seguitzla

2022-10-08

How do you find the range of the function y=f(x)=x^{2}-25 on the domain $-2\le x\le 3$?

### Answer & Explanation

aveanoon

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 ${x}^{2}$ that it has a minimum at x=0 and increases as you increase or decrease x from there. The same is the case with ${x}^{2}-25$. 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].

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