Quandaries
and Queries 

Hi, I hope you understand what I am trying to ask! Thanks! 

Hi Nicole, I think the confusion here arises from the fact that we are sometimes sloppy when we talk about functions. The sloppiness is that we write an expression for a function that describes how the function behaves, but we don't explicitly describe the domain of the function. We let the form of the function determine the domain. For example let and We usually simplify f(x) and write We then conclude that f(x) = g(x) but, in fact, this conclusion is incorrect. f(x) and g(x) are different functions since their domains are different. The domain of g(x) is all real numbers and the domain of f(x) is all real numbers except x = 2. The cancellation to get the last equality above is only valid if x is not 2. If x is 2 then the denominator is zero and we can't cancel it. The functions "behave the same" for all x where they both exist, but f(x) does not exist if x is 2. Also you say "if a rational function (by definition) has common factors in the numerator and the denominator then it is not a rational function". I prefer a definition that is inclusive. I would say that
This means, for example that 5 is a rational number since 5 can be written ^{5}/_{1}. Likewise I would say that
So, as above this means that g(x) = x + 2 is a rational function since it can be written Penny Hi Nicole, Section 4.4 of B. Thiessen's Math B30 text states, "For the purposes of our discussion we will define a rational function as a function that is formed by taking the ratio of two polynomial functions that have no common factor other than plus or minus 1." I believe that our Saskatchewan Math B30 course focuses on those rational functions that have no common factors in the numerator and denominator but this does not mean that a rational function with common factors isn't a rational function....the course simply has chosen to not use these kind of rational functions for the purposes of meeting the curriculum objectives. Perhaps this is to avoid overcomplicating the graphing procedure so students can focus on those graphs with asymptotic behaviour rather than those whose graphs have "holes" but would otherwise be continuous polynomial graphs. I hope this makes sense! Leeanne 

