Injections, Etc. 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.)

This discussion gives you a chance to practice the main proof styles for proving that functions are injections and surjections. It is based on Beginning Activity 2, part 3, from section 6.3 of our textbook.

To start, discuss how you might solve part 3 of Beginning Activity 2 (proving certain properties of the function g : ℝ → ℝ defined by g(x) = 5x + 3). Things you might talk about include how you would actually complete the proofs begun in the activity, but also what you are proving about g as a result. And of course asking and answering questions about what others are suggesting is a good way to contribute too.

If you get tired of talking about g, consider the similar function f : ℝ → ℝ defined by f(x) = 5x² + 3. Can you do proofs similar to those in the book’s activity about this function? Why or why not? What does your success or failure tell you about f?

Finally, in class on Friday we conjectured that if functions h and k can be composed, it doesn’t have to be the case that range(h○k) = range(h). See if you can come up with a counter-example to demonstrate this, i.e., an example that shows that the claim that for all composable functions h and k, range(h○k) = range(h) is false.