Answer:
[tex]\sf (q \circ q)(x) = \dfrac{x-9}{-9x + 82}[/tex]
Domain: [tex]\sf\left(-\infty, \dfrac{82}{9}\right) \cup \left(\dfrac{82}{9}, \infty\right) [/tex]
Step-by-step explanation:
To find the function [tex]\sf (q \circ q)(x)[/tex], where [tex]\sf q(x) = \dfrac{1}{x-9}[/tex], we need to perform function composition.
Function composition [tex]\sf (q \circ q)(x)[/tex] means applying the function [tex]\sf q[/tex] twice to [tex]\sf x[/tex]. In other words, it is [tex]\sf q(q(x))[/tex].
First, calculate [tex]\sf q(q(x))[/tex]:
[tex]\sf q(q(x)) = q\left(\dfrac{1}{x-9}\right) [/tex]
Substitute [tex]\sf q(x) = \dfrac{1}{x-9}[/tex] into [tex]\sf q(q(x))[/tex]:
[tex]\sf q\left(\dfrac{1}{x-9}\right) = \dfrac{1}{\left(\dfrac{1}{x-9}\right) - 9} [/tex]
Simplify the expression:
[tex]\sf q\left(\dfrac{1}{x-9}\right) = \dfrac{1}{\dfrac{1}{x-9} - 9} [/tex]
[tex]\sf q\left(\dfrac{1}{x-9}\right) = \dfrac{1}{\dfrac{1 - 9(x-9)}{x-9}} [/tex]
[tex]\sf q\left(\dfrac{1}{x-9}\right) = \dfrac{1}{\dfrac{1 - 9x + 81}{x-9}} [/tex]
[tex]\sf q\left(\dfrac{1}{x-9}\right) = \dfrac{1}{\dfrac{-9x + 82}{x-9}} [/tex]
[tex]\sf q\left(\dfrac{1}{x-9}\right) = \dfrac{x-9}{-9x + 82} [/tex]
Therefore, [tex]\sf \large\boxed{\boxed{(q \circ q)(x) = \dfrac{x-9}{-9x + 82}}}[/tex].
Next, let's determine the domain of [tex]\sf (q \circ q)(x)[/tex]. The domain consists of all [tex]\sf x[/tex] values for which the function is defined, i.e., where the denominator is not zero.
Find the domain:
The denominator [tex]\sf -9x + 82[/tex] must not be zero:
[tex]\sf -9x + 82 \neq 0 [/tex]
Solve for [tex]\sf x[/tex]:
[tex]\sf -9x \neq -82 [/tex]
[tex]\sf x \neq \dfrac{-82}{-9} [/tex]
[tex]\sf x \neq \dfrac{82}{9} [/tex]
Therefore, the domain of [tex]\sf (q \circ q)(x)[/tex] is all real numbers [tex]\sf x[/tex] such that [tex]\sf x \neq \dfrac{82}{9}[/tex].
In interval notation, the domain is:
[tex]\sf\large\boxed{ \boxed{\left(-\infty, \dfrac{82}{9}\right) \cup \left(\dfrac{82}{9}, \infty\right)}} [/tex]
This means [tex]\sf x[/tex] can take any value less than [tex]\sf \dfrac{82}{9}[/tex] or greater than [tex]\sf \dfrac{82}{9}[/tex], excluding [tex]\sf x = \dfrac{82}{9}[/tex].
