Suppose that the functions [tex][tex]$g$[/tex][/tex] and [tex][tex]$h$[/tex][/tex] are defined as follows.

[tex]\[
\begin{array}{l}
g(x)=-5+4x^2 \\
h(x)=5-6x
\end{array}
\][/tex]

(a) Find [tex][tex]$\left(\frac{g}{h}\right)(4)$[/tex][/tex].

(b) Find all values that are NOT in the domain of [tex][tex]$\frac{g}{h}$[/tex][/tex].

If there is more than one value, separate them with commas.

(a) [tex][tex]$\left(\frac{g}{h}\right)(4)=$[/tex][/tex]

(b) Value(s) that are NOT in the domain of [tex][tex]$\frac{g}{h}$[/tex][/tex]:



Answer :

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]