Let's break down the problem step by step:
### Part (a): Finding [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex]
The functions are defined as:
[tex]\[ f(x) = 2x^2 - 3 \][/tex]
[tex]\[ g(x) = 5x - 7 \][/tex]
First, we need to calculate [tex]\( f(-1) \)[/tex] and [tex]\( g(-1) \)[/tex]:
1. Calculate [tex]\( f(-1) \)[/tex]:
[tex]\[ f(-1) = 2(-1)^2 - 3 \][/tex]
[tex]\[ f(-1) = 2 \cdot 1 - 3 \][/tex]
[tex]\[ f(-1) = 2 - 3 \][/tex]
[tex]\[ f(-1) = -1 \][/tex]
2. Calculate [tex]\( g(-1) \)[/tex]:
[tex]\[ g(-1) = 5(-1) - 7 \][/tex]
[tex]\[ g(-1) = -5 - 7 \][/tex]
[tex]\[ g(-1) = -12 \][/tex]
Now, we can find [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex]:
[tex]\[ \left(\frac{f}{g}\right)(-1) = \frac{f(-1)}{g(-1)} \][/tex]
[tex]\[ \left(\frac{f}{g}\right)(-1) = \frac{-1}{-12} \][/tex]
[tex]\[ \left(\frac{f}{g}\right)(-1) = \frac{1}{12} \][/tex]
Hence,
[tex]\[ \left(\frac{f}{g}\right)(-1) = 0.08333333333333333 \][/tex]
### Part (b): Finding all values that are NOT in the domain of [tex]\(\frac{f}{g}\)[/tex]
Values that are not in the domain of [tex]\(\frac{f}{g}\)[/tex] occur when the denominator [tex]\( g(x) \)[/tex] is zero, i.e., [tex]\( g(x) = 0 \)[/tex].
The function [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = 5x - 7 \][/tex]
We need to solve for [tex]\( x \)[/tex] when [tex]\( g(x) = 0 \)[/tex]:
[tex]\[ 5x - 7 = 0 \][/tex]
[tex]\[ 5x = 7 \][/tex]
[tex]\[ x = \frac{7}{5} \][/tex]
Therefore, the value of [tex]\( x \)[/tex] that is not in the domain of [tex]\(\frac{f}{g}\)[/tex] is:
[tex]\[ x = \frac{7}{5} \][/tex]
### Final Answers
(a) [tex]\(\left(\frac{f}{g}\right)(-1) = 0.08333333333333333\)[/tex]
(b) Values that are NOT in the domain of [tex]\(\frac{f}{g}\)[/tex]: [tex]\(\frac{7}{5}\)[/tex]
