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

1 Answer
May 2, 2018

Answer:

Range: #[-25, -16]#

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(-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]#.