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

\[f(x) = \begin{cases} x^2 & \mathrm{if\ } x < 0 \\ x & \mathrm{if\ } 0 \le x \le 1 \\ 2x - 1 & \mathrm{if\ } 1 < x \end{cases}\]

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