How do you formally denote a family as a function? Is there any way to speak of a family by denoting it as a function? In that case, what denotation could we use? And after denoting the said family in the form of a traditional function, say ƒ for now, would I be right if I said that {xᵢ} = ranƒ?
 A: The word "family" has multiple interpretations. Anyway, a family is some "collection of objects". I think you understand it as an   indexed family of objects. These objects are elements of some set $X$. An indexed family is formally defined as a function $f : I \to X$. Here $I$ is the index set. With $x_i = f(i)$ one often writes $\{x_i\}$ or $\{x_i\}_{i \in I}$ for indexed families. Personally I do not like this notation because it uses the set braces $\{ \quad \}$ which suggests that an indexed family is itself a set. But this is not true, it contains more information than the image of the index function $f$. If we only know $f(I) = \{ x_i \mid i \in I\}$, we can neither reconstruct the index set $I$ nor the index function $f$. In particular $f$ can have the same value for more than one index, i.e. in an indexed family we can have repetitions.
So yes, you can consider the range $\operatorname{ran}(f)$ of $f$ which is nothing else than $f(I)$ and get $\operatorname{ran}(f) = \{ x_i \} = \{ x_i \mid i \in I\}$.
Note that each set $M$ can be regarded as an indexed family by "self-indexing". Simply take $I = M$ and $f = id : M \to M$.
Using indexed families is unnecessary in many cases, one could work simply with sets of objects. This means that often the range set $\{ x_i \} $ is completely sufficient for a given purpose.
