Proof by Cases 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.)

It’s often convenient to split a proof into multiple cases, so that you can use slightly different reasoning in each case. Section 3.4 of our textbook explores this idea. This discussion provides a chance to practice the idea in a slightly different context than the ones explored in the book.

In particular, consider the function

$$f(x)=\{\begin{array}{ll}{x}^2& \mathrm{if}x<0\\ x& \mathrm{if}0\le x\le 1\\ 2x-1& \mathrm{if}1<x\end{array}$$

Discuss how you would prove that for all real numbers x, f(x) ≥ 0.