To solve this problem, let’s break it down step-by-step:
### Part (a) Find [tex]\(\left(\frac{h}{g}\right)(3)\)[/tex]
First, we need to determine the individual values of [tex]\( h(x) \)[/tex] and [tex]\( g(x) \)[/tex] at [tex]\( x = 3 \)[/tex].
#### Step 1: Calculate [tex]\( h(3) \)[/tex]
Given:
[tex]\[ h(x) = (4 + x)(-2 + x) \][/tex]
Substitute [tex]\( x = 3 \)[/tex]:
[tex]\[ h(3) = (4 + 3)(-2 + 3) \][/tex]
[tex]\[ h(3) = 7 \cdot 1 = 7 \][/tex]
#### Step 2: Calculate [tex]\( g(3) \)[/tex]
Given:
[tex]\[ g(x) = -5 + x \][/tex]
Substitute [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = -5 + 3 = -2 \][/tex]
#### Step 3: Calculate [tex]\(\left(\frac{h}{g}\right)(3)\)[/tex]
[tex]\[ \left(\frac{h}{g}\right)(3) = \frac{h(3)}{g(3)} = \frac{7}{-2} = -\frac{7}{2} \][/tex]
Thus, the value of [tex]\(\left(\frac{h}{g}\right)(3)\)[/tex] is [tex]\(-\frac{7}{2}\)[/tex].
### Part (b) Find all values that are NOT in the domain of [tex]\(\frac{h}{g}\)[/tex]
The function [tex]\(\frac{h}{g}\)[/tex] is not defined where the denominator [tex]\( g(x) \)[/tex] is equal to zero.
#### Step 1: Set [tex]\( g(x) = 0 \)[/tex]
[tex]\[ g(x) = -5 + x = 0 \][/tex]
#### Step 2: Solve for [tex]\( x \)[/tex]
[tex]\[ x = 5 \][/tex]
Thus, the function [tex]\(\frac{h}{g}\)[/tex] is not defined at [tex]\( x = 5 \)[/tex].
### Summary
(a) [tex]\(\left(\frac{h}{g}\right)(3) = -\frac{7}{2}\)[/tex]
(b) The value that is NOT in the domain of [tex]\(\frac{h}{g}\)[/tex] is [tex]\( 5 \)[/tex].
