To Be or Not to Be...
As I entered a room to teach my next mathematics class, I discovered an interesting note on the blackboard:
f(x) = ln(x-3) - ln(2-x)
x-3>0 and 2-x>0
x>3 and x<2
Domain of f(x) is the empty set
Thus, f(x) is not a function!
My mind began to spin. Is it true that a function must have a non-empty domain? Could this f(x) be an example of a degenerate function?
I searched the definition of a function in mutiple texts, but none mentioned the possibility of an empty domain. I discussed the idea with other math teachers, and all first said "no, it is not a function" and later called me to recant their statement.
The ideas of several math teachers combined to produce this argument as to why f(x) has to be a function:
Suppose f(x) is not a function. Define g(x) = ln(x-3) + ln(2-x), which would not be a function for the same reason (i.e. empty domain). Then consider f(x) + g(x) = 2 ln(x-3), which is a function. So, our assumption about f(x) and g(x) must be wrong... otherwise something has been magically created out of nothing....but wait, isn't this the same as using two "untrue" algebraic statements to "falsely" create a "true" algebraic statement? So back to the drawing boards.....I welcome anyone's "proofs"....
Oh, by the way, the function f(x) is also 1-1 and onto....but the jury is still out as to whether or not it is differentiable (i.e. applying the definition is similar to looking for unicorns).