Sure, let's find each of the following step by step for the given function [tex]\( g(x) = \frac{x-4}{x+3} \)[/tex].
### Part (a): [tex]\( g(8) \)[/tex]
We need to find the value of [tex]\( g(8) \)[/tex].
Substitute [tex]\( x = 8 \)[/tex] into the function:
[tex]\[
g(8) = \frac{8 - 4}{8 + 3} = \frac{4}{11}
\][/tex]
So, [tex]\( g(8) = \frac{4}{11} \)[/tex].
### Part (b): [tex]\( g(4) \)[/tex]
We need to find the value of [tex]\( g(4) \)[/tex].
Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[
g(4) = \frac{4 - 4}{4 + 3} = \frac{0}{7} = 0
\][/tex]
So, [tex]\( g(4) = 0 \)[/tex].
### Part (c): [tex]\( g(-3) \)[/tex]
We need to find the value of [tex]\( g(-3) \)[/tex].
Substitute [tex]\( x = -3 \)[/tex] into the function:
[tex]\[
g(-3) = \frac{-3 - 4}{-3 + 3} = \frac{-7}{0}
\][/tex]
As the denominator is zero, [tex]\( g(-3) \)[/tex] is undefined.
### Part (d): [tex]\( g(-10.25) \)[/tex]
We need to find the value of [tex]\( g(-10.25) \)[/tex].
Substitute [tex]\( x = -10.25 \)[/tex] into the function:
[tex]\[
g(-10.25) = \frac{-10.25 - 4}{-10.25 + 3} = \frac{-14.25}{-7.25}
\][/tex]
Simplify the fraction:
[tex]\[
g(-10.25) = \frac{14.25}{7.25} = \frac{285}{145} = 1.9655
\][/tex]
Hence, [tex]\( g(-10.25) = 1.9655 \)[/tex].
### Part (e): [tex]\( g(x + h) \)[/tex]
We need to find the expression for [tex]\( g(x + h) \)[/tex].
Substitute [tex]\( x = x + h \)[/tex] into the function:
[tex]\[
g(x+h) = \frac{(x+h) - 4}{(x+h) + 3}
= \frac{x + h - 4}{x + h + 3}
\][/tex]
So, [tex]\( g(x+h) = \frac{x + h - 4}{x + h + 3} \)[/tex].
In summary:
a) [tex]\( g(8) = \frac{4}{11} \)[/tex]
b) [tex]\( g(4) = 0 \)[/tex]
c) [tex]\( g(-3) \)[/tex] is undefined
d) [tex]\( g(-10.25) = 1.9655 \)[/tex]
e) [tex]\( g(x+h) = \frac{x + h - 4}{x + h + 3} \)[/tex]
