# Concept clarification regarding functions [duplicate]

I'm currently learning about functions and I need a clarification regarding some basic concepts since every book I read seems to change something here and there.

The concepts of domain and codomain are quite clear and I have no issues about them. I have issues understanding the range and the image set.

First, I used to believe the range and the image set were much like the same thing. Now, after more reading it seems that range and image set are not exactly the same thing.

From what I have understood, the range is the image set of the domain. But I can have image sets of subsets of the domain (not equal to the domain) which will not be the range but just image sets of these subsets. Saying the same but differently, the range is always an image set, but an image set does not necessarily has to be the range.

My question is simple, is this understanding of the topic correct?

Typically we don't say "image set", but rather the "image of the set $$S$$ under $$f$$", which we may also write as $$f(S)$$.