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.

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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.

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Thank you. (And this discussion is going to get me asking them more questions about domain and range. It may also push me to integrate the units more.)

Chris, Just a point about the “Do we have to graph this question?” I think if a student gave me your answer, I might ask them how they know that the function doesn’t have negative values somewhere. For you and me, easy answer, but College Alg kids (I don’t think) know that for x>0, sin(x) < x … and in fact if they did, they might be fooled into thinking that sin(x) <x^2 (which it isn't between 0 and 1)

The function x^2 – sin x has a minimum near -1/4 and I'm not sure Alg prepares them to discover that… But now we get to add back the part under the cube root, which is going to be positive from 0 to 1. But is it enough??? It turns out it is, but it's the x^2 under there rather than the x^5 that makes it work (since it's larger from 0-1).

So for me, asking a kid in College algebra to figure out this range without a graph might be a stretch, but I do agree even at this level, they should be challenged to think about the overall behavior (domain and range) of functions before they draw the graph…(because as you well know, there are lots of functions that graphing technology won't show the true story at any common magnification… there are monsters out there, and they need to know that they should go to the graphing window when they are content they know what to expect.

Nice blog by the way. I like your thinking.