Surgery for Function Operations

My college algebra course boasts one of the driest textbooks on the planet. It’s one of those versions that has exercises from 1 to 99 for each section…brutal.   Can you relate?
The topics for college algebra are very standard and cover little more than what students should have encountered recently in their algebra 2 course. I therefore decided that this class would lend itself quite nicely testing out the theory that a high-level, rich question questioning can be facilitated from a traditional, drill-and-kill style textbook.

Previously, I recall that Operations on Functions was a particularly awful topic for both me and my students.  The textbook presents this concept in exactly the way you might think:

f(x) = [expression involving x]  and g(x) = [similar expression involving x]

Find f(x) + g(x), f(x) – g(x), f(g(x), f(x) *g(x), f(x)/g(x)…f(snoozefest)…you get the point.  It’s boring, they’ve done it before, and there’s not much high-level thinking involved.

Fortunately, it’s fixable by asking new questions from the same problems.  For example, have students choose a pair of functions from the book.  We have 99 choices after all!  For example, something quadratic and something linear,  like f(x) = x^2 + 1 and g(x) = 2x+4.

Here come the questions:

• Which of these function operations are commutative and which are not?  How do you know this?
• Does this work for all functions, or just the ones that you chose?
• For what values of x are the non-commutative function operations equal?
• What do you notice about those values of x for the different operations?
• Can you prove any of your results?
• How do the graphs of these new functions compare to the original graphs?

Compositions of functions are the most fun!  Here come some more:

• For which values of x is f(g(x)) > g(f(x)) for your specific functions?
• Both linear?
• How are the graphs of f(g(x)) and g(f(x)) related to both f(x) and g(x)?
• Don’t forget about f(f(x)) or g(g(x))! How do those relate to our original functions?
• What about g(g(g(x))) and g(g(g(g(x))))?
• What do you notice happening each time we compose the function with itself again?
• Can you generalize your conclusions based on the number of compositions and tell me what g(g(g…g(x)…)) would look like?
• What do you notice about each of these compositions?
• What do you notice about their graphs?

A personal favorite of mine is:  If 4x^2 + 16x + 17  =  f(g(x)), what could f(x) and g(x) have been?  This works really well with whiteboards and partners.

I might have students throw out any questions that they find interesting.  In fact, I’ll bet we can come up with at least 99 questions more intriguing than the ones given in the textbook.  Then let them choose which one(s) pique their curiosity.   Now hopefully we’ve taken the time that they would have spend doing 1-99 from a book and turned it into time better spent.