Rational functions questions and answers

Recent questions in Rational functions
Mackenzie Rios 2022-05-20 Answered

Find the domain and graph:
f ( t ) = t | t |
My book says to define it piecewise.
My questions:
1) Do all rational functions have to be defined piecewise, or just this one because there is an absolute value in the denominator, and the absolute value is always defined piecewise?
2) Are all rational functions defined piecewise in order to avoid having a denominator be equal to zero, is that the general reason for defining anything piecewise, to avoid having division by zero? (so then this would mean that the domain of a rational function is always defined piecewise, or only when we need to avoid having denominator be =0?)
This is how my book defines f(t) piecewise:
If t > 0, then | t | is t since t is already positive. For t > 0, simplify
f ( t ) = t | t | = t t = 1
If t < 0, then | t | is t since t is negative. For t < 0, simplify
f ( t ) = t | t | = t t = 1
(or for the last part, should the negative be inside the absolute value sign for t < 0, as in | t | instead of | t | ?)
So we define it piecewise to avoid having 0 in the denominator? Because isn't absolute value defined at 0, the absolute value is continuous everywhere, and thus defined at 0?
I'm confused about defining things piecewise, and how to know when to apply a piecewise attempt in order to define a function's domain.
Also, I'm confused about the second part of this, where t < 0 for | t | . How is it that here, | t | is t if absolute value is always positive?
Maybe I'm not understanding the absolute value concept correctly, because in grade school it has always been drilled into my head that |absolute value| just "turns things positive", so here I don't really understand how it can be negative.
I understand how the domain is ( , 0 ) ( 0 , ), because by using open intervals we're not letting it be exactly =0, but I'm confused about the whole piecewise thing.