Let's analyze the given problem in a step-by-step manner.
First, we need to find the price after applying a [tex]\(10 \%\)[/tex] discount on the original price [tex]\(x\)[/tex]:
[tex]\[
P(x) = 0.9x
\][/tex]
Next, the function [tex]\(C(x)\)[/tex] applies a discount of [tex]\(\$150\)[/tex] on the price [tex]\(x\)[/tex]:
[tex]\[
C(x) = x - 150
\][/tex]
Now, we need to find the composition function [tex]\(C(P(x))\)[/tex], which represents applying a [tex]\(10 \%\)[/tex] discount first and then using a [tex]$\$[/tex]150$ coupon.
Let's substitute [tex]\(P(x)\)[/tex] into [tex]\(C(x)\)[/tex]:
[tex]\[
C(P(x)) = C(0.9x) = 0.9x - 150
\][/tex]
So the correct composition function is:
[tex]\[
C(P(x)) = 0.9x - 150
\][/tex]
Upon reviewing the options:
[tex]\[
\begin{array}{l}
C(P(x)) = 1.9x - 150 \\
C(P(x)) = 0.9x - 150 \\
P(C(x)) = 1.9x - 150
\end{array}
\][/tex]
The right choice is:
[tex]\[
C(P(x)) = 0.9x - 150
\][/tex]
Therefore, the final answer is [tex]\(0.9, -150, -149.1\)[/tex].
