On Sunday, September 29, 2013 4:57:12 PM UTC-5, Hetware wrote:
> That is certainly not contained in any definition of /function/ I'm > > familiar with. A real valued function of real numbers is a set of > > ordered pairs of real numbers. The first number is often referred to as > > a member of the domain, whereas the second as a member of the range.
This is a perfectly good definition for a function.
The question then is: "What does "f(t) = (t^2-9)/(t-3) mean?"
It's certainly not a set of ordered pairs of real numbers, so it cannot be a function. It seems to be an expression or an equation of some sort, but it is certainly not a set of pairs of real numbers.
So then the question is: how do we interpret this expression as *describing* a set of ordered pairs of real numbers, i.e., a function?
The answer is that we interpret this as determining a set of ordered pairs as follows: the pair (a,b) belongs to the function f if and only if when we substitute a for t in the expression, the result of evaluating the resulting expression is b. This is the standard meaning and interpretation of this expression as describing a set of ordered pairs.
That requires us to substitute a in the expression *as is*. If this substitution results in something that cannot be evaluated for whatever reason, then that means that there is no pair with first component equal to a in the function f (that is, in the set that describes a).
The reason we don't want to interpret it as "if the the expression makes not sense as written but there is an alternative candidate then use the alternative candidate" is that there will be situation when we *don't* want that. Rather, we should do just like in a computer program: we do what we are told, nothing more and nothing less, and don't try to guess.
In the expression (t^2-9)/(t-3), substitution 3 for t yields an expression that makes no sense. Rather than try to guess what the answer "should" be, we do just like a computer program does, and simply say that the expression makes no sense.
> Can this equation be solved for t?
But that is not the question that is asked when you are describing a function. If a function is a set of ordered pairs, then "solving equations" is irrelevant to that set.