Sure, let's break down the problem step-by-step and evaluate the given functions for [tex]\( x = -8 \)[/tex]:
1. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = -8 \)[/tex]:
The function [tex]\( g(x) \)[/tex] is given by:
[tex]\[
g(x) = |x + 5|
\][/tex]
Substitute [tex]\( x = -8 \)[/tex] into the function:
[tex]\[
g(-8) = |-8 + 5| = |-3| = 3
\][/tex]
So, [tex]\( g(-8) = 3 \)[/tex].
2. Evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = -8 \)[/tex]:
The function [tex]\( h(x) \)[/tex] is given by:
[tex]\[
h(x) = \frac{1}{x + 7}
\][/tex]
Substitute [tex]\( x = -8 \)[/tex] into the function:
[tex]\[
h(-8) = \frac{1}{-8 + 7} = \frac{1}{-1} = -1
\][/tex]
So, [tex]\( h(-8) = -1 \)[/tex].
3. Evaluate [tex]\( (g - h)(x) \)[/tex] at [tex]\( x = -8 \)[/tex]:
The expression [tex]\( (g - h)(x) \)[/tex] denotes the difference between [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]. Thus:
[tex]\[
(g - h)(x) = g(x) - h(x)
\][/tex]
Substitute [tex]\( x = -8 \)[/tex]:
[tex]\[
(g - h)(-8) = g(-8) - h(-8) = 3 - (-1) = 3 + 1 = 4
\][/tex]
So, the value of [tex]\( (g - h)(-8) \)[/tex] is 4.
Therefore, the solution to the question is:
[tex]\[
(g-h)(-8) = 4
\][/tex]
