Let's define the composite function [tex]\(\left(\frac{q}{p}\right)(x)\)[/tex] as given by the problem statement.
To find [tex]\(\left(\frac{q}{p}\right)(x)\)[/tex], we need to use the functions [tex]\(q(x)\)[/tex] and [tex]\(p(x)\)[/tex] provided:
- [tex]\(p(x) = x^2 + 4x\)[/tex]
- [tex]\(q(x) = \sqrt{2 - x}\)[/tex]
The composite function [tex]\(\left(\frac{q}{p}\right)(x)\)[/tex] means we are dividing [tex]\(q(x)\)[/tex] by [tex]\(p(x)\)[/tex]. Therefore, the expression for [tex]\(\left(\frac{q}{p}\right)(x)\)[/tex] will be:
[tex]\[
\left(\frac{q}{p}\right)(x) = \frac{q(x)}{p(x)}
\][/tex]
Plugging the expressions for [tex]\(q(x)\)[/tex] and [tex]\(p(x)\)[/tex] into this formula, we get:
[tex]\[
\left(\frac{q}{p}\right)(x) = \frac{\sqrt{2 - x}}{x^2 + 4x}
\][/tex]
So, the required composite function [tex]\(\left(\frac{q}{p}\right)(x)\)[/tex] is:
[tex]\[
\left(\frac{q}{p}\right)(x) = \frac{\sqrt{2 - x}}{x^2 + 4x}
\][/tex]
This is the expression for [tex]\(\left(\frac{q}{p}\right)(x)\)[/tex].
