Help finding a function with a restricted domain and co domain

[Ash zak]

The problem says:

Give an example of a function with domain {100,101,...,200} and co domain {0,1,...,9} who's "rule"is not immediately discernible.

My first thought was to do the floor function of (2*x-200)/2 but that only works if the domain is {100,101,...,109} I have been trying to figure this out for over 2 hours. even a hint towards the right direction will be highly appreciated.

 

[Dr.Peterson]

A function doesn't have to have a "mathematical" rule. You could just list the domain elements, and put a randomo codomain element next to each of them, and you will have defined a function! I don't know if that is what is expected or not; the "rule" will not be discernible at all!

Do you know anything about the remainder (modulus) function? You could put just about anything inside that and get a "pseudorandom" function.

What is the context of your question? What have you learned besides the floor function?

 

[Ash zak]

Thanks for replying to my thread. Unfortunately I don't know anything about the remainder (modulus) function.
The course that i'm taking is a basic concepts course for secondary school teachers.

When you mention that I should list the domain and put a random codomain element next to it. I understand that it is considered a function. But I think my professor wants me to be able to show how I managed to get from one input in the domain to a the certain output I randomly put next to it. I hope that makes sense.

 

[Steven G]

The problem says:

Give an example of a function with domain {100,101,...,200} and co domain {0,1,...,9} who's "rule"is not immediately discernible.

My first thought was to do the floor function of (2*x-200)/2 but that only works if the domain is {100,101,...,109} I have been trying to figure this out for over 2 hours. even a hint towards the right direction will be highly appreciated.

1st you should realize that (2*x-200)/2= x-100. Do you see that? You said that you want the function to be the floor of x-100, let's write that as [x-100].
Now if we plug in 100, we get back 0, if we plug in 200, we get back 100. Hmm, we have the function going from 0 to 100, but we only want the function to go from 0 to 9. Maybe we should let f(x) = (9/100)[x-100]. Will this work???

Spending two hours on this was not necessary at all as Dr Peterson suggested to define the function for each number in the domain.
Please show us your work, that is give us x values so that f(x) =0, and f(x)=1, ..., f(x) = 9


If this was a test, then to be 100% sure that there is no way I could possibly be wrong I would use Dr Peterson's suggestion (which was my 1st thought). If there are no major loses for being wrong then I too would would have gone to the floor function.

 

[Dr.Peterson]

The reason I didn't suggest a small modification to your floor function idea is that it is immediately discernible.

Although you don't know the modulus function, you do know what a remainder is. What I would do is to start with any function you like (maybe y = x^2, so it's not linear) and then use the remainder after dividing by 10 -- that is, the last digit. Then the first few function values would be 0, 1, 4, 9, so you might call that discernible (or guessable); then you might change to a slightly more complicated starting function.

I strongly suspect, because of the codomain and because of your context, that something like this is what was intended.

 

[JeffM]

I HATE problems like this. "Not immediately discernible" is a subjective criterion. Those who have no prior knowledge of the floor function will certainly find any rule dependent on it to be not immediately discernible whereas those who do have prior knowledge may make a different finding.

 

[Dr.Peterson]

Agreed. It's a totally unfair problem, as it is not at all clear what sort of answer will be considered "correct", or even, as was asked, what is the "right direction". It's a very open-ended problem with no hint of an objective goal, and therefore very frustrating.

Given the context, I would have thought that the random idea could be just what was wanted (as it demonstrated what "function" really means, which is a "basic concept"); but maybe if we saw the chapter or other material that has been covered, we could make a better guess. It would still, however, be a guess.

On the other hand, if anyone saw the outputs of a function being "0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, ...", they might immediately see the pattern and be able to replicate it without needing the term "floor" (or even "rounded down"). I don't know whether "rule" has been defined as needing an algebraic expression (it shouldn't!), or what.