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Let us assume that I want to graph any of the functions below.

A) A can of soda costs $\$1$. Draw a graph depicting the total cost as a function of the number of cans you buy.

Comment: One cannot actually buy $4.6$ cans of soda.

B) Ellen is a member at a gym. She pays $\$30$ at the start of every month and she paid an initial fee of $\$10$ for a membership card. Draw a graph depicting the amount of money Ellen has paid after $x$ months.

Comment: The graph is in my opinion something like a step-function if interpreted as a function on the real numbers.

C) You bring a large cake into the classroom, intending to share it with the other students. Draw a graph depicting the amount of cake each student gets.

Comment: One cannot have $7.3$ students.

It is in my opinion rather hard to say exactly what the domains are in the examples above. However, I want to give some justification in the case that I should "approximate" the functions as being $\mathbb{R} \to \mathbb{R}$. Should I avoid giving word problems like these? Should I avoid mentioning this issue if no student points it out? Should I rather ask them to make a table and not to attempt to graph the functions at all?

Question

When working with 12-16 year olds, how should I depict graphs of functions that are technically not $\mathbb{R} \to \mathbb{R}$?

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    $\begingroup$ IMO, the graphs should be a set of isolated points in the plane. If that wasn't the intention, I would say there's a problem with the redaction. $\endgroup$
    – FormerMath
    Apr 19 at 18:57
  • $\begingroup$ In B - is it possible the membership could be prorated if joining in the middle of the month? In that case it could be a continuous function and not isolated points. $\endgroup$
    – Amy B
    Apr 20 at 19:07
  • $\begingroup$ Basically you want to "graph" a sequence $u_n=f(n)$. Just draw the graph of $f$ and mark the points with integer abscissa with a red pen. $\endgroup$
    – user5402
    Apr 20 at 20:29
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In all of these examples, I would draw a line unless that idealization led to missing something essential in the problem. These idealizations happen all the time, and they tend to be very useful. E.g., for the flow of water, we normally don't need to consider that water consists of discrete molecules. Ditto for examples like immigration, fluctuations in the population of a certain species of flower, or photons hitting a solar cell.

On a philosophical level, number systems like the rational numbers, real numbers, and hyperreal numbers are all just ways of modeling reality. The model never corresponds perfectly with reality. Nobody has ever measured a number in real life that could be definitively classified as irrational or irrational, for example.

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  • $\begingroup$ “classified as irrational or irrational”? Clarify, please. $\endgroup$ Apr 20 at 10:33
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    $\begingroup$ @IncnisMrsi When you are holding a physical object in your hand, it is impossible to determine whether its length in centimeters is a rational number. $\endgroup$
    – Chris Cunningham
    Apr 20 at 12:32
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    $\begingroup$ @Incnis Mrsi: I think there is a typo and one of the appearances of "irrational" should be "rational". Thus, the intended meaning was probably "(definitively classified as irrational) OR (definitively classified as rational)". Note, by the way, that "definitively classified as (irrational OR rational)" is probably not in doubt (although I guess there might be a more subtle issue of whether the results of measurements are correctly modeled by real numbers, as opposed to, say, fuzzy real numbers or some kind of quantum wave function thingie). $\endgroup$ Apr 20 at 12:37
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I would do points, rather than a connected curve for A. Except if you are looking at large amounts, I wouldn't be so fussy and just draw a line. This actually has some use if you consider say different rates sold by a monopolist practicing price descrimination, for instance. (I think this is beyond your audience though.)

For B, if you are interested in total spend, a connected stepwise curve is good because it emphasizes the time aspect (i.e. you've still spent $40 halfway through the first month. If you are looking at the individual spend points, I quite like the "arrow" graphic that is used in engineering economics for project spend points (important for eventual discounting to NPV).

See here:

https://www.google.com/search?q=arrow%20graphic%20discounted%20cash%20flow&tbm=isch&tbs=rimg:CSyBEnHYY2UPYUhr0s4VTtuo

But again, this is really a bit beyond your audience. Really, I would not emphasize that stuff so much. Even if you are doing practical problems in commercial arithmetic, you can just solve the problems with linear (or whatever) approximations and then go in afterwards and either pick a set point one higher or lower (based on the logic of the situation). E.g. Number of men to dig a ditch by a deadline. If the answer comes out 7.3, than "8" is the right response. In other words, I would worry about other stuff more. But anyhow...here's a response to the question.

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