Answer :
To determine the value of [tex]\(\left(\frac{f}{g}\right)(8)\)[/tex] given the functions [tex]\(f(x) = 3 - 2x\)[/tex] and [tex]\(g(x) = \frac{1}{x+5}\)[/tex], we need to follow these steps:
1. Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 8\)[/tex]:
[tex]\[ f(8) = 3 - 2 \cdot 8 \][/tex]
[tex]\[ f(8) = 3 - 16 \][/tex]
[tex]\[ f(8) = -13 \][/tex]
2. Evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 8\)[/tex]:
[tex]\[ g(8) = \frac{1}{8 + 5} \][/tex]
[tex]\[ g(8) = \frac{1}{13} \][/tex]
[tex]\[ g(8) \approx 0.07692307692307693 \][/tex]
3. Calculate the quotient [tex]\( \left( \frac{f}{g} \right)(8) \)[/tex]:
[tex]\[ \left( \frac{f}{g} \right)(8) = \frac{f(8)}{g(8)} \][/tex]
[tex]\[ \left( \frac{f}{g} \right)(8) = \frac{-13}{\frac{1}{13}} \][/tex]
[tex]\[ \left( \frac{f}{g} \right)(8) = -13 \times 13 \][/tex]
[tex]\[ \left( \frac{f}{g} \right)(8) = -169 \][/tex]
Thus, the value of [tex]\(\left(\frac{f}{g}\right)(8)\)[/tex] is [tex]\(-169\)[/tex].
Therefore, the correct answer is:
[tex]\[ -169 \][/tex]
1. Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 8\)[/tex]:
[tex]\[ f(8) = 3 - 2 \cdot 8 \][/tex]
[tex]\[ f(8) = 3 - 16 \][/tex]
[tex]\[ f(8) = -13 \][/tex]
2. Evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 8\)[/tex]:
[tex]\[ g(8) = \frac{1}{8 + 5} \][/tex]
[tex]\[ g(8) = \frac{1}{13} \][/tex]
[tex]\[ g(8) \approx 0.07692307692307693 \][/tex]
3. Calculate the quotient [tex]\( \left( \frac{f}{g} \right)(8) \)[/tex]:
[tex]\[ \left( \frac{f}{g} \right)(8) = \frac{f(8)}{g(8)} \][/tex]
[tex]\[ \left( \frac{f}{g} \right)(8) = \frac{-13}{\frac{1}{13}} \][/tex]
[tex]\[ \left( \frac{f}{g} \right)(8) = -13 \times 13 \][/tex]
[tex]\[ \left( \frac{f}{g} \right)(8) = -169 \][/tex]
Thus, the value of [tex]\(\left(\frac{f}{g}\right)(8)\)[/tex] is [tex]\(-169\)[/tex].
Therefore, the correct answer is:
[tex]\[ -169 \][/tex]
