Graphing Piecewise Function

[nycmath]

Precalculus
Michael Sullivan
Edition 10
Page 88, Example 3

Before graphing calculators and the internet, graphing piecewise functions by hand was a task for most students. Let's pretend we are back to a period in history without graphing calculators and Desmos. How do we graph piecewise functions by hand?

Sullivan provides example 3 on page 88 and the steps to graph by hand on page 89. I am not taking anything away from Sullivan but his explanation for graphing f on page 89 makes no sense to me and probably to many others as well. I want you to read what Sullivan said on page 89. See attachment.

A. Can you explain what Sullivan is talking about?
B. How do we graph piecewise functions by hand?

Graphing one "piece" of a function at a time is challenging but also fun. Does anyone here know a better way?

[Eigenvalue]

Here is my work for the question:

[nycmath]

Here is my work for the question:

Very good. Why is there a hole at the points (1, 1) and (1, -1)?