How do you find the range of the function y=f(x)=x^2−25 on the domain −2≤x≤3?

May 2, 2018

Range: $\left[- 25 , - 16\right]$

Explanation:

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 $f \left(- 2\right)$, $f \left(3\right)$ 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 $f \left(x\right)$ is increasing as we get more distant from $x = 0$. We have $f \left(- 2\right) = 4 - 25 = - 21$ and $f \left(3\right) = 9 - 25 = - 16$. The greater value occurs at $f \left(3\right) = - 16$.

Since our minimum output is $- 25$ and our maximum is $- 16$, and we hit every value in between, our range is $\left[- 25 , - 16\right]$.