Answer:
One possible set of functions is
[tex]q(x) = \sqrt{x} -5[/tex] and [tex]r(x) = 3x^5[/tex]
Step-by-step explanation:
The way to tackle problems of this sort is to remember that the inner function is executed first and its output is fed to the outer function
Inner function is [tex]r(x)[/tex]
Outer function is [tex]q(x)[/tex]
Set [tex]r(x)[/tex] as a function of x. This can be done by a convenient substitution Let's set [tex]r(x) = 3x^5[/tex]
We have
[tex]q(r(x) = \sqrt{3x^5} + 5[/tex]
Simply replace [tex]3x^5[/tex] with x to get
[tex]q(x) = \sqrt{x} - 5[/tex]
So two possible functions that satisfy the composition
[tex]q(r(x)) =[/tex] [tex]\sqrt{3x^5} + 5[/tex]
are
[tex]q(x) = \sqrt{x} -5[/tex] and
[tex]r(x) = 3x^5[/tex]
We can verify this by working out [tex]q(r(x))[/tex] using the above functions. Remember it is the inner function that is executed first
[tex]r(x) =3x^5[/tex]
[tex]q(r(x)) = q(3x^5)[/tex]
But [tex]q(x) = \sqrt{x} - 5[/tex]
Substitute x with [tex]3x^5[/tex] to get [tex]\sqrt{3x^5} - 5[/tex]
