Texas Mathematics Teacher Fall/Winter 2020 - 10

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.

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| 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
Texas Mathematics Teacher Fall/Winter 2020

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