Sure, let's tackle each part of the question step-by-step.
### Part (a) Find [tex]\(\left(\frac{g}{h}\right)(4)\)[/tex]
Given the functions:
[tex]\[ g(x) = (x + 2)(x + 5) \][/tex]
[tex]\[ h(x) = -7x - 7 \][/tex]
To find [tex]\(\left(\frac{g}{h}\right)(4)\)[/tex], we first need to evaluate [tex]\(g(4)\)[/tex] and [tex]\(h(4)\)[/tex]:
1. Evaluate [tex]\(g(4)\)[/tex]:
[tex]\[ g(4) = (4 + 2)(4 + 5) \][/tex]
[tex]\[ g(4) = 6 \cdot 9 \][/tex]
[tex]\[ g(4) = 54 \][/tex]
2. Evaluate [tex]\(h(4)\)[/tex]:
[tex]\[ h(4) = -7(4) - 7 \][/tex]
[tex]\[ h(4) = -28 - 7 \][/tex]
[tex]\[ h(4) = -35 \][/tex]
Now, we compute [tex]\(\frac{g(4)}{h(4)}\)[/tex]:
[tex]\[ \left(\frac{g}{h}\right)(4) = \frac{g(4)}{h(4)} = \frac{54}{-35} = -1.542857142857143 \][/tex]
Thus:
[tex]\[ \left(\frac{g}{h}\right)(4) = -1.542857142857143 \][/tex]
### Part (b) Find all values that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]
The domain of [tex]\(\frac{g}{h}\)[/tex] is all values [tex]\(x\)[/tex] such that [tex]\(h(x) \neq 0\)[/tex]. We need to find when [tex]\(h(x) = 0\)[/tex]:
[tex]\[ h(x) = -7x - 7 \][/tex]
Set [tex]\(h(x)\)[/tex] to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ -7x - 7 = 0 \][/tex]
[tex]\[ -7x = 7 \][/tex]
[tex]\[ x = -1 \][/tex]
So, the value that makes [tex]\(h(x) = 0\)[/tex] is [tex]\(x = -1\)[/tex]. This is the value that is NOT in the domain of [tex]\(\frac{g}{h}\)[/tex].
Thus:
[tex]\[ \text{Value(s) that are NOT in the domain of } \frac{g}{h} : -1 \][/tex]
### Final Answers:
(a) [tex]\(\left(\frac{g}{h}\right)(4) = -1.542857142857143\)[/tex]
(b) Value(s) that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex] : [tex]\(-1\)[/tex]
