Range: can mean codomain or image (don't use the word range)
Image: Let A and B be sets and f:A\rightarrow B be a function. Let X \subseteq A.
Definition: image of X under f is Im_f(X)=\{b\in B | (\exists a \in X)(f(a) = b)\}
When X = A, image of f is Im_f(A) \subseteq B
(\forall X\subseteq A)(Im_f(X) \subseteq B)
think of Im_f: \mathcal{P}(A) \rightarrow \mathcal{P}(B)
Preimage: Let A and B be sets and f:A\rightarrow B be a function. Let Y \subseteq B.
Definition: preimage of Y under f is PreIm_f(Y)=\{a\in A | f(a) \in Y)\} \subseteq A
When Y = B, preimage of f is PreIm_f(B) \subseteq A
think of PreIm_f: \mathcal{P}(B) \rightarrow \mathcal{P}(A)
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