Functions Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Sections 6.1 and 6.2 of our textbook introduce functions, an idea that you have most likely seen before, at least at some level. What you may not have seen in those sections is some of the terminology and notation associated with functions. In this discussion, I hope to reinforce some of that terminology and notation, while extending it to include function composition (an idea not addressed yet in the textbook, but which you have probably encountered before — if nothing else, you saw it when you learned about the chain rule in calculus, since the chain rule exists to find derivatives of compositions of functions).

To start, see if you can define what “function composition” is, give some examples of it, etc.

Then discuss, i.e., suggest ideas, ask questions about, etc., how the textbook’s ideas about functions apply to compositions of functions. For example, what does composing two functions require of or do to their domains, codomains, and ranges? What would an arrow diagram of a composition of functions look like? Does composition make sense for functions defined as sequences, graphs, or other alternative forms the book mentions? What do compositions look like in those settings? Other interesting questions or thoughts that occur to you?