Certainly! Let's break this problem down step-by-step.
### Part (a): Find [tex]\(\left(\frac{g}{h}\right)(3)\)[/tex]
1. Define the functions [tex]\(g(x)\)[/tex] and [tex]\(h(x)\)[/tex]:
[tex]\[
g(x) = -7 + 2x^2
\][/tex]
[tex]\[
h(x) = 8 - 5x
\][/tex]
2. Evaluate [tex]\(g(3)\)[/tex]:
[tex]\[
g(3) = -7 + 2(3)^2 = -7 + 2 \cdot 9 = -7 + 18 = 11
\][/tex]
3. Evaluate [tex]\(h(3)\)[/tex]:
[tex]\[
h(3) = 8 - 5(3) = 8 - 15 = -7
\][/tex]
4. Calculate [tex]\(\left(\frac{g}{h}\right)(3)\)[/tex]:
[tex]\[
\left(\frac{g}{h}\right)(3) = \frac{g(3)}{h(3)} = \frac{11}{-7} = -\frac{11}{7} \approx -1.5714285714285714
\][/tex]
So, [tex]\(\left(\frac{g}{h}\right)(3) = -1.5714285714285714\)[/tex].
### Part (b): Find all values that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]
For [tex]\(\frac{g}{h}\)[/tex] to be defined, [tex]\(h(x)\)[/tex] must not be zero because division by zero is undefined.
1. Find when [tex]\(h(x) = 0\)[/tex]:
[tex]\[
8 - 5x = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
5x = 8
\][/tex]
[tex]\[
x = \frac{8}{5}
\][/tex]
This means that the function [tex]\(\frac{g}{h}\)[/tex] is not defined when [tex]\(x = \frac{8}{5}\)[/tex].
So, the value that is NOT in the domain of [tex]\(\frac{g}{h}\)[/tex] is:
[tex]\[ \frac{8}{5} \][/tex]
### Final Answers
(a) [tex]\(\left(\frac{g}{h}\right)(3) = -1.5714285714285714\)[/tex]
(b) Value(s) that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]: [tex]\(\frac{8}{5}\)[/tex]
