I happened across a review sheet for another instructor’s College Algebra exam today. I know not whose, nor do I wish to know. I just want to use it as an example of what my poor students have to go through.
There were eight tasks on the review sheet. I would like my students to have the skills represented in those tasks for sure. But I wouldn’t happy with just those skills.
So here are the tasks. Each is followed up by the sort of question I would ask my students on an exam. Pity them.
- Original: Find the domain and range of
. Follow up: Give two more functions: one that has the same range as f but a different domain, and one that has the same domain as f but a different range.
- Original: Is
even or odd or neither. Follow up: Can a function be both?
- Original: Solve the absolute value inequality
and graph the solutions. Follow up: How do these solutions relate to the function
?
- Original: Graph the function
. Then find the intervals on which f is increasing and decreasing. Are there any local maxima or minima? If yes, where are they located? Follow up: Choose two points near a maximum or minimum value (if such a value exists). Find and comment on the average rate of change between these two points.
- Original: If
, find the net change and the average rate of change between
and
Follow up: Why are these values different? BONUS: Give a new function for which these values are equal between the same two points.
- Original: If
, write the transformations that yield
. Also graph both
and
on the same coordinate axes. Follow up: How many solutions are there to this system of equations?
- Original: If
and
, then what are
and
? Also list the domain for each case. Follow up: Choose one of the four functions you wrote. Write its inverse (if such a thing exists).
- Original: Graph this piecewise function
Follow up: There is a gap in the graph. Change the second piece of the function to eliminate this gap.
Overthinking my teaching 
Ok…more stuff to steal. Thanks. And I like the way you think.
However, I am curious about number 7. I will look it up later if you don’t respond, but what kind of function has no inverse? I know some inverses are not functions, but does that mean that there is no inverse? I’m probably splitting hairs, but my inverse function knowledge is a little rusty and were going to talk about them next week. So if there is a function that does not have an inverse, what kind of function is it?
True that, Don. I think I meant inverse function but was sloppy (as are most College Algebra texts) and used inverse as shorthand. If I were to give that task, I would be perfectly happy with a response that pointed out that such a thing must always exist-even if it’s not a function. And I would be happy with a student who understood inverse to mean inverse function.
And then I would share these two responses in class, invite discussion and summarize as you have here. Again-a window into my College Algebra classroom. Those poor kids sometimes just want right and wrong answers and points, not ambiguous quiz questions that lead to further discussion. I win most of em over by the end of the semester. Not all, but most.
One more thing. Yes, all functions have inverse relations. Since a function is officially a set of ordered pairs, the inverse swaps the order of the elements. We have dealt with that this semester and approached inverse from the perspective of “switching the range and the domain”. This leads to interesting ambiguities, about which I will need to write more sometime soon.