MHB Anatomy of piece-wise functions

[samir]

Hi!

I'm looking at some piece-wise function right now and I can't help but wonder what all these parts are called. I'm learning to use and write this type of functions now and I think I have a pretty good understanding of how they work. I even took the extra step of learning some "set builder notation" and inevitably learned to use and bend some basic "well defined" sets for my need and use them in my expressions. This is not required in my math class, but it's a great deal of fun! It immediately came to a good use in my work. Bear in mind that I am still just a beginner at this. What I'm looking to learn now is some basic terminology really.

Here is the function I'm referring to.

$$f(x)=\begin{cases}\frac{x+4}{x}, & x\leq 2 \\ x^{3}+1, & x \gt 2 \end{cases}$$

• What is the function name? If it's anything I have learned it's that $f$ is the name of the function. So I know the answer to this one myself. I still wanted to post it here because this thread is supposed to be the "anatomy of piece-wise functions". We can't afford loose ends.
  
• What is the function rule? Is everything on the right hand side of the equality sign the function rule?
  
• What does the left brace represent? Does it have any special meaning? Like the opening brace of a set? Why is there no right brace then? So it's a no then?... it's not comparable to set notation?
  
• What do we call the expressions that go to the right of the brace? Are they function rules? Are they "sub-rules"? I have seen referred to them as "branches". Is this correct? Or is this only relevant when we have graph of the function?
  
• What do we call the relations that sit to the right of each expression? Are they conditions? Are they arguments? What are they? Do we have a proper term for them?
  
• What do we call the point at which we change our attention from one expression or "sub-rule" to the other? That is, what do we call $x=2$ in this example? Can we call it critical point? I feel like it should be given a name because it is a very interesting point. It entirely changes our perspective on the entire function!

So if someone could fill me in on these, please do.

This is how I would call these things.

Function name: $f$

Function rule: $\begin{cases}\frac{x+4}{x}, & x\leq 2 \\ x^{3}+1, & x \gt 2 \end{cases}$

Function sub-rule 1: $\frac{x+4}{x}$

Function sub-rule 2: $x^{3}+1$

Sub-rule 1 condition (or condition 1): $x\leq 2$

Sub-rule 2 condition (or condition 2): $x \gt 2$

Critical point: $x=2$

 

[Bushy]

Sub-rule 1 condition (or condition 1): $x\leq 2$

Sub-rule 2 condition (or condition 2): $x \gt 2$

call these inequality

 

[Deveno]

As you may recall, it turns out that functions are a special kind of set, a subset of the cartesian product of two sets. The first set is called the domain, and the second set is called the co-domain.

So if our domain is $A$, and our co-domain is $B$, we have:

$f = \{(a,b) \in A \times B: b = f(a)\}$.

This isn't "quite right", because it's a "circular definition", we are defining $f$, in terms of $f$. If we have a definite rule, we can avoid this-like we can define the "squaring function" as:

$f = \{(x,y) \in \Bbb R \times \Bbb R: y = x^2\}$.

In general, in the absence of a definite "rule", we have to define a function like so:

A function $f$ is a subset of $A \times B$ such that if $(a_1,b_1)$ and $(a_1,b_2)$ are in $f$, then $b_1 = b_2$.

The important thing to take away from this, really, is that a function $f:A \to B$ is a subset of $A \times B$.

Now, suppose we have two functions:

$f_1 : A_1 \to B$
$f_2: A_2 \to B$.

Suppose further that $A_1 \cup A_2 = A$, and $A_1 \cap A_2 = \emptyset$.

Then it can be proven that $f = f_1 \cup f_2$ is a function $f: A \to B$ (this is called "pasting together disjoint domains", or "piecewise definition").

Typically, the things you refer to as "conditions" are called *sub-domains* (the domains of $f_1$ and $f_2$ in my example are subsets of the domain of $f$). We want these to be disjoint, or else our function may be "ill-defined" (we can't decide on a SINGLE value for $f(x)$ when $x$ lies in two different sub-domains that overlap, unless the rules happen to agree on that overlap (intersection)-very unlikely).

Usually, we want the sub-domains to be disjoint *intervals* (this goes back to topological considerations, which unfortunately are too involved to get into, here-suffice to say, the internal structure of the real numbers is "rich"). This allows us to "stitch together" locally continuous functions, that may have some jump discontinuities in them. Even with said discontinuities, we can still sometimes do things that "make sense" like calculate the area under a discontinuous curve.

As you might well have guessed, the sub-domains give us "sub-functions". Calling them "branches" is a nice term-I'd keep using that if I were you.

The left brace signifies an array follows. It's purely a formatting convention.

Unfortunately, critical point already has another definition in common use-I would call it an "exceptional" point myself, but I don't think this is standard terminology.