Function, domain

topsquark

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reference: https://www.freemathhelp.com/forum/threads/consider-y-−√x-is-y-a-function-of-x-why-or-why-not-is-it-true-that-y-is-always-less-than-x-show-that-it-is-or-give-a-counter-example.118955/#post-472024

\(\displaystyle y=-\sqrt{x}\) is not a function on the set of real numbers.
I feel like we're being unnecessarily picky here. There is no subset of the real numbers where the vertical line test actually fails in this example. A function is still a function on the real numbers even if there are subsets of real numbers that the function cannot be evaluated at. For example, f(x) = 1/x is considered to be a function on the set of real numbers, even if it cannot be evaluated at x = 0. Right? At least I've never run into anyone who has said that 1/x is not a function over the reals.

-Dan
 

pka

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I feel like we're being unnecessarily picky here. There is no subset of the real numbers where the vertical line test actually fails in this example. A function is still a function on the real numbers even if there are subsets of real numbers that the function cannot be evaluated at. For example, f(x) = 1/x is considered to be a function on the set of real numbers, even if it cannot be evaluated at x = 0. Right? At least I've never run into anyone who has said that 1/x is not a function over the reals.
Oh my Lord Dan. Are you now part of the ill-prepared mathematics-education community? Gee, I hope not.
Please read my reply #16. Not knowing the exact definitions does sink many students of national tests.
Please note that I am not blaming the students in all cases, but it is the fault of the MathEd community.
People like me work of the testing companies. We expect correct application of fundamental concepts such function.
 

Dr.Peterson

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It's true that the full definition of a function includes its domain and codomain. But at this level, the emphasis is on having one value for any (valid) input; beginning students have enough trouble understanding that central concept, without getting tangled up in less essential details.

In this case, the question didn't ask about being a function over the reals! We can interpret it, "Is this a function [over some domain that is a subset of the reals]", and the answer there is simple: yes! Bringing in issues of domain at this point doesn't help the student. There is time later in the course (or a subsequent course) to formalize everything.

An "incomplete" question is not "absolutely meaningless". It's just inadequate for an upper-level math course, which this isn't.
 

topsquark

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Oh my Lord Dan. Are you now part of the ill-prepared mathematics-education community?
I did read post #16. Apparently I missed something. (I still am, I think.)

Hey! I'm being a Physicist over here! I have been thinking about it overnight and I think that it is true that the domain of 1/x, not being the whole of the real numbers, should not be refered to as "over the reals." But I still don't recall anyone actually telling me that this doesn't make it a "function." (Then again we don't talk much about domains and ranges in Physics, either.)

Thanks for the correction!

-Dan
 
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