Hetware wrote: >quasi wrote: >> Hetware wrote: >>> >>> So the answer is consensus among mathematicians holds that >>> F(t) = (t^2 - 9)/(t - 3) is undefined at t = 3? >> >> Yes. >> >>> Perhaps what I should have said at the outset is something >>> along the lines of: on any given day, if I'm setting up an >>> equation in physics, and produce an expression such as >>> F(t) = (t^2 - 9)/(t - 3), I treat it as t + 3. >> >> Then _define_ it as F(t) = t + 3. >> >> If simplifying (t^2 - 9)/(t - 3) to t + 3 is correct in the >> context of your application, then simplify it in advance before >> defining the function. >> >>> and do not expect any adverse consequence from doing so. >> >> If the physical context makes removable discontinuities >> impossible, then _remove_ them. Don't leave them there in >> the definition of the function. > >I am not familiar with a definition of /function/ which tells >me the order in which sub-expressions should be evaluated.
Let f be the function defined by
f(t) = (t - 3)/(t - 3)
In a Precalculus or Calculus context, the domain of f is the set of all real numbers for which the expression is defined.
For a particular value of t, f(t) is the value, if any, obtained by substituting that value of t into the expression. If after the substitution, the evaluation yields a result which is undefined, then that value of t is regarded as not in the domain of f.
Thus, f(3) = (3 - 3)/(3 - 3) = 0/0 which is undefined, so 3 is not in the domain of f.
The concept of function evaluation is _direct_ _substitution_.