Common Core Standards: Math
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
Domain is to a function as trails are to a mountain. Certain trails get to certain places on the mountaintop, just like certain inputs can give you certain outputs for the function. The domain of a mountain is the set of all the possible trails we can take. Mathematically, the domain (n) of a function f(n), is the collection of all the different x values we can input into our function.
You could arguably say that any function can take any domain. After all, the x-axis does extend to infinity and beyond.
But we all know there aren't an infinite number of marked trails on the mountain. There may be an infinite number of ways to get to the top of the mountain (bungee jumping and teleportation come to mind), but there are limitations as to how many trails we can have. For instance, we need a whole number of trails because half a trail can't get us anywhere. Not to mention the negative numbers (what would a negative number of trails mean?).
So, here we are, trying to make our way up the metaphorical mountain, and our domain of trails shrunk from infinity down to positive whole numbers, probably not exceeding half a dozen or so. While this is a very simplistic way to look at domain, it's the right idea to get your students thinking about it. You may need to scale it up for more complex problems, though.
Although there are some functions whose domains are limited (like f(x) = 1⁄x, where x ≠ 0), many domains are restricted because they don't make sense in terms of a particular context (like having a negative number of trails that lead to a mountaintop). Students should be able to determine the appropriate domains based on the context of the problem.
Hopefully faster than it takes to climb that mountain.