To solve this question, let's proceed step-by-step:
### Part (a) Calculate [tex]\(\left(\frac{g}{h}\right)(5)\)[/tex]
1. First, we need to find the values of [tex]\(g(5)\)[/tex] and [tex]\(h(5)\)[/tex].
[tex]\[
g(x) = (6 + x)(-5 + x)
\][/tex]
Plugging in [tex]\(x = 5\)[/tex]:
[tex]\[
g(5) = (6 + 5)(-5 + 5) = 11 \cdot 0 = 0
\][/tex]
[tex]\[
h(x) = 8 - x
\][/tex]
Plugging in [tex]\(x = 5\)[/tex]:
[tex]\[
h(5) = 8 - 5 = 3
\][/tex]
2. Now, we need to find the value of [tex]\(\left( \frac{g}{h} \right)(5)\)[/tex]:
[tex]\[
\left(\frac{g}{h}\right)(5) = \frac{g(5)}{h(5)} = \frac{0}{3} = 0
\][/tex]
So, [tex]\(\left(\frac{g}{h}\right)(5) = 0\)[/tex].
### Part (b) Find all values that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]
1. The function [tex]\(\frac{g}{h}\)[/tex] is undefined at points where [tex]\(h(x) = 0\)[/tex] since division by zero is undefined.
2. To find such [tex]\(x\)[/tex] values, solve for where [tex]\(h(x) = 0\)[/tex]:
[tex]\[
8 - x = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = 8
\][/tex]
3. Thus, the value that is not in the domain of [tex]\(\frac{g}{h}\)[/tex] is [tex]\(x = 8\)[/tex].
So, summarizing the solutions:
(a) [tex]\(\left(\frac{g}{h}\right)(5) = 0\)[/tex]
(b) Value(s) that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex] : [tex]\(8\)[/tex]
