First, we need to determine the values of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at [tex]\( x = 5 \)[/tex].
1. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 5 \)[/tex]:
Given [tex]\( f(x) = 7 + 4x \)[/tex],
[tex]\[
f(5) = 7 + 4(5)
\][/tex]
Calculate the value inside the parentheses:
[tex]\[
4(5) = 20
\][/tex]
Then add 7:
[tex]\[
f(5) = 7 + 20 = 27
\][/tex]
So, [tex]\( f(5) = 27 \)[/tex].
2. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 5 \)[/tex]:
Given [tex]\( g(x) = \frac{1}{2x} \)[/tex],
[tex]\[
g(5) = \frac{1}{2(5)}
\][/tex]
Calculate the value inside the parentheses:
[tex]\[
2(5) = 10
\][/tex]
Then find the value of the fraction:
[tex]\[
g(5) = \frac{1}{10} = 0.1
\][/tex]
So, [tex]\( g(5) = 0.1 \)[/tex].
3. Calculate [tex]\( \left( \frac{f}{g} \right)(5) \)[/tex]:
To find [tex]\( \left( \frac{f}{g} \right)(5) \)[/tex], we must divide [tex]\( f(5) \)[/tex] by [tex]\( g(5) \)[/tex]:
[tex]\[
\left( \frac{f}{g} \right)(5) = \frac{f(5)}{g(5)}
\][/tex]
Substitute [tex]\( f(5) \)[/tex] and [tex]\( g(5) \)[/tex] into the equation:
[tex]\[
\left( \frac{f}{g} \right)(5) = \frac{27}{0.1}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{27}{0.1} = 27 \div 0.1 = 270
\][/tex]
So, the value of [tex]\( \left( \frac{f}{g} \right)(5) \)[/tex] is [tex]\( 270 \)[/tex].
Thus, the correct answer is [tex]\( 270 \)[/tex].
