Question

Angelica is working with function machines. She has the two machines shown at right. She wants to put them in order so that the output of the first machine becomes the input of the second. She wants to use a beginning input of 6.
a. In what order must she put the machines to get a final output of 5?
b. Is it possible for her to find an input that will get a final output of -5? If so, show how she could do that. If not, explain why not.

Answer 1

Given the two functions as $y = \sqrt{x-5}$ and $y = x^2 - 6$.
We can rewrite the functions as $f (x) = \sqrt{x-5}$ and $g (x)=x^2 - 6$

To arrange the functions such that the output of the first machine becomes the input of the second, we have f ( g (x) ) or g ( f(x) )
$f ( g (x) )= \sqrt{x^2-6-5} = \sqrt{x^2-11}$
and
$g ( f (x) )=( \sqrt{x-5} )^2-6=x-5-6=x-11$

Given an input of 6,
$f ( g (6) )=\sqrt{6^2-11} =\sqrt{36-11}=\sqrt{25}=5$
and
$g ( f (6) )=x-11=6-11=-5$

Therefore, to get a final output of 5, she will put the function machine with the function $y=x^2-6$ as the first machine and the function machine with the function $y=\sqrt{x-5}$ as the second machine.


b. Using the same input of 6 also results in -5 as the square reoot of 25 is both 5 and -5.
Also, rearranging the maching such that the function machine with the function $y=\sqrt{x-5}$ is the first machine and the function machine with the function $y=x^2-6$ is the second machine will also result in a final output of -5 with an input of 6.