Stranger Domains
You look west. You're having a nightmare. You find yourself teaching a college math course. Your coffee is cold and outside is cold and you're cold because the room is cold and you ask what few students are looking up "What is the domain?". That's when the most horrific thing happens:

"the x's.", they drone.
...and well, that's what they were told. (To be fair, it's not a total travesty, more of a consequence of us always making \( x \) "the variable".)
Recall from the lesson on functions that functions are rules that assign an object to another (single) object. The objects we are talking about live in the following three areas:
Recall from the lesson on functions that functions are rules that assign an object to another (single) object. The objects we are talking about live in the following three areas:
THE DOMAINThe domain of a function is the set of objects we pick from to apply the rule to. If the rule doesn't make sense with an object, it is not in the domain, and we says the rule is "undefined" with this object.
Example: If I'm assigning students to desks, I cannot assign a rather large bird to a desk. Rather large birds are not students (hygiene), and so the rule doesn't make sense with them. R.L.B.s are not in my domain. (UNDEFINED). 

Mathematically, we write down these notions in the following way:
\[ function name : domain \to codomain \]
\[ function name : domain \to codomain \]
Consider the function \( f(x) = \frac{1}{x} \).
DOMAIN: My rule is division by \( x \), and division makes sense as long as I'm dividing by a number that isn't zero, thus my domain is any nonzero number. CODOMAIN: When I divide 1 by a real number, I get a real number. Thus my codomain is the real numbers. (I'm not expecting to divide 1 by something and get banana.) RANGE: When I divide 1 by a real number, I cannot possibly get zero, so even though zero is in the codomain, it is not in the range (the set of numbers I can actually obtain through this rule). We would write: \[ f : \mathbb{R}/ \left\{0 \right\} \to \mathbb{R} \] 
Consider you own a Dog Wash, and your function makes dirty dogs clean. (Yes I am writing a hit R&B song right now).
DOMAIN: My rule makes dirty dogs clean, so that certainly my domain can't have cats. I could certainly wash a clean dog again (I'm not paying for it), so it is evident that my domain is "DOGS". CODOMAIN: This is also dogs! After a dog comes out of my dog wash I hope it hasn't become a reclining sofa. RANGE: If my domain is all dogs, my range this time will also be all dogs. In other words, I can't put in all kinds of dogs and get out only St. Bernards (otherwise some shady stuff is going on in there). We would write: \[ dogwash : dogs \to dogs \] 