If you have a function that describes something that can only be positive - for example, the number of items produced - is it correct to say that input values that result in negative output values are not in the domain of the function?  I've heard this before, but it seems a bit odd to me, since I've always thought that the domain was the set of values for which the function is defined, not those for which the function will produce an output that is sensible in the context in which the function is being used.

Yes.  Note that the function in your example is not defined for f(x) < 0. The domain is then x ≥ 0.

This is a matter of context. Mathematically, your definition is correct, but things get a little hazier when describing real processes. Saying it is out of the "domain" is not exactly correct, but you know what they mean. Restricting a function to certain inputs is perfectly normal.

I'm not sure that it's this simple... if a function is not defined when f(x) < 0, then clearly it is not defined for values of x that result in f(x) < 0.  But are the types of functions I'm talking about not defined for negative values of f(x)?  Everything that I have seen says that the domain of a a function excludes only those values of x for which the function does not produce a real number, or is otherwise not defined; for example, some piecewise functions.  But in my example, nothing in the function itself excludes negative values of f(x); it is only our desire to use that function for a practical application that does so.



Also, what if we have a function that consists of only positive exponents, and no negative constants, and for which our application dictates that the only sensible outputs are non-negative?  Then, even a negative value of x will result in a zero or positive value of f(x).



 

The keyword is "defined".  The limit is placed on an applicable general function and becomes part of the definition of the applicable particular function for that particular application only.   So, one of your original exmples requires a range = f(x) ≥ 0 and a domain of x ≥ 0.  You could also use absolute values in the definition.  

I don't understand your "also ?".  The equation is being defined(although not clearly) to model some process.  f(x)=Ae^(bx) ?  The domain is ± ∞ and A is restricted to A ≥ 0.  ?

OK thanks, it's starting to sink in and make sense now.  





The second paragraph was just an illustration that negative values of x could result in values of f(x) that are in the allowable range; it's not really relevant to my question.





 

Wow.  So wrong.  Much sadness.

Counterexample: f(x) = 1

OP, the domain of a function can be almost whatever you want it to be.

For example, f(x)=x and domain = {1,2,3}.

Then for all values of x != 1,2, or 3, my function is not defined.

Note, that if I actually plugged in a value for x, I would still get an output, but that output is not an output of my function as defined.

It is common in some applications to ignore irrelevant parts of the domain for your functions.  For example, position of a thrown baseball after time 0. There is no negative time, so our relevant domain is D = {x|x>=0}.

There are also functions that have potential inputs that would result in no outputs, and so their domains must exclude those inputs.  

For example, f(x) = 1/x. This function cannot have an input of 0, and therefore its domain cannot have 0 in it.

OP, back to the counterexample I started this post with, and relevant to your second-to-last post, a function like f(x) = e^x will result in f(x) > 0 for all x.  Same with f(x) = e^(-x). Just like f(x) = 1 in that f(x) > 0 for all x.

Ridiculous.  The example is the number of items produced, which must be zero or positive and the domain is resources consumed.  OP asked if it was correct to restrict the domain to those values which result in realistic output values, 0 or positive numbers.  The answer is yes.

Your post implies the domain is positive because the range is positive.

That's not true.

The domain is zero or positive, because the resources to make an item must be zero or positive.  One can not make something with or from negative resources or ingredients.