Proper definition of a function I'm sorry if this is a pedantic question, but I want to be sure I'm using terminology correctly.
Without a second thought, I would make statements of the following form: "Consider the function $f: A \to B$ defined by $f(a) = b$."
This isn't fully correct, however, because the full definition of the function $f$ should be the domain, codomain, and the rule. Without one, the definition is ambiguous.
Am I incorrect on this, or is this a commonly accepted shorthand? Would it make more sense to say "consider the function $f: A \to B$ given by" or "governed by"?

Saying "given by", "verifying", "satisfying", etc. is more suitable for defining functions using words. Otherwise, remember that you can also use the notation:
\begin{align*}
f: & A  \longrightarrow  B \\
& a  \longmapsto f(a)=b
\end{align*}
And we'll say it's well defined if any point $a\in A$ has an image through $f$ that satisfies $f(a)\in B$, and also that it gives no contradiction, to be said: $a=a'$ implies $f(a)=f(a')$. If that's not verified, we say it's not well defined, and you may better change it's definition to make clear where does $f$ have sense. For example, the function
\begin{align*}
f: & \mathbb{R}  \longrightarrow  \mathbb{R}_+ \\
& x  \longmapsto f(x)=1/x
\end{align*}
isn't really well defined, since $x=0$ has no image, and not al images from points in $\mathbb{R}$ are positive, so a right definition of it will be
\begin{align*}
f:  \mathbb{R}\setminus\{0\} & \longrightarrow  \mathbb{R} \\
 x \ \ \ & \longmapsto  \ f(x)=1/x
\end{align*}
though sometimes domain and codomain specifications are omitted because they can be deduced by the rule or the context
