To solve the given problem, we are given two functions [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]. Let's start by finding [tex]\(\left(\frac{g}{h}\right)(5)\)[/tex] and then determine the values that are not in the domain of [tex]\(\frac{g}{h}\)[/tex].
### Part (a)
First, we need to evaluate [tex]\( g(5) \)[/tex] and [tex]\( h(5) \)[/tex]:
1. Calculate [tex]\( g(5) \)[/tex]:
[tex]\[ g(x) = (4 + x)(2 + x) \][/tex]
Substitute [tex]\( x = 5 \)[/tex]:
[tex]\[ g(5) = (4 + 5)(2 + 5) = 9 \times 7 = 63 \][/tex]
2. Calculate [tex]\( h(5) \)[/tex]:
[tex]\[ h(x) = -2 - 7x \][/tex]
Substitute [tex]\( x = 5 \)[/tex]:
[tex]\[ h(5) = -2 - 7 \times 5 = -2 - 35 = -37 \][/tex]
3. Find [tex]\( \left(\frac{g}{h}\right)(5) \)[/tex]:
[tex]\[ \left(\frac{g}{h}\right)(5) = \frac{g(5)}{h(5)} = \frac{63}{-37} = -1.7027027027027026 \][/tex]
Thus, the answer to part (a) is:
(a) [tex]\(\left(\frac{g}{h}\right)(5) = -1.7027027027027026\)[/tex]
### Part (b)
To find the values that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( h(x) = 0 \)[/tex]. This is because the function [tex]\(\frac{g}{h}\)[/tex] is undefined where [tex]\( h(x) = 0 \)[/tex].
1. Solve [tex]\( h(x) = 0 \)[/tex]:
[tex]\[ h(x) = -2 - 7x = 0 \][/tex]
[tex]\[ -2 = 7x \][/tex]
[tex]\[ x = -\frac{2}{7} \][/tex]
Thus, the value that is not in the domain of [tex]\( \frac{g}{h} \)[/tex] is [tex]\( x = -\frac{2}{7} \)[/tex].
(b) Value(s) that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]: [tex]\(-\frac{2}{7}\)[/tex]
Therefore, the values for parts (a) and (b) are:
(a) [tex]\(\left(\frac{g}{h}\right)(5) = -1.7027027027027026\)[/tex]
(b) Value(s) that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]: [tex]\(-\frac{2}{7}\)[/tex]
