Provide "acknowledged body of knowledge" related to the concept. Emphasize the most significant aspects of the concept in an organized, organic manner. Present information sequentially so students see continuity. Draw attention to important, discrete details; don't swamp students with myriad facts.

Teacher directed instruction with examples

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.