Let's go through the questions step by step.
### Part (a): Find [tex]\(\left(\frac{g}{h}\right)(4)\)[/tex]
First, we need to evaluate both [tex]\( g(4) \)[/tex] and [tex]\( h(4) \)[/tex].
1. Evaluate [tex]\( g(4) \)[/tex]:
Given the function:
[tex]\[ g(x) = -5 + 4x^2 \][/tex]
Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[ g(4) = -5 + 4(4^2) \][/tex]
[tex]\[ g(4) = -5 + 4(16) \][/tex]
[tex]\[ g(4) = -5 + 64 \][/tex]
[tex]\[ g(4) = 59 \][/tex]
2. Evaluate [tex]\( h(4) \)[/tex]:
Given the function:
[tex]\[ h(x) = 5 - 6x \][/tex]
Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[ h(4) = 5 - 6(4) \][/tex]
[tex]\[ h(4) = 5 - 24 \][/tex]
[tex]\[ h(4) = -19 \][/tex]
3. Compute [tex]\(\left(\frac{g}{h}\right)(4)\)[/tex]:
Using the values found:
[tex]\[ \left(\frac{g}{h}\right)(4) = \frac{g(4)}{h(4)} = \frac{59}{-19} \][/tex]
[tex]\[ \left(\frac{g}{h}\right)(4) = -\frac{59}{19} \][/tex]
[tex]\[ \left(\frac{g}{h}\right)(4) \approx -3.1052631578947367 \][/tex]
Thus, the answer for part (a) is:
[tex]\[ \left(\frac{g}{h}\right)(4) = -3.1052631578947367 \][/tex]
### Part (b): Find all values that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]
For the function [tex]\(\frac{g}{h}\)[/tex] to be undefined, the denominator [tex]\( h(x) \)[/tex] must be zero. We need to find the values of [tex]\( x \)[/tex] for which [tex]\( h(x) = 0 \)[/tex].
[tex]\[ h(x) = 5 - 6x \][/tex]
Set the equation to zero:
[tex]\[ 5 - 6x = 0 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 6x = 5 \][/tex]
[tex]\[ x = \frac{5}{6} \][/tex]
Thus, the value that is not in the domain of [tex]\(\frac{g}{h}\)[/tex] is:
[tex]\[ \frac{5}{6} \][/tex]
### Summary
- (a) [tex]\(\left(\frac{g}{h}\right)(4) = -3.1052631578947367\)[/tex]
- (b) Value(s) that are NOT in the domain of [tex]\(\frac{g}{h}\)[/tex]: [tex]\(\frac{5}{6}\)[/tex]
