Sure! Let's go through the problem step by step.
### Part (a): Find [tex]\(\left(\frac{h}{g}\right)(5)\)[/tex]
1. Evaluate [tex]\(h(5)\)[/tex]:
[tex]\[
h(x) = x + 3
\][/tex]
Substituting [tex]\(x = 5\)[/tex]:
[tex]\[
h(5) = 5 + 3 = 8
\][/tex]
2. Evaluate [tex]\(g(5)\)[/tex]:
[tex]\[
g(x) = (x + 6)(x - 4)
\][/tex]
Substituting [tex]\(x = 5\)[/tex]:
[tex]\[
g(5) = (5 + 6)(5 - 4) = 11 \cdot 1 = 11
\][/tex]
3. Calculate [tex]\(\left(\frac{h}{g}\right)(5)\)[/tex]:
[tex]\[
\left(\frac{h}{g}\right)(5) = \frac{h(5)}{g(5)} = \frac{8}{11} \approx 0.7272727272727273
\][/tex]
Therefore, [tex]\(\left(\frac{h}{g}\right)(5) = \frac{8}{11} \approx 0.7272727272727273\)[/tex].
### Part (b): Find all values that are NOT in the domain of [tex]\(\frac{h}{g}\)[/tex]
To find the values that are NOT in the domain of [tex]\(\frac{h}{g}\)[/tex], we need to determine where the function [tex]\(g(x)\)[/tex] is equal to zero because division by zero is undefined.
1. Solve [tex]\(g(x) = 0\)[/tex]:
[tex]\[
g(x) = (x + 6)(x - 4)
\][/tex]
Set the expression equal to zero:
[tex]\[
(x + 6)(x - 4) = 0
\][/tex]
This product is zero when either factor is zero:
[tex]\[
x + 6 = 0 \quad \text{or} \quad x - 4 = 0
\][/tex]
Solving these equations gives:
[tex]\[
x = -6 \quad \text{or} \quad x = 4
\][/tex]
Thus, the values that are NOT in the domain of [tex]\(\frac{h}{g}\)[/tex] are [tex]\(-6\)[/tex] and [tex]\(4\)[/tex].
### Final Answers:
(a) [tex]\(\left(\frac{h}{g}\right)(5) = \frac{8}{11} \approx 0.7272727272727273\)[/tex]
(b) Values that are NOT in the domain of [tex]\(\frac{h}{g}\)[/tex]: [tex]\(-6, 4\)[/tex]
