The range is the collection of all function outputs that result from a given domain of inputs. In this case, if collect all the results of f(−2), f(3) and all the values of x in between, we've collected the range.
Remember from the graph of x2 that it has a minimum at x=0 and increases as you increase or decrease x from there. The same is the case with x2−25. The minimal value it can take is −25, which it takes precisely when x=0. Zero is in our given domain, so we know that the minimum value of the range is −25.
To find the maximum, it suffices to plug in the endpoints, since we know f(x) is increasing as we get more distant from x=0. We have f(−2)=4−25=−21 and f(3)=9−25=−16. The greater value occurs at f(3)=−16.
Since our minimum output is −25 and our maximum is −16, and we hit every value in between, our range is [−25,−16].
