If [tex]\( f(x) = 7 + 4x \)[/tex] and [tex]\( g(x) = \frac{1}{2x} \)[/tex], what is the value of [tex]\( \left(\frac{f}{g}\right)(5) \)[/tex]?

A. [tex]\( \frac{11}{2} \)[/tex]
B. [tex]\( \frac{27}{10} \)[/tex]
C. 160
D. 270



Answer :

To solve for [tex]\(\left(\frac{f}{g}\right)(5)\)[/tex] given the functions [tex]\(f(x) = 7 + 4x\)[/tex] and [tex]\(g(x) = \frac{1}{2x}\)[/tex], we need to follow these steps:

1. Evaluate [tex]\(f(5)\)[/tex]:
[tex]\[ f(x) = 7 + 4x \][/tex]
Substitute [tex]\(x = 5\)[/tex]:
[tex]\[ f(5) = 7 + 4 \cdot 5 = 7 + 20 = 27 \][/tex]

2. Evaluate [tex]\(g(5)\)[/tex]:
[tex]\[ g(x) = \frac{1}{2x} \][/tex]
Substitute [tex]\(x = 5\)[/tex]:
[tex]\[ g(5) = \frac{1}{2 \cdot 5} = \frac{1}{10} = 0.1 \][/tex]

3. Compute [tex]\(\left(\frac{f}{g}\right)(5)\)[/tex]:
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \][/tex]
Substitute [tex]\(x = 5\)[/tex]:
[tex]\[ \left(\frac{f}{g}\right)(5) = \frac{f(5)}{g(5)} = \frac{27}{0.1} \][/tex]
To perform the division:
[tex]\[ \frac{27}{0.1} = 27 \times 10 = 270 \][/tex]

So, the value of [tex]\(\left(\frac{f}{g}\right)(5)\)[/tex] is 270.

Among the given options, the correct answer is:
[tex]\[ \boxed{270} \][/tex]