Given the two functions as [tex] y = \sqrt{x-5} [/tex] and [tex] y = x^2 - 6 [/tex].
We can rewrite the functions as [tex] f (x) = \sqrt{x-5} [/tex] and [tex] g (x)=x^2 - 6 [/tex]
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) )
[tex]f ( g (x) )= \sqrt{x^2-6-5} = \sqrt{x^2-11} [/tex]
and
[tex]g ( f (x) )=( \sqrt{x-5} )^2-6=x-5-6=x-11[/tex]
Given an input of 6,
[tex]f ( g (6) )=\sqrt{6^2-11} =\sqrt{36-11}=\sqrt{25}=5[/tex]
and
[tex]g ( f (6) )=x-11=6-11=-5[/tex]
Therefore, to get a final output of 5, she will put the function machine with the function [tex]y=x^2-6[/tex] as the first machine and the function machine with the function [tex]y=\sqrt{x-5}[/tex] 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 [tex]y=\sqrt{x-5}[/tex] is the first machine and the function machine with the function [tex]y=x^2-6[/tex] is the second machine will also result in a final output of -5 with an input of 6.
