Functions

• high school edition: a function takes in a real number and outputs a real number
  • except when there are vertical asymptotes or other discontinuities
• in programming, functions
  • might take in inputs
  • might return values
  • might have side effects
  • might never return anything
  • might crash
  • might return different values when called multiple times
• high level intuition: a function $f$ takes in one thing $x$ and produces exactly one output $f(x)$.
  • in mathematics, functions are deterministic. just a simplifying assumption to make functions easier to work with.

Domains and Codomains

• a function $f$ can only be applied to elements of its domain
• for any $x$ in the domain, $f(x)$ belongs to the codomain.
• asymmetry
  • the function must be defined for every element of the domain
  • however, the output of the function. must always be in the codomain, but not all elements of the codomain must be produced as outputs.
• example: $f(x) = |x|$
  • domain is $\mathbb R$, codomain is also $\mathbb R$. Everything produced is a real number, but not all real numbers can be produced.
• notation for a function whose domain is $A$ and whose codomain is $B$, we write $f:A\to B$.
  • think of this like a “function prototype” in C++.
  • the term range is ambiguous: it could mean codomain or image (set of things that can be produced by $f$.) therefore, this term will NOT be used in this class.

Official Rules for Functions

• Formally speaking, we say that $f:A\to B$ if the following two rules hold:

1. $f$ must obey its domain/codomain rules:

$$ \forall a \in A.~\exists b \in B.~ f(a) = b $$

1. $f$ must be deterministic:

$$ \forall a_1 \in A.~\forall a_2 \in A.~(a_1=a_2 \to f(a_1)=f(a_2)) $$

• can a function have an empty domain (yes!), can a function have an empty codomain (no!)

defining functions

• specify the domain, specify the codomain, and give a rule used to evaluate the function.
• e.g. $f: \mathbb Z \to \mathbb Z$, where $f(x) = x^2 +3x-15$