A Reflection on the Process and Value of Studying Our Own Professional Growth Our class discussion about switching the independent and dependent variables led beautifully into our study of the symbolic algebraic process for determining the inverse of a function. Two students volunteered to find the inverses of two functions on the board: f (x) = 3x - 4 and +2 f (x) = x___ x - 5 . Since the second equation was quite difficult, the class rallied behind the volunteer, and eventually were ___ + 2 (see Figure 4). My able to find f -1 (x) = 5x x-1 students demonstrated great problem-solving techniques and teamwork as they addressed the issue of having two terms with y. Although we had practiced a few simple examples of method three, the students preferred methods one and two for solving the matching activity. Their common mistake was to swap the graphs circled in Figure 5, indicating that had trouble interpreting the slope and y-intercepts of a linear function. Again, they clearly were struggling with retention, and reviewing previous concepts was necessary. I decided to skip the closure activity because time constraints unfortunately did not allow for it prior to administering the post-test. Student 1: You should put both y's on the left. Student 2 [at whiteboard]: Why? Student 3: The y needs to be by itself, so put the y's on the left-everything else on the right, and then we'll figure out what comes next. Figure 5: Matching Activity with Mistakes Circled Figure 4: Student 2's Work at the Whiteboard Before the matching activity, we reviewed methods for determining if two functions are inverses. As a class, we recalled this list: 1. Determine if the graphs are reflections over y = x. 2. Pick an equation and solve for its inverse. 3. Calculate the composition of the functions: (f ° g)(x) = x. 10 | Fall/Winter 2020 Step 6: Administer the Post-Test and Analyze the Results On the post-test, I was able to see improvements in all areas (see Figure 2). All students were able to find an inverse graphically and algebraically and to recognize that for a function and its inverse, the domain and range switch. Each student now answered with a correct response for question three which asked students to explain the relationship between inverses. The student whose answer to question three on the pretest was "a one to one which means you want to see if 2 functions are inverses of each other," now wrote, "a way of showing when the dependent and independent variables switch and depend on each other in another way." The student's growth from pre to post was considerable. Another student wrote, "Inverses are opposite functions that are related." Other answers varied by explaining the relationship between domains and ranges to explaining that an inverse reflects its function over the line y = x. Students used words like "opposite" and "undo." Texas Mathematics Teacher

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