1. The answer you have provided to your own question #1 is not the answer to the question you have asked. You have explained a procedural step to find the composite function f(g(x)) (when it exists) when f(x) and g(x) are given. But your question (which is not clearly stated), we think, is about the conditions under which a composite function exists. Assuming it is “What are the conditions required for f(g(x)) to exist?”, the answer is as follows:

          Conditions for f(g(x)) to exist are,

          * the range of g(x) must be a subset of the domain of f(x)

                Or

           *f(g(x)) must be defined with the domain of g(x) restricted so that its range is a subset of the domain of f(x).

     2. This has been already answerd in the response to #1.

     3. This question makes no sense. (what’s k?)

~MetaHub Panel for the Teachers’ Forum

What you should be teaching your students is not that “the sum of rational numbers is rational”, but, “the sum of two rational numbers is rational” which you can extend, using the principle of mathematical induction, to “the sum of a *finite* number of rational numbers is rational”.

It should be clear that this claim cannot be extended to *infinite* sums blindly. In fact, by definition, an infinite sum is actually the limit of a finite sum (partial sums).

“e” is a counterexample for this. The infinite sum of rational numbers need not necessarily be a rational number. 

~MetaHub Panel for Teachers’ Forum

You are most welcome Sujeewani!

This collection is the same as the collection of odd numbers. For grade 6, according to their syllabus content, it will be suitable for their to grade level to say that the smallest such positive integer is 1. That is, the smallest positive odd number.

Keep sending your questions or doubts in teaching. We are delighted to help! 🙂