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I mostly tutor students ranging from beginning algebra to calculus level. I think the explaining the range as "the set of all possible outputs" would not really cut it for someone struggling with math at that level. Usually I default to:

"Draw a rough sketch of the graph, including all the shifts, stretches, etc. The range is all the portions on the $y$ axis which have a piece of the graph to the left/right of them."

I feel that not only is this explanation very good, but many times when I ask the students to draw a rough sketch of the graph, they have no idea how; so now I have to reteach the student how to graph equations. However, I am not a private tutor; I work for my school, and I am not allowed to spend more than $15-30$ minutes with a student at a time, so I do not have time (nor want to) reteach the student half a chapter just so they can solve one problem. How can I explain the range in a better and less time-consuming way?

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    $\begingroup$ I wouldn't use your definition. It shall rather be a graphical explanation for the case the function is real valued. Take the function which assigns to each person its name. What is the range of this function? $\endgroup$ – Stephan Kulla Sep 10 '16 at 11:58
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    $\begingroup$ Same as the range of a rocket. It's what you can hit either with the rocket or with an input to your function, say the rocket x. Then you could say the city "-1" is safe from the rocket "x^2". $\endgroup$ – Chris C Sep 11 '16 at 15:22
  • $\begingroup$ Not an answer, because I only want to answer the question that matters ("wat should I do") and not the question that you asked. You will find yourself in many such situations, with range, with domain, etc. If you have 15-30 minute per student and they are struggling, the most useful thing to do is to explain them how to work. Sure, give an intuition of range, but most importantly learn them what to do when they are stuck: identify the problematic notion, look for the definition, try apply it, look at the example treated in the lecture, if a problematic notion is used look it up... $\endgroup$ – Benoît Kloeckner Sep 15 '16 at 12:10
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I tend to speak of input and output and sometimes mention vending machines as an analogy. There is a range of products available from a soda machine. Typically it might include Pepsi. It wouldn't include a snickers bar -- not a possible output of a soda machine. An important part of understanding a vending machine is knowing what it vends. Similarly, an important part of understanding a function is knowing what values it produces. This also provides a possible answer to your other question about teaching domains. A vending machine will accept some combinations of coins but not others.

Obviously this doesn't directly help with the problem of e.g. finding domain of a simple polynomial or rational function. But, if the students have a good mental model for what a domain is then they can perhaps better understand it in a concrete case. After mentioning vending machines, if often use $f(x) = x^2$ as a simple example. Like a vending machine that only gives diet products (as might be found in a gym), this is a function that only gives nonnegative numbers.

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I think the problem is, in part, that you are coming at this from the wrong perspective.

They have a problem regarding ranges, so you've become fixed on helping them understand ranges. Then, when you discover they have a more fundamental problem... you are still fixated on your goal of teaching them ranges, rather than helping them with their actual problems.

But what they need is help with their actual problems.

So, if a student can't graph functions, you really do need to help them with graphing functions. And with the aim of learning how to graph functions, rather than how to fake it enough to replicate your solution to the range problem they've been assigned.

If you're good, you can even slip in range-related ideas as you do so — e.g. commenting how the graph $y = x^2 - 1$ has $y$ start at $0$, dip down to $-1$, and then back to $0$ as $x$ varies over the interval $[-1,1]$.

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  • $\begingroup$ The problem is though that I work at a community college, so may students are not very motivated to learn; they just want to get their work done as fast as possible. This, coupled with the fact I am only allowed to tutor them for 15-30 minutes at a time, creates a problem $\endgroup$ – Ovi Sep 9 '16 at 22:57
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    $\begingroup$ Unfortunately, you're not a miracle worker. It's impossible for you to help an unmotivated student learn something that they don't yet have (and don't care to gain) the tools to understand. $\endgroup$ – rnrstopstraffic Sep 10 '16 at 5:33
  • $\begingroup$ @Ovi: First of all: It's great how motivated you are to teach your students the concept of functions. I agree to rnrstopstraffic that motivating the students is the main task I would concentrate on... BTW, I also noticed in my teachings that motivating the students is something I want to concentrate more on... $\endgroup$ – Stephan Kulla Sep 10 '16 at 12:24

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