Quick Question on Pre-image Terminology Sorry for the daft question, but, is the following a correct thing to say?
"The preimage of a function f is a function iff for any element b in the range, there exists exactly one a in the domain such that f(a) = b"
"A function is injective iff its pre-image is a function."
Thank you.
 A: Good question; terminology is confusing.  It seems that you are using the term "preimage" incorrectly.
Suppose $f:A \to B$ is a function.  For some set $D\subseteq B$, we say that $f^{-1}(D)$ is the preimage of $B$ under $f$.
The relation $f^{-1}: B \to A$ given by $f^{-1}(b) = \{a \in A: f(a) = b\}$ can be called the inverse (relation) of $f$.  So, to restate what you meant correctly:
"The inverse relation of a function $f$ is a function iff for any $b$ in the (image), there exists exactly one $a$ such that $f(a) = b$"
"A function is injective iff its inverse relation is a function".
A: You are misusing the term "preimage". I think you mean "inverse". If $f:X\rightarrow Y$ and $U\subset Y$, then the preimage of $U$ (under $f$) is the set $f^{-1}(U)=\{x\in X:f(x)\in U\}$.
If you replace the word "preimage" with the word "inverse", the your statements are correct. I presume that by "range" you mean "image" (in my example, this would be the set $f(X)\subset Y$, which is not necessarily all of $Y$ unless $f$ is also surjective).
