Let's start by evaluating the functions and then proceed to find the answers to the specific parts of the question.
### Part (a) Calculate [tex]\(\left(\frac{f}{g}\right)(2)\)[/tex]
First, we need to evaluate [tex]\( f(2) \)[/tex] and [tex]\( g(2) \)[/tex].
Given:
[tex]\[ f(x) = 2x^2 - 5 \][/tex]
[tex]\[ g(x) = 4x - 6 \][/tex]
1. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 2(2)^2 - 5 = 2 \cdot 4 - 5 = 8 - 5 = 3
\][/tex]
2. Calculate [tex]\( g(2) \)[/tex]:
[tex]\[
g(2) = 4(2) - 6 = 8 - 6 = 2
\][/tex]
Now, we need to calculate [tex]\( \left(\frac{f}{g}\right)(2) \)[/tex]:
[tex]\[
\left(\frac{f}{g}\right)(2) = \frac{f(2)}{g(2)} = \frac{3}{2} = 1.5
\][/tex]
So, [tex]\(\left(\frac{f}{g}\right)(2) = 1.5\)[/tex].
### Part (b) Find all values that are NOT in the domain of [tex]\(\frac{f}{g}\)[/tex]
The domain of [tex]\(\frac{f}{g}\)[/tex] is determined by the values of [tex]\(x\)[/tex] for which [tex]\(g(x) \neq 0\)[/tex]. Therefore, we need to find the values of [tex]\(x\)[/tex] that make [tex]\(g(x) = 0\)[/tex].
Set [tex]\( g(x) = 0 \)[/tex]:
[tex]\[
4x - 6 = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
4x = 6
\][/tex]
[tex]\[
x = \frac{6}{4}
\][/tex]
[tex]\[
x = \frac{3}{2}
\][/tex]
Thus, the value [tex]\( x = \frac{3}{2} \)[/tex] is not in the domain of [tex]\(\frac{f}{g}\)[/tex].
### Summary
(a) [tex]\(\left(\frac{f}{g}\right)(2) = 1.5\)[/tex]
(b) The value that is NOT in the domain of [tex]\(\frac{f}{g}\)[/tex] is:
[tex]\[
\frac{3}{2}
\][/tex]
