Teach 180: Connecting Inverses (Day 49)

Anytime you can help students to connect new learning to previous learning, I believe that the understanding of the new concept is stronger. Today I set out to purposefully connect the concept of function inverse to the familiar concept of inverse operations. I began by having each table find f(g(x)) and g(f(x)) for the two cards at their table. Then, I had each table report out on their results. (You can see the pairs of functions at the left.) Me: "What????? Wait a minute. You mean each of you got that f(g(x)) = x and g(f(x)) = x. I wonder if that always happens. Hmmmm. Let's switch g(x) cards and see if that will happen again." So, I had the student groups swap their g(x) functions and calculate f(g(x)) and g(f(x)) again. This time no groups got a result of x. Me: "Well, that's thoroughly disappointing. I was hoping we would get x again. I wonder why we didn't. OK, let's look at the initial pairs we had." Next, we wrote the original pairs of functions on the board and I asked students what they noticed about each pair of functions. One student quickly realized that one pair of functions had squaring and square rooting and she described those operations as "opposites". After that other students noted the other inverse operations they saw, like multiplying by 2 and dividing by 2, and adding 3 and subtracting 3. I pointed out that not only did each pair of functions have inverse operations, but that they were done in an inverse or opposite order.  For example, in the first pair of cards for f(x) we would add 3 and then square the result.  For the inverse function g(x), we would do the inverse of those operations in the opposite order.  This means add 3, then square becomes square root, then subtract 3. Our next class will meet on Tuesday (there is no school on Friday due to Parent-Teacher conferences and no class on Monday due to the new bell schedule).  For now, I am going to ask students to create inverse functions by using inverse operations.  We'll graph our functions and inverses in desmos to see that the composition of a function and its' inverse is the identity function. We will also note that if a and b are numbers that are defined for both functions that f(a) = b and g(b) = a.  This will also be easy to show on desmos. I will give my students "easy" inverses to create initially, but then they will be given ones that are more challenging, such as find the inverse of f(x) = (1 + x)/x.  This will lead to a need for the formal way to find an inverse - replacing f(x) with y, switching x and y, and solving for y.