Here is something cool that happened in College Algebra today. We were doing a short thing to summarize our domain and range work before moving on.
A student asked, Is the only way to find range to make a graph?
This stopped me in my tracks. I had not really thought about the knowledge I draw on when identifying the range of a function, and the question cut to the heart of the matter.
My gut instinct answer was yes. But I wanted to explore that a little. I concocted a silly function to do so. . I wanted to say that I would need to graph that to know its range.
But the longer I looked at it, the more clear it was that I knew a lot about this silly thing without graphing it. The term dominates, for instance, in the long run, so I know it goes to infinity on both sides of the y-axis. I could see that 0 is in both the domain and the range.
But I wasn’t 100% sure whether there were any negative values for the function.
Later in the day, this got me thinking about end behavior. This is why we teach that end behavior silliness, right? It’s not about end behavior, it’s about knowing what values can come out of a function, and having a basis for knowing this.
I am brainstorming here. The point is that the student question showed a sign of her learning, and it pushed me to rethink something too. Win-win.
Another cool thing happened, too. We were comparing and , looking for sameness and difference. I had to push to get domain and range on the table.
We agreed that the two functions have the same domain—all real numbers. We were split on whether they have the same range.
But not for the reason I expected. Not at all.
A student argued that The only time when they are the same is when x=2. Therefore they do not have the same range.
My students found this argument compelling.
Ignore the second intersection point in the left half-plane. Focus on the essence of the argument.
Do these functions have the same range? is interpreted as Do these functions intersect?
That seems like a useful insight into the mind of a College Algebra student.