Let's solve the given problems step by step.
### Part (a): Finding [tex]\(\left(\frac{f}{g}\right)(1)\)[/tex]
First, we need to evaluate [tex]\(f(1)\)[/tex] and [tex]\(g(1)\)[/tex] and then find their quotient.
1. Evaluate [tex]\(f(1)\)[/tex]:
[tex]\(f(x) = 3x^2 - 7\)[/tex]
[tex]\[
f(1) = 3(1)^2 - 7 = 3 \cdot 1 - 7 = 3 - 7 = -4
\][/tex]
2. Evaluate [tex]\(g(1)\)[/tex]:
[tex]\(g(x) = -2x + 5\)[/tex]
[tex]\[
g(1) = -2(1) + 5 = -2 \cdot 1 + 5 = -2 + 5 = 3
\][/tex]
3. Find the quotient [tex]\(\left(\frac{f}{g}\right)(1)\)[/tex]:
[tex]\[
\left(\frac{f}{g}\right)(1) = \frac{f(1)}{g(1)} = \frac{-4}{3} = -\frac{4}{3}
\][/tex]
So, [tex]\(\left(\frac{f}{g}\right)(1) = -\frac{4}{3}\)[/tex].
### Part (b): Finding the values that are NOT in the domain of [tex]\(\frac{f}{g}\)[/tex]
The function [tex]\(\frac{f}{g}\)[/tex] is undefined when [tex]\(g(x) = 0\)[/tex]. So, we need to find the value of [tex]\(x\)[/tex] for which [tex]\(g(x) = 0\)[/tex].
1. Solve [tex]\(g(x) = 0\)[/tex]:
[tex]\(g(x) = -2x + 5\)[/tex]
[tex]\[
-2x + 5 = 0
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
-2x = -5 \implies x = \frac{5}{2}
\][/tex]
So, the value that is NOT in the domain of [tex]\(\frac{f}{g}\)[/tex] is [tex]\(\frac{5}{2}\)[/tex].
### Summary of Answers
(a) [tex]\(\left(\frac{f}{g}\right)(1) = -\frac{4}{3}\)[/tex]
(b) Value(s) that are NOT in the domain of [tex]\(\frac{f}{g}\)[/tex]: [tex]\(\frac{5}{2}\)[/tex]
Thus, the final answers are:
(a) [tex]\(\left(\frac{f}{g}\right)(1) = -\frac{4}{3}\)[/tex]
(b) Value(s) that are NOT in the domain of [tex]\(\frac{f}{g}\)[/tex]: [tex]\(\frac{5}{2}\)[/tex]
