I mostly tutor students ranging from beginning algebra to calculus level. I think the explaining the range as "the set of all possible outputs" would not really cut it for someone struggling with math at that level. Usually I default to:
"Draw a rough sketch of the graph, including all the shifts, stretches, etc. The range is all the portions on the $y$ axis which have a piece of the graph to the left/right of them."
I feel that not only is this explanation very good, but many times when I ask the students to draw a rough sketch of the graph, they have no idea how; so now I have to reteach the student how to graph equations. However, I am not a private tutor; I work for my school, and I am not allowed to spend more than $15-30$ minutes with a student at a time, so I do not have time (nor want to) reteach the student half a chapter just so they can solve one problem. How can I explain the range in a better and less time-consuming way?