not have listed.

Assessment: Quality of group participation and understanding of the basic concept.

Assessment, Phase One, Level of Engagement, Fascination:

3. Image: Creating a Mental Picture

Objective: To use a dramatization to emphasize the concept of the function as a MAPPING from x to y, and the inverse function as a mapping from y to x.

Activity: You will need two robes, one black and one white, two cardboard squares with a string through them to allow them to be worn around the neck, and two signs, one labeled "X," and the other labeled "Y." On one of the cardboard squares write: f(x) = x + 5, and on the other write: f-1(y) = y - 5. Draw two large chalk circles on the floor of the classroom; label one circle "X" and the other "Y." Choose nine students to participate in the demonstration. One student will be the FUNCTION (dressed in white, and wearing the f(x) = x + 5 square around her neck). One student will be the INVERSE FUNCTION (dressed in black, and wearing the f-1(y) = y - 5 square around his neck). Two of the students will be standing in the "X" circle, and should hold the "X" sign TOGETHER, and the other five students should stand off to the side. No one should be in the "Y" circle. Now the FUNCTION should go to the "X" circle, take the two students with her to the five students standing off to the side, and take away their "X" sign. Then she should bring all seven students to the "Y" circle, and give them a "Y" sign to all hold TOGETHER. This is very important since there are not seven y's, but one y, represented by the seven students. Tell the class that the FUNCTION has mapped the x = 2 into a y = 7. Now have the INVERSE FUNCTION go to the "Y" circle, take the seven students off to the side, take away their "Y" sign and bring the ORIGINAL two students back to the "X" circle and give them back their "X" sign to hold together. Tell the class that the INVERSE FUNCTION had mapped the y = 7 into an x = 2 and furthermore, has undone what the FUNCTION has done. In other words, you started with two students, went through a mapping and an inverse mapping and ended up where you began.

Assessment: Quality of class participation and level of understanding of the concept.

Assessment, Phase Two, Seeing the Big Picture:

4. Inform: Receiving Facts & Knowledge

Objective: To show the students graphically the relationship between a function and its inverse. To teach the students how to solve for the inverse function, given a function algebraically. Activity One: Draw the function y = 3x on the blackboard, being careful to use accurate scaling. Tell the students you will give them a value of x, and that they must tell you, USING THE GRAPH, the value of y which corresponds to it. Do this for several values of x. Stress that the graph, which is simply a pictorial representation of the function, allows them to find the value of y associated with each value of x. The graph represents the mapping of all possible values of x into all resulting values of y. Next, tell them that you will give them various values of y and that, USING THE GRAPH, they must tell you the corresponding values of x. Do this for several values of y. Point out to them that they have been using the inverse function to find out which values of x match the values of y which you have given them. Ask them how it is possible for the SAME GRAPH to act as a function in one case (given x, predict y), and as an inverse function in another case (give y, predict x)!! Stress that, in the case of the FUNCTION, you input x and get out y; whereas, with the INVERSE FUNCTION, you input y and get out x. To show them what the graph of the inverse function looks like, draw another set of axes on the board and redraw the function y = 3x on it, only rotated 90 degrees counterclockwise, so that the positive y axis points left, and the positive x axis points up. Since the usual convention is to have the positive axes pointing right and up, draw a third set of axes, with a mirror reflection (in the y axis) of the function you have drawn in the second set of axes. Now the positive x axis should still point up, but the positive y axis will now point to the right. The function you should have on the axes should be x = 1/3y, the inverse function of y = 3x! It is important to emphasize to the students that with the inverse function, you input y and get out x! Although it seems confusing, the inverse function is simply the original function looked at from a different point of view (start with y instead of starting with x), which is why if you simply turn around the original graph you can get the inverse graph. Many people find this confusing. IT IS IMPORTANT TO KNOW THAT MATHEMATICIANS, ONCE THEY ROTATED THE ORIGINAL GRAPH TO GET THE GRAPH OF THE INVERSE (X AXIS POINTING UP, AND THE Y AXIS POINTING TO THE RIGHT), ARBITRARILY DECIDED THAT THE Y AXIS MUST ALWAYS POINT UP AND THE X AXIS MUST ALWAYS POINT TO THE RIGHT. In other words, the mathematicians arbitrarily interchange the x and y variables. Therefore, instead of writing the inverse as x = 1/3y, they write it as y = 1/3x. That is why, in any math book you will ever read, the inverse of y = 3x is written as y = 1/3x, instead of as x = 1/3y, which is the only way it makes intuitive sense, as a mapping, or algebraically. Note: Some teachers may be confused by this, but in all the years I have taught it this way, the students found it to be very understandable, as long as you point out that mathematicians have ARBITRARILY decided to write the inverse of y = 3x as y = 1/3x, instead of as x = 1/3y. Once you have taken the students through finding the inverse graph in this long way, you should give them another simple example to work through this way, in groups of four, to see if they have really grasped the idea. After they have done this, you are ready to show them the "Mirror" method of graphing the inverse function. That is, given the graph of a function, draw the line y = x (lightly, in pencil) on the same graph. The graph of the inverse function will be the mirror image of the original function, treating the line y = x as the mirror. This is the fastest way to graph inverse functions, and the students will love it, but to teach them this way before teaching them WHAT IT MEANS is meaningless! Activity Two: This is the algebraic way of finding the inverse function, given the original function. Write the function y = 2x - 4 on the board, and show the students how to solve for x where x = 1/2(y+4) then (being arbitrary again) switch the x with the y to obtain y = 1/2(x+4). This is the inverse function. Do this for several simple functions, in order to stress how to SOLVE for the inverse functions. When you want to have the students find the inverse functions like y = x2, it will be necessary to review the concept of one-to-one functions, and emphasize to the students that, in order to have an inverse function, given a value of y, it has to be possible to find a SINGLE value of x which corresponds to it. In the case of y = x2, this is not possible unless you restrict x to x >0: thus y = x2, x > 0 has one inverse and y = x2, x < 0 has another inverse. Finally, after spending some time having students work through some algebraic examples by solving for the inverse function, you may want to teach them the TRICK of switching the x and the y in the original function to obtain the inverse. However, first make them work the problems out the long way, to obtain an understanding of what is happening. Note: Some teachers teach this trick to students first, thereby totally bypassing the intuitive understanding of inverse functions. This is a calculation SHORT-CUT, and should not be confused with a true understanding of inverse functions.

Assessment: An objective test covering graphs and algebraic solutions for inverse functions.

Assessment, Phase Three, Success with Acquiring Knowledge:

5. Practice: Developing Skills

Objective: An objective test covering graphs and algebraic solutions for inverse functions.

Activity: Text problems and teacher-prepared worksheets

Assessment: Quality of the worksheets and homework should be evaluated with an eye towards the PROCESS used, and not just to see if the answer is correct. If the process is totally correct, and there is a simple addition mistake, the students should get most of the credit. Higher mathematics should emphasize process as well as accuracy.

Assessment, Phase Four, Success with Acquiring Skills:

6. Extend: Extending Learning to the Outside World

Objective: That the students further deepen their understanding of the concepts and skills taught by having them assume the role of teacher in composing a quiz for fellow students.

Activity: Divide the class into groups of four. Have each group design a quiz for their fellow students. Encourage them to be as creative as possible in designing their quiz, i.e., don't have all the functions to invert by linear functions. Take the best parts (perhaps all) from the different quizzes and combine them into a test to be given to the entire class. (At this point, the teacher can ditto all the groups' quizzes, and let all the students examine them. Then, if one question is not understood, the student who wrote it can teach it to the class. After all, the purpose is that ALL the students understand inverse functions. What better way than to design a test and to teach. Also, the "product" of teaching shifts them to understanding, and the test does not become an end in itself.)

Assessment: The quality of the questions and the test results

7. Refine: Refining the Extension

Objective: To help students to see the usefulness of what they have learned.

Activity: The teacher conducts a lecture/discussion emphasizing the many uses of inverse functions in the world. (Note from Bernice McCarthy: I applaud this departure from 4MAT. Here this teacher is moving back to the combined methodology of Quadrants One and Two in order to take the students into an awareness of the usefulness of what they have just learned.) Clearly, since the students are in a class of precalculus, they will probably be taking many upper level math courses in the future, certainly at least two terms of calculus. Therefore, the future usefulness of inverse functions for these students is obvious. Also, in this age of high technology, it is becoming increasingly necessary for students to be familiar with mathematics and the natural sciences. So again, inverse functions will be very useful to them. In my lecture/discussion, I emphasize these points. But the students find it more interesting to discuss more specific uses of inverse functions that they can understand immediately, rather than some time "in the future." So we talk about the field of communication theory (telephones, telegraphs, radios, stereos and television). It is impossible to pick up a bare wire and talk into it and communicate with someone miles away. However, in a telephone, a transducer transforms sound waves into electrical impulses (the FUNCTION). These impulses can be sent to someone miles away. When these signals reach the other person, an inverse transducer (the INVERSE FUNCTION) inverse transforms the electrical signals into sound waves which can be understood by the other person. All forms of long distance communication use this principle. For those who are interested in codes, the same principle is used, although on a much more sophisticated level: a device is necessary to encode a message, and another device is necessary to inverse code the message at the other end. Cable TV companies also use this principle to ensure that everyone with a TV set does not see their programs without charge. They broadcast their programs on a UHF station; theoretically, it should be possible to tune in on their programs. So they use a coding device to scramble their signal. An inverse coding device is necessary to unscramble the signal, which they supply when one subscribes to their service. Without the inverse coder, if you try to illegally tune in on their programs, all you will see is "snow." In more advanced mathematics, many equations cannot be solved directly (especially in the field of differential equations). The Laplace transform is a method of transforming a differential equation into an algebraic expression. This expression can be easily manipulated algebraically, then the inverse transform gives you the answer to the original differential equation. Whichever examples the teacher decides to use to illustrate inverse functions, it is very important to show the students the usefulness of what they have learned, even if they cannot exactly understand all the details. They enjoy the feeling of having learned something useful. Some teachers tell their students they will find out "one day" how useful their learning will be. I believe this only decreases their willingness to learn. Have the students write a short essay on the usefulness they feel inverse functions will have in their own lives.

Assessment: Quality of the essays.

8. Perform: Creative Manifestation of Material Learned

Objective: To further reinforce student awareness of the usefulness of what they have just learned.

Activity: Students share the essays they have written.

Assessment: Quality of understanding and class participation.

Assessment, Phase Five,Performance, Creative Use of Material Learned: